Elastic Antiproton-Nucleus Scattering from Chiral Forces
Matteo Vorabbi Michael Gennari Paolo Finelli Carlotta Giusti Petr Navrátil










^1National Nuclear Data Center, Bldg. 817, Brookhaven National Laboratory, Upton, New York 11973-5000, USA


^2TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia, V6T 2A3, Canada


^3University of Victoria, 3800 Finnerty Road, Victoria, British Columbia V8P 5C2, Canada


^4Dipartimento di Fisica e Astronomia, 
Università degli Studi di Bologna and 

INFN, Sezione di Bologna, Via Irnerio 46, I-40126 Bologna, Italy


^5Dipartimento di Fisica,  
Università degli Studi di Pavia and 

INFN, Sezione di Pavia,  Via A. Bassi 6, I-27100 Pavia, Italy






 

Elastic scattering of antiprotons off ^4He, ^12C, and ^16,18O is described for the first time with a consistent
microscopic approach based on the calculation of an optical potential (OP) describing the antiproton-target interaction. The OP is derived using
the recent antiproton-nucleon (p̅N) chiral interaction to calculate the p̅N t matrix, while the target densities are computed with the ab initio no-core shell
model using chiral interactions as well. Our results are in good agreement with the existing experimental data and the results computed at different chiral orders of
the p̅N interaction display a well-defined convergence pattern.


25.43.+t; 24.10.Ht; 25.40.Cm; 25.40.Dn; 11.10.Ef

Elastic Antiproton-Nucleus Scattering from Chiral Forces
    Petr Navrátil^2
    December 30, 2023
========================================================





With the Facility for Antiproton and Ion Research (FAIR) construction under way <cit.> and the recent antiProton Unstable Matter Annihilation (PUMA) 
proposal <cit.>, scientific interest in new experiments on antiproton scattering off nuclear targets
(nucleons and nuclei) will experience a renaissance. 

In the past, there has been a lot of activity in the antiproton physics at the Low Energy Antiproton Ring (LEAR) at CERN as
well as at KEK in Japan and Brookhaven National Laboratory (BNL) in the USA. At LEAR, in particular, several measurements of cross sections have been made for antiproton
elastic and charge-exchange scattering reactions at antiproton momenta in the
range 100 MeV/c ≤ p ≤ 2 GeV/c <cit.>.

The dominant feature of antiproton-proton scattering at low energies, i.e., the energy region on which our Letter is focused, is the annihilation process that, due to its large
cross section, greatly reduces the probability of rescattering processes. Antiproton-nucleus (p̅A) scattering is thus likely to be described by simple reaction mechanisms
without the complication of multiple scattering processes, which makes it a very clean method to study nuclear properties. 
In fact, the pronounced diffraction structure of the differential cross sections (in contrast with elastic proton scattering) is commonly interpreted as a consequence of the role played by
the strong absorptive potential driven by the annihilation of nucleons and antinucleons. 
Antiproton absorption is surface-dominated <cit.> and is strongly sensitive to nuclear radii. 
The exchange mechanism and the antisymmetrization between the projectile and the target constituents are not relevant in the p̅A interaction, while
the role played by the three-body forces involving an antiproton and two nucleons (p̅NN) still remains an open question.



From the theoretical point of view, the description of antiproton-nucleon (p̅N) processes was mainly based on long-range meson exchanges, with the addition of
phenomenological models for annihilation contributions. Several approaches have been proposed over the last forty years. One of the most successful potentials is the model
proposed by Dover and Richard <cit.> who were inspired by the Paris potential. Other antinucleon-nucleon (N̅N) interactions, based
on the meson theory, were also derived <cit.>, where the N̅N potential of Ref. <cit.> was used to
study p̅A quasibound states <cit.>. A more general approach <cit.> was also employed to provide a partial-wave analysis
of antiproton-proton data.
A similar situation is found for p̅A scattering processes. In the 80s, several nonrelativistic and relativistic calculations were performed with different
approaches which made use of an optical potential (OP) <cit.> but required some phenomenological input. A summary of all these calculations can be found
in Ref. <cit.>. Even in recent years new phenomenological models have been
presented <cit.>.

Because of the tremendous advances in computational techniques achieved in the past decades, it is now possible to compute the OP for p̅A scattering in a fully
microscopic and consistent way. The purpose of this Letter is to construct the first fully microscopic OP for elastic p̅A scattering using the most recent techniques in nuclear
physics, in particular, the application of chiral p̅N potentials combined with nuclear densities obtained from ab initio calculations with chiral two- (NN) and
three-nucleon (3N) interactions. The results for the elastic differential cross sections produced with our OP will be then tested against the existing experimental data.
For such a purpose, we adopt a scheme analogous to that employed in Ref. <cit.>, where a microscopic OP for proton-nucleus (pA) elastic scattering has been
derived within the Watson multiple scattering theory <cit.> at the first order term of the spectator expansion <cit.> and assuming the impulse
approximation.
Recently, interest in the microscopic calculation of the OP for nucleon-nucleus (NA) processes produced several new papers and a very recent review can be found in
Ref. <cit.>. Here we mention the work of Burrows et al. <cit.>, which improved the calculation of the OP including the coupling between the target
nucleon and the residual nucleus, the work of Arellano and Blanchon <cit.> on the irreducible nonlocality of the OP, the work of
Whitehead et al. <cit.> based on the calculation of the nucleon self-energy within the framework of the improved local density approximation, and
the work of Kohno <cit.> on the Pauli rearrangement potential.

In the present work the OP is computed in momentum space as


    U ( q, K; E )    = ∑_N=p,n∫ d  P η ( q, K, P) 
       × t_p̅N[  q , 1/2( A+1/A K + √(A-1/A) P) ; E ] 
       ×ρ_N (  P + √(A-1/A) q/2 ,  P - √(A-1/A) q/2)   ,


where q and K represent the momentum transfer and the average momentum, respectively. Here P is an integration variable, t_p̅N is the p̅N
free t matrix and ρ_N is the one-body nuclear density matrix. The parameter η is the Møller factor, which imposes the Lorentz invariance of the flux when we pass from
the p̅A to the p̅N frame in which the t matrices are evaluated. Finally, E is the energy at which the t matrices are evaluated and it is fixed at one-half 
the kinetic energy of the incident antiproton in the laboratory frame. 

The calculation of Eq.(<ref>) requires two basic ingredients: the p̅N scattering matrix and the one-body nuclear density of the target. The calculation of the
density matrix is performed using the same approach followed in Ref. <cit.>, where one-body translationally invariant (trinv) densities were computed within the
ab initio no-core shell model <cit.> (NCSM) approach using NN and 3N chiral interactions as the only input.
The NCSM method is based on the expansion of the nuclear wave functions in a harmonic oscillator basis and it is thus characterized by the harmonic oscillator
frequency ħΩ and the parameter N_max, which specifies the number of  nucleon excitations above the lowest energy configuration allowed by the Pauli principle.
In the present work we used the NN chiral interaction developed by Machleidt et al. <cit.> up to the fifth order (N^4LO) with
a cutoff Λ = 500 MeV.
In addition to the NN interaction, we also employed the 3N force to compute the one-body densities of the target nuclei. We adopted the 3N chiral
interaction derived up to third order (N2LO), which employs a simultaneous local and nonlocal regularization with the cutoff values of
650 and 500 MeV, respectively <cit.>. The interaction is also renormalized using the similarity renormalization group (SRG) technique that
evolves the bare interaction at the desired resolution scale λ_SRG and ensures a faster convergence of our calculations. The densities have been computed
using ħΩ = 20 MeV and N_max = 14 for ^4He and ħΩ = 16 MeV and N_max = 8 for ^12C and ^16,18O.
For all these calculations we always adopted λ_SRG = 2.0 fm^-1. Finally, the importance-truncated NCSM
basis <cit.> was used for the ^12C and ^16,18O calculations at N_max = 8. We refer the reader to
Ref. <cit.> for all the details about the calculation of the densities and the removal of the center-of-mass contributions.

The same NN interaction was used in Ref. <cit.> to compute the pA scattering matrix. 
The p̅N interaction is different from the
proton-nucleon (pN) one and in the present case it is not possible to compute the t_p̅N matrix with the same potential adopted for the calculation of the density.
For such calculation we use the first p̅N interaction at the next-to-next-to-next-to-leading order (3lo) in chiral perturbation theory (ChPT)
recently derived by Dai, Haidenbauer, and Meißner <cit.>. In recent years, approaches based on ChPT had a great success, especially in the NN
sector <cit.>. They are able to include symmetries and symmetry-breaking patterns of low energy QCD
and, at the same time, provide a reliable framework to express the NN force in terms of a series of pion-exchange and contact interaction terms.
Two-body and many-body contributions naturally arise from the same prescriptions. The NN reaction matrix is obtained solving a regularized Lippmann-Schwinger equation
for the bare NN potential. We refer the reader to Refs. <cit.> for a complete survey of ChPT and to
Refs. <cit.> for the recent developments. Higher-order corrections to Eq. (<ref>) are very difficult to
estimate <cit.>, in particular in a consistent picture along the chiral expansion of the NN potential, but surely deserve future studies.




In comparison with conventional NN scattering, some issues must be addressed in the case of N̅N scattering. The main difference is that in the N̅N case the
annihilation channel is available because the total baryon number is zero. For low-momentum protons, elastic p̅N scattering requires a higher number of partial waves
compared to the pN counterpart. All phase shifts are complex because of the annihilation process and both isospin 0 and 1 contribute in each partial
wave <cit.>. As a consequence, a treatment of p̅N scattering is intrinsically more complicated than the usual NN system.

A conventional way to relate the NN interaction to the N̅N counterpart is G parity, i.e., a combination of charge conjugation and rotation in isospin space <cit.>.
It connects the pion-exchange physics, so even in the N̅N case the long-range physics is completely determined by chiral dynamics. In Ref. <cit.>, Dai et al.
developed a p̅N potential at 3lo in analogy with the corresponding NN potential presented in Refs. <cit.>, 
with the same power counting and a regularization scheme in the coordinate space. It seems that such a local scheme could avoid problems with the long-range part of the
interaction due to pion exchange that, of course, should not be affected by any regularization procedure.
We are aware of the many 
theoretical aspects beyond the regularization procedures (see Ref. <cit.> and references therein) 
and more studies will be needed in the future.
In Ref. <cit.>, five different potentials are provided with different values of the 
coordinate space cutoff R, that reproduce with almost the same quality the N̅N phase shifts. 
In the present work we employ the R=0.9 fm version.









In Fig. <ref> our results for the differential cross sections of elastic antiproton scattering off ^4He and ^12C, computed at the antiproton laboratory
energy of 180 MeV, and ^16,18O  at 178 MeV are presented and compared with the experimental data.
Our model provides a very good description of the data for all the target nuclei considered. In particular, it is remarkable the agreement in correspondence of the first minimum of
the diffraction pattern for all the targets and the general reproduction of the data for ^18O, since this is an sd nucleus and is on the borderline of applicability of the NCSM.

One of the advantages of using a NN or a N̅N interaction in the ChPT scheme is the ability to estimate the theoretical error associated with the truncation of the potential
at a certain order of the chiral expansion. In Fig. <ref> we display the convergence pattern of the differential cross section for the ^12C(p̅,p̅) ^12C
reaction computed at different chiral orders. For a consistent comparison, all the calculations have been performed with the p̅N and NN interactions at the same order
in the chiral expansion. For the calculation of the density at N^2LO and N^3LO we included the 3N force at N^2LO with the couplings c_D and c_E
constrained to the triton half-life and binding energy. This produced two more fits of these parameters <cit.>, different from those employed with the NN N^4LO
interaction, to be used with the NN interaction at the same chiral order. All these results are displayed in Fig. <ref>. As can be seen in the figure, at the leading order
(LO) the calculated cross section is in clear disagreement with data and has a minimum at about 33^∘, which is more than 2 orders of magnitude lower than the experimental
one, which is positioned at about 23^∘. A bit better result is obtained at NLO, where the first minimum is shifted towards smaller angles but the agreement with the experimental
cross section is still poor. At N^2LO the minimum is increased by about 2 orders of magnitude, close to the experimental value, but in comparison with the experimental cross
section the calculated cross section is shifted towards larger angles and the agreement with data remains poor. Only at the N^3LO the first minimum is well reproduced and the
general agreement with data is quite good. It is interesting to note how the differences between the results at different orders decrease going from LO to N^3LO, which reflects the
improvement and confirms a well-defined convergence pattern.
Similar results were found in Refs. <cit.>, where a similar analysis was performed for pA elastic
scattering using several chiral NN interactions at N^3LO and 4lo.
The conclusion is that, for energies around 200 MeV, a good description of the experimental data is obtained with NN or N̅N interactions up to at
least N^3LO. However, the choice of a different fitting procedure <cit.> can produce an interaction capable to describe the
experimental data already at N^2LO, as recently showed in Ref. <cit.> for the NA case.

All the results presented so far were obtained with target densities computed using NN and 3N interactions renormalized via the SRG. To assess the impact of the SRG
procedure in our calculations, we display in Fig. <ref> the results for the differential cross section and analyzing power for ^4He computed with the bare NN and
3N interactions and the same values of N_max and ħΩ. The results are also compared with the ones in Fig. <ref>. As can be inferred from the
figure, the resulting densities produce the same results with minor differences at large scattering angles. Unfortunately, this is the only fully consistent calculation that we can perform
at the moment, since, in general, the usage of the bare interaction requires higher values of the N_max parameter for a complete convergence of the structure calculations and
this is computationally prohibitive for heavier systems like carbon or oxygen.

Finally, in Fig. <ref> we display our predictions for the analyzing power of ^12C and ^16,18O, computed at the same energies and with the same inputs of
Fig. <ref>. We also show the only available experimental data <cit.> obtained on carbon targets as part of the LEAR run of experiments.
The measured asymmetries are small, statistically compatible with zero, and suggest smaller polarisation parameters than those predicted by some N̅N phenomenological
potential models (see Fig. 11 of Ref. <cit.>). Our predictions, on the other hand, are consistent with measurements.



In summary, a fully microscopic OP for p̅A scattering has been derived within the Watson multiple scattering theory using the N̅N, NN, and
3N chiral interactions as the only input for our calculations. The new N̅N interaction derived up to N^3LO has been used in our
calculations to obtain the t_p̅N scattering matrix needed in Eq.(<ref>). We tested our OP in comparison with the available experimental data for antiproton
elastic scattering off ^4He, ^12C, and ^16,18O. Our results are in good agreement with data and are able to reproduce the correct angular position of the diffraction
minima. The OP has been also computed using the p̅N interaction at lower orders in
the chiral expansion to test the convergence of our results. As obtained in previous pA calculations, also in this case for a good description of the data it is mandatory to use an
interaction derived at least up to N^3LO. As a concluding remark, we mention that at this stage new questions arise about the importance of p̅NN
interactions in both structure and reaction calculations.



The authors are deeply grateful to L.-Y. Dai, J. Haidenbauer and U. Meißner who kindly provided the potentials derived in Ref. <cit.>.
We are also grateful to A. Rotondi, A. Larionov, and H. Lenske for providing the antiproton-nucleus experimental data.
Finally, we would like to thank S. Quaglioni for providing the c_D and c_E parameters used in the convergence analysis.
The work at Brookhaven National Laboratory was sponsored by the Office of Nuclear Physics, Office of Science of the U.S. Department of Energy under
Contract No. DE-AC02-98CH10886 with Brookhaven Science Associates, LLC.
The work at TRIUMF was supported by the NSERC Grant No. SAPIN-2016-00033. TRIUMF receives federal funding via a contribution agreement with the National
Research Council of Canada.
Computing support came from an INCITE Award on the Titan supercomputer of the Oak Ridge Leadership Computing Facility (OLCF) at ORNL, from Westgrid and
Compute Canada.










55
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

[fai()]fair
  @noop  howpublished
  <https://www.gsi.de/en/researchaccelerators/fair.htm>NoStop
[Obertelli(2018)]Obertelli:2622466
  author author A. Obertelli, http://cds.cern.ch/record/2622466 title PUMA: antiprotons and radioactive nuclei, type
  Tech. Rep. number CERN-INTC-2018-023. INTC-M-018 (institution CERN, address Geneva, year
  2018)NoStop
[Aumann et al.(2019)Aumann,
  Bartmann, Bouvard, Boine-Frankenheim, Broche, Butin,
  Calvet, Carbonell, Chiggiato,
  De Gersem, De Oliveira, Dobers, Ehm, Ferreira Somoza, Fischer, Fraser, Friedrich, Grenard, Hupin, Johnston, Kubota, Gomez-Ramos, Indelicato,
  Lazauskas, Malbrunot-Ettenauer, Marsic, Müller, Naimi, Nakatsuka, Necca, Neyens, Obertelli, Ono, Pasinelli, Paul, Pollacco, Rossi, Scheit, Seki, Schmidt, Schweikhard, Sels, Siesling, Uesaka, Wada, Wienholtz, Wycech, and Zacarias]Aumann:2691045
  author author T. Aumann, author W. Bartmann,
  author A. Bouvard, author O. Boine-Frankenheim, author A. Broche, author
  F. Butin, author D. Calvet, author J. Carbonell, author P. Chiggiato, author H. De Gersem, author R. De Oliveira, author T. Dobers, author F. Ehm, author J. Ferreira Somoza, author J. Fischer, author M. Fraser,
  author E. Friedrich, author J.-L. Grenard, author
  G. Hupin, author K. Johnston, author Y. Kubota, author M. Gomez-Ramos, author P. Indelicato, author R. Lazauskas, author S. Malbrunot-Ettenauer, author N. Marsic, author W. Müller, author S. Naimi, author N. Nakatsuka, author R. Necca, author G. Neyens, author A. Obertelli, author Y. Ono, author S. Pasinelli, author N. Paul,
  author E. C. Pollacco, author D. Rossi, author
  H. Scheit, author R. Seki, author A. Schmidt, author L. Schweikhard, author S. Sels,
  author E. Siesling, author T. Uesaka, author
  M. Wada, author F. Wienholtz, author S. Wycech,  and author S. Zacarias, http://cds.cern.ch/record/2691045
  title PUMA: antiprotons and radioactive nuclei,  type Tech. Rep. number CERN-SPSC-2019-033.
  SPSC-P-361 (institution CERN, address Geneva,  year 2019)NoStop
[Amsler and Myhrer(1991)]annurev.ns.41.120191.001251
  author author C. Amsler and author F. Myhrer, 10.1146/annurev.ns.41.120191.001251 journal journal Annual Review of Nuclear and Particle
  Science volume 41, pages 219
  (year 1991), http://arxiv.org/abs/https://doi.org/10.1146/annurev.ns.41.120191.001251
  https://doi.org/10.1146/annurev.ns.41.120191.001251 NoStop
[Gastaldi and Klapisch(1984)]nla.cat-vn777045
  author author U. Gastaldi and author R. Klapisch, @noop  title Physics at LEAR with low-energy cooled antiprotons (publisher Plenum Press New York, year 1984) pp. pages xv, 898 pNoStop
[Klapisch et al.(1990)Klapisch, Richard, and Bradamante]nla.cat-vn257837
  author author R. Klapisch, author J.-M. Richard,  and author F. Bradamante, @noop  title Antiproton-nucleon and antiproton-nucleus interactions  (publisher Plenum Press New York, year 1990)  pp. pages ix, 331 pNoStop
[Walcher(1988)]annurev.ns.38.120188.000435
  author author T. Walcher, 10.1146/annurev.ns.38.120188.000435 journal journal Annual Review of Nuclear and Particle
  Science volume 38, pages 67 (year 1988), http://arxiv.org/abs/https://doi.org/10.1146/annurev.ns.38.120188.000435
  https://doi.org/10.1146/annurev.ns.38.120188.000435 NoStop
[Dover et al.(1992)Dover,
  Gutsche, Maruyama, and Faessler]DOVER199287
  author author C. Dover, author T. Gutsche,
  author M. Maruyama,  and  author A. Faessler, https://doi.org/10.1016/0146-6410(92)90004-L journal
  journal Progress in Particle and Nuclear Physics volume 29, pages 87  (year
  1992)NoStop
[Adachi and Geramb(1987)]ADACHI1987461
  author author S. Adachi and author H. V. Geramb, https://doi.org/10.1016/0375-9474(87)90581-1
  journal journal Nuclear Physics A volume 470, pages 461  (year
  1987)NoStop
[Dover and Richard(1980)]PhysRevC.21.1466
  author author C. B. Dover and author J. M. Richard, 10.1103/PhysRevC.21.1466 journal
  journal Phys. Rev. C volume 21,  pages 1466 (year 1980)NoStop
[Dover and Richard(1982)]PhysRevC.25.1952
  author author C. B. Dover and author J. M. Richard, 10.1103/PhysRevC.25.1952 journal
  journal Phys. Rev. C volume 25,  pages 1952 (year 1982)NoStop
[Hippchen et al.(1989)Hippchen, Holinde, and Plessas]PhysRevC.39.761
  author author T. Hippchen, author K. Holinde,   and author W. Plessas, 10.1103/PhysRevC.39.761 journal journal
  Phys. Rev. C volume 39, pages 761
  (year 1989)NoStop
[El-Bennich et al.(2009)El-Bennich, Lacombe, Loiseau, and  Wycech]PhysRevC.79.054001
  author author B. El-Bennich, author M. Lacombe,
  author B. Loiseau,  and  author S. Wycech, 10.1103/PhysRevC.79.054001 journal journal Phys. Rev. C volume 79, pages 054001 (year 2009)NoStop
[Hrtánková and Mareš(2018)]HRTANKOVA201845
  author author J. Hrtánková and author J. Mareš, https://doi.org/10.1016/j.nuclphysa.2017.09.011 journal
  journal Nuclear Physics A volume
  969, pages 45  (year 2018)NoStop
[Zhou and Timmermans(2012)]PhysRevC.86.044003
  author author D. Zhou and author R. G. E. Timmermans, 10.1103/PhysRevC.86.044003 journal journal Phys. Rev. C volume
  86, pages 044003 (year 2012)NoStop
[Hodgson(1963)]hodgson1963
  author author P. Hodgson, https://books.google.it/books?id=7sq6AAAAIAAJ title The Optical Model of Elastic Scattering (publisher Clarendon Press, Oxford, year 1963)NoStop
[Heiselberg et al.(1989)Heiselberg, Jensen, Miranda, and  Oades]Heiselberg1989
  author author H. Heiselberg, author A. S. Jensen, author A. Miranda,   and author G. C. Oades, http://stacks.iop.org/1402-4896/40/i=2/a=002 journal
  journal Physica Scripta volume 40,  pages 141 (year 1989)NoStop
[Zhang et al.(1996)Zhang,
  Liu, Robson, and Li]PhysRevC.54.332
  author author Y.-s. Zhang, author J.-f. Liu,
  author B. A. Robson,  and  author Y.-g. Li, 10.1103/PhysRevC.54.332 journal journal
  Phys. Rev. C volume 54, pages 332
  (year 1996)NoStop
[Gaitanos and Kaskulov(2015)]GAITANOS2015181
  author author T. Gaitanos and author M. Kaskulov, https://doi.org/10.1016/j.nuclphysa.2015.04.006 journal
  journal Nuclear Physics A volume
  940, pages 181  (year 2015)NoStop
[Friedman et al.(2015)Friedman, Gal, Loiseau, and Wycech]FRIEDMAN2015101
  author author E. Friedman, author A. Gal,
  author B. Loiseau,  and  author S. Wycech, https://doi.org/10.1016/j.nuclphysa.2015.08.010 journal journal Nuclear Physics A volume 943, pages 101  (year
  2015)NoStop
[Larionov and Lenske(2017)]LARIONOV2017450
  author author A. Larionov and author H. Lenske, https://doi.org/10.1016/j.nuclphysa.2016.10.006
  journal journal Nuclear Physics A volume 957, pages 450  (year
  2017)NoStop
[Gennari et al.(2018)Gennari, Vorabbi, Calci, and  Navratil]Gennari:2017yez
  author author M. Gennari, author M. Vorabbi,
  author A. Calci,  and author P. Navratil, 10.1103/PhysRevC.97.034619 journal journal Phys.
  Rev. volume C97, pages 034619
  (year 2018), http://arxiv.org/abs/1712.02879
  arXiv:1712.02879 [nucl-th] NoStop

[Riesenfeld and Watson(1956)]Riesenfeld:1956zza
  author author W. B. Riesenfeld and author K. M. Watson, 10.1103/PhysRev.102.1157 journal
  journal Phys. Rev. volume 102,  pages 1157 (year 1956)NoStop

[Chinn et al.(1995a)Chinn, Elster,
  Thaler, and Weppner]PhysRevC.52.1992
  author author C. R. Chinn, author C. Elster,
  author R. M. Thaler,  and  author S. P. Weppner, 10.1103/PhysRevC.52.1992 journal journal Phys. Rev. C volume 52, pages 1992 (year 1995a)NoStop
[Dickhoff and Charity(2019)]Dickhoff2018
  author author W. Dickhoff and author R. Charity, https://doi.org/10.1016/j.ppnp.2018.11.002
  journal journal Progress in Particle and Nuclear
  Physics volume 105, pages 252 
  (year 2019)NoStop
[Burrows et al.(2019)Burrows, Elster, Weppner, Launey, Maris, Nogga, and Popa]Burrows:2018ggt
  author author M. Burrows, author C. Elster,
  author S. P. Weppner, author K. D. Launey, author
  P. Maris, author A. Nogga,  and author G. Popa, 10.1103/PhysRevC.99.044603
  journal journal Phys. Rev. C volume 99, pages 044603 (year
  2019)NoStop
[Arellano and Blanchon(2018)]Arellano2018
  author author H. F. Arellano and author G. Blanchon, 10.1103/PhysRevC.98.054616 journal journal Phys. Rev. C volume
  98, pages 054616 (year 2018)NoStop
[Whitehead et al.(2018)Whitehead, Lim, and Holt]Whitehead:2018bfs
  author author T. R. Whitehead, author Y. Lim,   and author J. W. Holt,  @noop    (year 2018), http://arxiv.org/abs/1812.08725 arXiv:1812.08725 [nucl-th] NoStop

[Kohno(2018)]PhysRevC.98.054617
  author author M. Kohno, 10.1103/PhysRevC.98.054617 journal
  journal Phys. Rev. C volume 98,  pages 054617 (year 2018)NoStop
[Barrett et al.(2013)Barrett, Navrátil, and Vary]BARRETT2013131
  author author B. R. Barrett, author P. Navrátil,  and author J. P.  Vary, https://doi.org/10.1016/j.ppnp.2012.10.003 journal journal Progress in Particle and Nuclear Physics volume 69, pages 131  (year
  2013)NoStop
[Entem et al.(2015)Entem,
  Kaiser, Machleidt, and Nosyk]Entem:2014msa
  author author D. R. Entem, author N. Kaiser,
  author R. Machleidt,  and  author Y. Nosyk, 10.1103/PhysRevC.91.014002 journal journal Phys. Rev. volume C91, pages
  014002 (year 2015), http://arxiv.org/abs/1411.5335
  arXiv:1411.5335 [nucl-th] NoStop

[Entem et al.(2017)Entem,
  Machleidt, and Nosyk]Entem:2017gor
  author author D. R. Entem, author R. Machleidt,   and author Y. Nosyk, 10.1103/PhysRevC.96.024004 journal journal Phys. Rev. C volume 96, pages 024004 (year 2017)NoStop
[Navrátil(2007)]Navratil2007
  author author P. Navrátil, 10.1007/s00601-007-0193-3 journal journal Few-Body Syst. volume
  41, pages 117 (year 2007)NoStop
[Gysbers et al.(2019)Gysbers, Hagen, Holt, Jansen, Morris, Navrátil, Papenbrock, Quaglioni, Schwenk,
  Stroberg, and Wendt]Gysbers2019
  author author P. Gysbers, author G. Hagen,
  author J. D. Holt, author G. R. Jansen, author
  T. D. Morris, author
  P. Navrátil, author
  T. Papenbrock, author
  S. Quaglioni, author
  A. Schwenk, author S. R. Stroberg,  and author K. A. Wendt, 10.1038/s41567-019-0450-7
  journal journal Nature Physics volume 15, pages 428 (year
  2019)NoStop
[Roth and Navrátil(2007)]PhysRevLett.99.092501
  author author R. Roth and author P. Navrátil, 10.1103/PhysRevLett.99.092501 journal journal Phys. Rev. Lett. volume 99, pages 092501 (year
  2007)NoStop
[Roth(2009)]PhysRevC.79.064324
  author author R. Roth, 10.1103/PhysRevC.79.064324 journal
  journal Phys. Rev. C volume 79,  pages 064324 (year 2009)NoStop
[Dai et al.(2017)Dai,
  Haidenbauer, and Meißner]Dai2017
  author author L.-Y. Dai, author J. Haidenbauer,   and author U.-G. Meißner,  10.1007/JHEP07(2017)078 journal journal Journal of High Energy Physics volume
  1707, pages 78 (year 2017)NoStop
[Epelbaum et al.(2015a)Epelbaum, Krebs,  and Meißner]Epelbaum:2014sza
  author author E. Epelbaum, author H. Krebs,   and author U. G. Meißner,  10.1103/PhysRevLett.115.122301 journal
  journal Phys. Rev. Lett. volume 115,  pages 122301 (year 2015a), http://arxiv.org/abs/1412.4623 arXiv:1412.4623 [nucl-th] NoStop

[Epelbaum et al.(2015b)Epelbaum, Krebs,  and Meißner]Epelbaum:2014efa
  author author E. Epelbaum, author H. Krebs,   and author U. G. Meißner,  10.1140/epja/i2015-15053-8 journal journal Eur. Phys. J. volume A51, pages 53 (year 2015b), http://arxiv.org/abs/1412.0142 arXiv:1412.0142 [nucl-th] NoStop

[Bernard et al.(1995)Bernard, Kaiser, and Meißner]bernard_ChPT_1995
  author author V. Bernard, author N. Kaiser,   and author U.-G. Meißner,  10.1142/S0218301395000092 journal journal International Journal of Modern Physics E volume 04, pages 193 (year 1995), http://arxiv.org/abs/https://doi.org/10.1142/S0218301395000092
  https://doi.org/10.1142/S0218301395000092 NoStop
[Kolck(1999)]VanKolck1999337
  author author U. V. Kolck, http://dx.doi.org/10.1016/S0146-6410(99)00097-6
  journal journal Progress in Particle and Nuclear
  Physics volume 43, pages 337 
  (year 1999)NoStop
[Epelbaum et al.(2009)Epelbaum, Hammer, and Meißner]Epelbaum:2008ga
  author author E. Epelbaum, author H.-W. Hammer,  and author U.-G. Meißner, 10.1103/RevModPhys.81.1773 journal journal Rev. Mod. Phys. volume
  81, pages 1773 (year 2009), http://arxiv.org/abs/0811.1338 arXiv:0811.1338 [nucl-th] NoStop

[Machleidt and Entem(2011)]Machleidt:2011zz
  author author R. Machleidt and author D. R. Entem, 10.1016/j.physrep.2011.02.001 journal journal Phys. Rept. volume
  503, pages 1 (year 2011), http://arxiv.org/abs/1105.2919 arXiv:1105.2919 [nucl-th] NoStop

[Chinn et al.(1995b)Chinn, Elster,
  Thaler, and Weppner]PhysRevC.51.1418
  author author C. R. Chinn, author C. Elster,
  author R. M. Thaler,  and  author S. P. Weppner, 10.1103/PhysRevC.51.1418 journal journal Phys. Rev. C volume 51, pages 1418 (year 1995b)NoStop
[Batusov et al.(1990)Batusov
  et al.]Batusov:1990ge
  author author Yu. A.  Batusov et al., @noop  journal
  journal Sov. J. Nucl. Phys. volume
  52, pages 776 (year 1990), note
  [Yad. Fiz.52,1222(1990)]NoStop

[Garreta et al.(1984)Garreta, Birien, Bruge, Chaumeaux, Drake, Janouin, Legrand, Lemaire, Mayer, Pain, Peng, Berrada, Bocquet, Monnand, Mougey, Perrin, Aslanides, Bing, Lichtenstadt, and Yavin]GARRETA198464
  author author D. Garreta, author P. Birien,
  author G. Bruge, author A. Chaumeaux, author
  D. Drake, author S. Janouin, author D. Legrand, author M. Lemaire, author B. Mayer, author J. Pain, author J. Peng, author M. Berrada,
  author J. Bocquet, author E. Monnand, author
  J. Mougey, author P. Perrin, author E. Aslanides, author O. Bing, author J. Lichtenstadt,  and author A. Yavin, https://doi.org/10.1016/0370-2693(84)91552-1
  journal journal Physics Letters B volume 149, pages 64  (year
  1984)NoStop
[Bruge et al.(1986)Bruge,
  Chaumeaux, Birien, Drake,
  Garreta, Janouin, Legrand,
  Lemaire, Mayer, Pain,
  Berrada, Bocquet, Monnand,
  Mougey, Perrin, Aslanides,
  Bing, Lichtenstadt, Yavin,  and Peng]BRUGE198614
  author author G. Bruge, author A. Chaumeaux,
  author P. Birien, author D. Drake, author
  D. Garreta, author S. Janouin, author D. Legrand, author M. Lemaire, author B. Mayer, author J. Pain, author M. Berrada, author J. Bocquet,
  author E. Monnand, author J. Mougey, author
  P. Perrin, author E. Aslanides, author O. Bing, author J. Lichtenstadt, author A. Yavin,
   and author J. Peng, https://doi.org/10.1016/0370-2693(86)90676-3 journal
  journal Physics Letters B volume
  169, pages 14  (year 1986)NoStop
[Bydžovský et al.(1991)Bydžovský, Mach, and Nichitiu]PhysRevC.43.1610
  author author P. Bydžovský, author R. Mach,  and author F. Nichitiu, 10.1103/PhysRevC.43.1610 journal journal Phys. Rev. C volume
  43, pages 1610 (year 1991)NoStop
[Epelbaum et al.(2005)Epelbaum, Glöckle, and Meißner]chiralepelbaum_n3lo
  author author E. Epelbaum, author W. Glöckle,  and author U.-G. Meißner, 10.1016/j.nuclphysa.2004.09.107 journal journal Nucl. Phys. volume
  A747, pages 362 (year 2005), http://arxiv.org/abs/nucl-th/0405048 arXiv:nucl-th/0405048 [nucl-th]
  NoStop

[Hammer et al.(2019)Hammer,
  König, and van
  Kolck]hammer2019nuclear
  author author H. W. Hammer, author S. König,   and author U. van Kolck,  @noop    (year 2019), http://arxiv.org/abs/1906.12122 arXiv:1906.12122 [nucl-th] NoStop
[Quaglioni()]quaglioni
  author author S. Quaglioni, @noop  howpublished private
  communicationNoStop
[Vorabbi et al.(2016)Vorabbi, Finelli, and Giusti]Vorabbi:2015nra
  author author M. Vorabbi, author P. Finelli,   and author C. Giusti, 10.1103/PhysRevC.93.034619 journal journal Phys. Rev. volume C93, pages
  034619 (year 2016), http://arxiv.org/abs/1510.05928 arXiv:1510.05928 [nucl-th] NoStop

[Vorabbi et al.(2017)Vorabbi, Finelli, and Giusti]Vorabbi:2017rvk
  author author M. Vorabbi, author P. Finelli,   and author C. Giusti, 10.1103/PhysRevC.96.044001 journal journal Phys. Rev. volume C96, pages
  044001 (year 2017), http://arxiv.org/abs/1710.00716 arXiv:1710.00716 [nucl-th] NoStop

[Ekström et al.(2013)Ekström, Baardsen, Forssén,
  Hagen, Hjorth-Jensen, Jansen,
  Machleidt, Nazarewicz, Papenbrock, Sarich, and Wild]PhysRevLett.110.192502
  author author A. Ekström, author G. Baardsen,
  author C. Forssén, author G. Hagen, author
  M. Hjorth-Jensen, author
  G. R. Jansen, author
  R. Machleidt, author
  W. Nazarewicz, author
  T. Papenbrock, author
  J. Sarich,  and author
  S. M. Wild, 10.1103/PhysRevLett.110.192502 journal journal
  Phys. Rev. Lett. volume 110, pages
  192502 (year 2013)NoStop
[Birsa et al.(1985)Birsa, Bradamante, Dalla
  Torre-Colautti, Giorgi, Martin,
  Penzo, Schiavon, Villari, Degli Agosti, Heer,
  Hees, Lechanoine-Leluc, Onel, Rapin, Bos, Kluyver, Kunne, Linssen,
  Beard, Bugg, Hall,
  Clough, and Shypit]1985PhLB..155..437B
  author author R. Birsa, author F. Bradamante, author S. Dalla
  Torre-Colautti, author M. Giorgi, author A. Martin,
  author A. Penzo, author P. Schiavon, author
  A. Villari, author
  S. Degli Agosti, author
  E. Heer, author R. Hees, author C. Lechanoine-Leluc, author Y. Onel, author D. Rapin, author K. Bos, author J. C.  Kluyver, author R. A.  Kunne, author L. Linssen, author C. Beard,
  author D. V. Bugg, author J. R. Hall, author
  A. S. Clough,  and author
  R. L. Shypit, 10.1016/0370-2693(85)91601-6 journal journal
  Physics Letters B volume 155, pages
  437 (year 1985)NoStop



