Scattering of charmed baryons on nucleons

Scattering of charmed baryons on nucleons

J. Haidenbauer j.haidenbauer@fz-juelich.de G. Krein gkrein@ift.unesp.br Institute for Advanced Simulation, Institut für Kernphysik, and Jülich Center for Hadron Physics, Forschungszentrum Jülich, D-52425 Jülich, Germany Instituto de Física Teórica, Universidade Estadual Paulista
Rua Dr. Bento Teobaldo Ferraz, 271 - Bloco II, 01140-070, São Paulo, SP, Brazil
Abstract

Chiral effective field theory is utilized for extrapolating results on the interaction, obtained in lattice QCD at unphysical (large) quark masses, to the physical point. The pion-mass dependence of the components that constitute the potential up to next-to-leading order (pion-exchange diagrams and four-baryon contact terms) is fixed by information from lattice QCD simulations. No recourse to SU(3) or SU(4) flavor symmetry is made. It is found that the results of the HAL QCD Collaboration for quark masses corresponding to MeV imply a moderately attractive interaction at MeV with scattering lengths of  fm for the as well as the partial waves. For such an interaction the existence of a charmed counterpart of the hypertriton is unlikely but four- and/or five-baryons systems with a baryon could be indeed bound.

keywords:
Charm hyperon-nucleon interaction, Chiral effective field theory
Pacs:
14.20.Lq, 12.39.Fe, 13.75.Ev, 21.30.Fe

1 Introduction

The dynamics of hadrons with different flavor degrees of freedom provides different windows for our understanding of the underlying theory of strong interaction, quantum chromodynamics (QCD). With regard to hadrons with charm, so far spectroscopy has been the most visible and definitely by far the most interesting branch of research. Indeed, the large number of structures observed in experiments at energies above the open charm production threshold provides a challenge for our standard (but obviously naive) picture that mesons are composed out of quark-antiquark pairs and baryons out of three quarks. See Refs. Guo:2017 (); Lebed:2016hpi (); Chen:2016 (); Esposito:2016 () for recent overviews and discussions of these structures, commonly referred to as X, Y and Z states.

Some proposals for experiments at sites such as INFN Feliciello:2012 (), J-PARC JPARC (), and FAIR CBM (); Wiedner:2011 () aim at exploring also other aspects of charm physics. Specifically, so-called charm factories would allow the targeted production of charmed hadrons like the -meson or the and hyperons and a study of their interaction with ordinary hadrons. The expectation of possible experiments in the not so far future has triggered a variety of theoretical investigations. In particular, it has led to a renewed interest in the interaction of the with nucleons and with nuclei over the last few years Liu:2011xc (); Huang:2013 (); Gal:2014 (); Garcilazo:2015qha (); Maeda:2015hxa (); Shyam:2016uxa (); Ohtani:2017wdc (). Those studies join the ranks of a long history of speculations about bound nuclear systems involving the , the lightest charmed baryon Dover:1977jw (); Bando:1981ti (); Bando:1983yt (); Gibson:1983zw (); Bunyatov:1992in (); Tsushima:2002ua (); Tsushima:2003dd (); Kopeliovich:2007kd () — see also the recent reviews Hosaka:2016ypm (); Krein:2017usp (). Indeed, in most of the investigations so far, the interaction (), derived within the meson-exchange framework Liu:2011xc (); Dover:1977jw (); Bando:1981ti (); Bando:1983yt (); Gibson:1983zw () or in the constituent quark model Maeda:2015hxa (); Froemel:2004ea (), turns out to be strongly attractive.

Interestingly, quite the opposite picture emerged from recent (2+1)-flavor lattice QCD (LQCD) simulations by the HAL QCD Collaboration Miyamoto:2016 (); Miyamoto:2016A (); Miyamoto:2017 (). Pertinent calculations, performed for unphysical quark masses corresponding to pion masses of MeV111The Gell-Mann-Oakes-Renner relation states that the squared pion mass is proportional to the average light quark mass. Therefore, the notions “quark mass” and “pion mass” are used synonymously., suggest that the and interactions could be much less attractive than predicted by the phenomenological potentials mentioned above. While initial preliminary studies Miyamoto:2016 (); Miyamoto:2016A () indicated an extremely weak interaction, the recently published final LQCD results Miyamoto:2017 () rectify that conjecture and imply a somewhat stronger, though still only moderately attractive interaction.

In the present work we provide predictions for the interaction at the physical point based on the LQCD simulations by the HAL QCD Collaboration Miyamoto:2017 (). The extrapolation of the LQCD results, available for MeV, to MeV is performed within the framework of chiral effective field theory (EFT) Epelbaum:2008 (); Machleidt:2011 (). Thereby we follow a strategy that has been already employed by us in the past in the analysis of other LQCD results on baryon-baryon interactions in the strangeness sector, notably the  Haidenbauer:2011a (), Haidenbauer:2011b (), and Haidenbauer:2017 () systems: At first chiral EFT is utilized to establish a potential for baryon and meson masses that correspond to those in the lattice simulation. In particular, open parameters are determined by a fit to pertinent LQCD results (phase shifts, scattering lengths). Then the potential is extrapolated to the physical point. Thereby the pion-mass dependence of the ingredients (pseudoscalar-meson exchange, four-baryon contact terms) is taken into account explicitly, and in line with chiral EFT.

The paper is structured as follows: In Sect. 2 we provide an outline of the employed formalism. Results for phase shifts (for the and partial waves) are reported in Sect. 3, for pion masses corresponding to those in the LQCD simulation and for the physical value, MeV. The corresponding scattering lengths are evaluated too and turn out to be in the order of  fm. Finally, consequences of our results for the existence of bound hypernuclei are discussed. The paper closes with a short summary.

2 Formalism

The interaction is constructed by using chiral EFT as guideline. Thereby we follow closely our application of this scheme to the and systems in Refs. Polinder (); Haidenbauer:2013 () where corresponding potentials have been obtained up to NLO in the Weinberg power counting Epelbaum:2008 (); Machleidt:2011 (). In this framework the potential is given in terms of pion exchanges and a series of contact interactions with an increasing number of derivatives. The latter represent the short-range part of the baryon-baryon force and are parameterized by low-energy constants (LECs), that need to be fixed in a fit to data. Both classes of contributions depend on the quark mass (or, equivalently, the pion mass). At LO the only quark-mass dependence of the potential is through the pion mass that appears in the propagator of the pion-exchange potential, cf. below. However, at NLO the contact terms as well as the pion coupling constants depend on the pion mass. For details, we refer to Refs. Beane:2002vs (); Beane03 (); Epe02 (); Epe02a (); Baru:2015 (); Baru:2016 (), where the quark mass dependence of the interaction has been investigated; see also Ref. Petschauer:2013 ().

Let us start by introducing the contact interaction that we employ. For the partial waves considered in the present study (, -) it is given by

(1)

with and being the initial and final center-of-mass momenta in the or systems. The quantities , , , are the aforementioned LECs that need to be fixed by a fit to lattice data (phase shifts). The ansatz Eq. (1) is motivated by the corresponding expression in the standard Weinberg counting up to NLO Epe02 (); Petschauer:2013 () but differs from it by the terms proportional to which are nominally of higher order. Nevertheless, we include these terms because they allow us to obtain an optimal description of the LQCD results at MeV as well as at MeV and, thereby, enable us a better constrained extrapolation to lower pion masses. Contrary to our study of the interaction in Refs. Polinder (); Haidenbauer:2013 (), here we do not impose SU(3) (or SU(4)) flavor symmetry. In any case, given that there is no pertinent information on the channel from LQCD in Ref. Miyamoto:2017 (), it is impossible to fix the LECs for the and transitions and, therefore, they are set to zero.

The contribution of pion exchange to the potential is given by

(2)

where stand for and/or , is the transferred momentum, , and is a pertinent isospin factor Polinder (). As already mentioned, we do not assume the validity of SU(4) flavor symmetry in the present study. This concerns also the coupling constants . The coupling constant can be determined from the experimentally known decay rate, see Refs. Albertus:2005 (); Can:2016 (). With regard to the coupling constant we resort to LQCD results Alexandrou:2016 (). Besides their value at the physical point, we need also the dependence of the and coupling constants as well as the one for the vertex,

(3)

LQCD results for the dependence of the pion decay constant are readily available in the literature, e.g. in Ref. Durr:2013 (). From that reference, one deduces the values MeV at MeV, MeV at MeV, and MeV at MeV. For obtaining the latter value, a linear dependence of has been assumed, as suggested by Fig. 5 in Ref. Durr:2013 (). LQCD results for the dependence of the the axial-vector strengths can be found in Ref. Alexandrou:2016 () though only up to MeV. For a rather moderate increase with is suggested by LQCD Alexandrou:2016 (). is found to be practically independent of , though, unfortunately, the lattice results do not match well with the known value at the physical point. There is no information on the variation of the vertex with . Because of these reasons we neglect the dependence of the ’s on in our calculation and assume that . Specifically, we use  PDG (),  Alexandrou:2016 () and  Albertus:2005 (); Can:2016 (). In any case, the by far strongest dependence of on comes via Durr:2013 () and this circumstance is adequately taken into account in our calculation.

Since under the assumption that isospin is conserved, there is no one-pion exchange contribution to the potential. However, we include the coupling of to via pion exchange, which is known to play an important role in case of the and systems Haidenbauer:2013 (). The resulting effective two-pion exchange contribution to the potential is generated by solving a coupled-channel Lippmann-Schwinger (LS) equation, see below. In this context let us note that the pertinent contribution would arise anyway at NLO, even in a single-channel treatment. In principle, at NLO there are further contributions from two-pion exchange Haidenbauer:2013 (). However, in the present study we omit those for simplicity reasons and assume that they are effectively absorbed into the contact terms. Furthermore, contributions from and meson exchanges that would arise under the assumption of SU(3) (or SU(4)) symmetry, are likewise delegated to the contact interactions.

In our standard calculation the potential is set to zero. As already mentioned above, in the extraction of the and phase shifts from lattice data in Ref. Miyamoto:2017 () the coupling of the system to is not taken into account explicitly and no results for phase shifts are provided. However, we perform test calculations with the pion-exchange contribution to the potential turned on in order to see its effect on our results. A further issue that we have ignored in our derivation of the interaction is the role of heavy quark spin symmetry Liu:2011xc (); Lu:2017 (). Indeed the and thresholds are just about MeV apart PDG () so that the coupling between those systems could be important Liu:2011xc (). However, since already the channel is not explicitly considered in Ref. Miyamoto:2017 (), there is no point in including in our analysis and, therefore, we omitted it likewise in our study. However, it is important to notice that, if there are any effects of it, these are absorbed into the LECs. In any case, since MeV, there should be less influence on the amplitude anyway. Indeed, because of the larger mass difference  MeV, as compared to  MeV, one would expect that even the channel coupling to plays a less important role for the charm sector than in the strangeness sector.

The reaction amplitudes are obtained from the solution of a coupled-channel LS equation for the interaction potentials, which is given in partial-wave projected form by

(4)

Here, the label indicates the particle channels and the label the partial wave. is the pertinent reduced mass. The on-shell momentum in the intermediate state, , is defined by . Following the practice of the HAL QCD Collaboration Miyamoto:2017 (), phase shifts will be given as functions of the kinetic energy in the center-of-mass (cm) frame, .

Since the integral in the LS equation (4) is divergent for the chiral potentials specified above, a regularization needs to be introduced. We utilize here the same prescription as in our studies, where the potentials in the LS equation are cut off in momentum space by multiplication with a regulator function, , so that the high-momentum components of the baryon and pseudoscalar meson fields are removed. In the present study we employ the cut-off scales - MeV, in line with the range that yielded optimal and stable results in our NLO study of the and interactions Haidenbauer:2013 (). The variations of the results with the cutoff reflect uncertainties that will be indicated by bands in the plots we show in the next section.

In the analysis of the LQCD simulations we follow closely the strategy of our previous works in Refs. Haidenbauer:2011a (); Haidenbauer:2011b (); Haidenbauer:2017 (): (i) The LECs, i.e. the only free parameters in the potential, are determined by a fit to LQCD results (phase shifts) employing the inherent baryon and meson masses of the lattice simulation; (ii) Results at the physical point are obtained via a calculation in which the pertinent physical masses of the mesons are substituted in the evaluation of the potential and those of the baryons in the baryon-baryon propagators appearing in the LS equation. The baryon masses corresponding to the LQCD simulations at and  MeV are taken from Ref. Miyamoto:2017 (). For the calculation at the physical point we use the masses from the PDG PDG (), i.e. MeV, MeV.

Alternative ways to implement the quark mass dependence of the nuclear forces can be found in Refs. Baru:2015 (); Berengut:2013 ().

3 Results

LQCD results for phase shifts are available for the and partial waves for MeV Miyamoto:2017 (). We determine the LECs of the contact interaction, cf. Eq. (1), by a fit to the lattice data at the two lower pion masses. Specifically, the pion-mass dependence exhibited by the LQCD simulation is exploited to determine the LECs and that encode the pion-mass dependence of the contact interaction. The fits are done to the phase shifts, generated from the parameterized version of the potentials provided in Ref. Miyamoto:2017 (), for energies up to  MeV. Alternative fits taking into account the HAL QCD results up to  MeV were performed too. In this context it should be noted that applications of chiral EFT to scattering and specifically to the state reveal that NLO interactions are expected to provide quantitative results up to roughly MeV Epelbaum:2008 (); Machleidt:2011 ().

Figure 1: Results for the partial wave. Left: Phase shifts of the HAL QCD Collaboration Miyamoto:2017 () at MeV (blue open circles) and MeV (green filled circles), as a function of the cm kinetic energy, together with our fits (lines/bands). The broader (red) band is the prediction for MeV. Right: Dependence of the scattering length on the pion mass. The bands represent the cutoff variation MeV, see text.

Results for the partial wave are presented in Fig. 1. The phase shifts for pion masses , and MeV are shown on the left side while the dependence of the scattering length on the pion mass is depicted on the right side. The bands represent the dependence of the results on variations of the cutoff . One can see that the lattice results at  MeV are reproduced quantitatively by our potential up to cm kinetic energies of around  MeV, as expected for an NLO interaction, while those at  MeV are remarkably well described over the whole energy range shown. In both cases the cutoff dependence is negligibly small at low energies so that the bands are hardly visible. The phase shift obtained from our interaction when extrapolated to the physical point (red bands) do exhibit a noticeable but still rather moderate cutoff dependence. A maximum of around 20 degrees of the phase shift is predicted in this case. The pion-mass dependence of the scattering length, shown on the right-hand side of Fig. 1, is fairly smooth and almost linear in . Only close to the physical point a somewhat stronger dependence is visible. The value predicted at MeV is  fm.

Figure 2: Results for the partial wave. Left: Phase shifts of the HAL QCD Collaboration Miyamoto:2017 () at MeV (blue open circles) and MeV (green filled circles), as a function of the cm kinetic energy, together with our fits (lines/bands). The broader (red) band is the prediction for MeV. Right: Dependence of the scattering length on the pion mass. The bands represent the cutoff variation MeV, see text.

Results for the partial wave are presented in Fig. 2. Again phase shifts as well as the pion-mass dependence of the scattering length are shown. Those are very similar to the ones of the state, even on a quantitative level. Indeed, it was already noted by the HAL QCD Collaboration that the potentials they extracted for and  MeV are almost identical Miyamoto:2017 () and we see that this characteristic feature persists even in our extrapolation to the physical point. This is certainly a remarkable feature in view of the fact that in case of the partial wave there is a coupling to the induced by the tensor force, and this coupling is taken into account in our analysis.

For completeness we have summarized the results for the and scattering lengths, and , in Table. 1. From those numbers one can see that the variation in the extrapolated values due to the employed regularization scheme is in the order of fm. Additional fits to the phase shifts where the considered energy range was extended up to MeV led to changes in the scattering lengths at the physical point of around fm, with a clear tendency to smaller values. Further exploratory fits carried out by us indicate that variations in the scattering length of  fm at  MeV amount to differences of about  fm at MeV. Combining these observations with the uncertainty of  fm given by the HAL QCD Collaboration for their result at MeV suggests that the scattering lengths at the physical point could be about  fm larger, i.e. as large as  fm. Finally, we performed calculations with the pion-exchange contribution (2) to the potential included. Adding its contribution within our coupled-channel framework (4) has a negligible effect on the results at MeV and MeV. However, it leads to noticeable variations in the predictions for MeV, which amount to roughly  fm in the scattering lengths. Obviously, additional information on the phase shifts, as had been provided in the preliminary study of the HAL QCD Collaboration Miyamoto:2016A (), and promised in Ref. Miyamoto:2017 () for the future, would be helpful for reducing the uncertainty in the extrapolation. In any case, a more quantitative estimate of the overall uncertainty should be attempted, once lattice data with better statistics are available and/or results for pion masses closer to the physical point.

MeV MeV MeV MeV
HAL QCD Miyamoto:2017 ()
our results
HAL QCD Miyamoto:2017 ()
our results
Table 1: scattering lengths in the () and () partial waves (in fm) for different pion masses. Predictions for the corresponding effective range parameters (in fm) at the physical point are likewise given.

Clearly, and in line with the LQCD results for larger pion masses, we do not get any bound states. However, what are the consequences of the results presented in the preceding section for the possible existence of bound nuclei? Let us look at the strangeness sector and, specifically, at the lightest nucleus that is experimentally observed, namely the hypertriton . The experimental value for the binding energy is MeV, which implies a separation energy for the of only MeV. Faddeev calculations of the coupled three-body systems for realistic potentials have been reported in Refs. Nogga:2002 (); Haidenbauer:2007 (); Nogga:2013 (). Those suggest that, for interactions which provide sufficient attraction so that the hypertriton is bound, the scattering lengths are in the order of to fm for the state and to fm for .

The scattering lengths for the system at the physical point, deduced from the the LQCD simulations of the HAL QCD Collaboration, are to fm for the as well as the states. Thus, in the case of the partial wave the value is considerably smaller than its counterpart in the strangeness sector while for the channel the difference is less dramatic. Since the average (or ) potential that is relevant for the hypertriton is dominated by the spin-singlet channel, i.e. , see Ref. Gibson:1994 () for details, it seems rather unlikely that an only moderately attractive singlet interaction can support the existence of a bound state. Even effects due to a reduction of the kinetic energy associated with the induced by the larger mass of the as compared to the , emphasized in several works in the past Garcilazo:2015qha (); Dover:1977jw (); Gibson:1983zw (), cannot compensate for this large difference in the strength. Actually, in Ref. Gibson:1983zw () an estimate for the binding energy of charmed three- and four-body systems is provided based on an exact solution of corresponding scattering equations. Interestingly, Model 4 considered in that work yields scattering lengths very similar to the ones of our analysis, namely fm and fm (cf. Table I in that work). For this interaction no bound state is found for the nucleus. However, the 4-body systems and could be already bound, though possibly only weakly, even when considering the additional repulsive effect from the Coulomb force due to the charge of the as discussed in Ref. Gibson:1983zw ().

With regard to recent few-body calculations, the interaction employed in Ref. Maeda:2015hxa () is strongly attractive and leads already to bound states. The binding energies for are then in the order of MeV. Those results do not allow us to draw any conclusions about what would happen for significantly less attractive interactions like the one we deduce from the analysis of the LQCD simulations by the HAL QCD Collaboration. The situation might be different for the three-body calculation in Ref. Garcilazo:2015qha (), where a charmed hypertriton with (and total isospin ) is predicted. For that state the spin-triplet interaction is dominant Miyagawa:1995 (). Though no actual results are given, possibly the potentials employed in this calculation are the same as those described in Ref. Gal:2014 (), cf. Table III of that reference. If so one would expect perhaps likewise a () bound state for a interaction that produces scattering lengths of  fm as the one deduced by us from the HAL QCD results.

4 Summary

In the present Letter, we have used the framework of chiral effective field theory to extrapolate lattice QCD results for the interaction at  MeV by the HAL QCD Collaboration Miyamoto:2017 () to the physical point. Thereby, we have followed a strategy employed previously in the strangeness sector Haidenbauer:2011a (); Haidenbauer:2011b (); Haidenbauer:2017 (). However, contrary to the calculations performed in those works, no recourse to SU(3) or SU(4) flavor symmetry has been made in the present study. Furthermore, in the present work the pion-mass dependence of all components that constitute the potential up to next-to-leading order (pion-exchange diagrams and four-baryon contact terms) are taken into account. Information from lattice QCD simulations is utilized to implement these features.

Our analysis points to a moderately attractive interaction at the physical point with scattering lengths of  fm for the as well as for the partial waves. Such an interaction lends to the possibility of bound four- and/or five-baryons systems with a baryon and presumably of heavier hypernuclei. On the other hand, two-body () bound states as advocated in some recent investigations based on phenomenological potentials Liu:2011xc (); Maeda:2015hxa () can be definitely excluded, even if one considers uncertainties in the extrapolation of the lattice results. Also the existence of a hypertriton-like three-body bound state (with ) is rather unlikely.

Acknowledgments
This work was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq, 305894/2009-9 (G.K.) and Fundação de Amparo à Pesquisa do Estado de São Paulo - FAPESP, 2013/01907-0 (G.K.)

References

  • (1) F.-K. Guo, C. Hanhart, U.-G. Meißner, Q. Wang, Q. Zhao and B.-S. Zou, arXiv:1705.00141 [hep-ph], Rev. Mod. Phys., in print.
  • (2) R. F. Lebed, R. E. Mitchell and E. S. Swanson Prog. Part. Nucl. Phys. 93 (2017) 143.
  • (3) H.-X. Chen, W. Chen, X. Liu, Y.-R. Liu and S.-L. Zhu, Rept. Prog. Phys. 80 (2017) 076201.
  • (4) A. Esposito, A. Pilloni and A. D. Polosa, Phys. Rept. 668 (2016) 1.
  • (5) A. Feliciello, Nucl. Phys. A 881 (2012) 78.
  • (6) H. Noumi et al., Proposal P50, J-PARC PAC-16, 2013
    http://j-parc.jp/researcher/Hadron/en/pac_1301/pdf/P50_2012-19.pdf
  • (7) B. Friman, C. Hohne, J. Knoll, S. Leupold, J. Randrup, R. Rapp and P. Senger, Lect. Notes Phys. 814 (2011) 1.
  • (8) U. Wiedner, Prog. Part. Nucl. Phys. 66 (2011) 477.
  • (9) Y. R. Liu and M. Oka, Phys. Rev. D 85 (2012) 014015
  • (10) H. Huang, J. Ping and F. Wang, Phys. Rev. C 87 (2013) 034002.
  • (11) A. Gal, H. Garcilazo, A. Valcarce and T. Fernández-Caramés, Phys. Rev. D 90 (2014) 014019.
  • (12) H. Garcilazo, A. Valcarce and T. F. Caramés, Phys. Rev. C 92 (2015) 024006.
  • (13) S. Maeda, M. Oka, A. Yokota, E. Hiyama and Y. R. Liu, PTEP 2016 (2016) 023D02.
  • (14) R. Shyam and K. Tsushima, Phys. Lett. B 770 (2017) 236.
  • (15) K. Ohtani, K. J. Araki and M. Oka, arXiv:1704.04902 [hep-ph].
  • (16) C. B. Dover and S. H. Kahana, Phys. Rev. Lett. 39 (1977) 1506.
  • (17) H. Bando and M. Bando, Phys. Lett. 109B (1982) 164.
  • (18) H. Bando and S. Nagata, Prog. Theor. Phys. 69 (1983) 557.
  • (19) B. F. Gibson, G. Bhamathi, C. B. Dover and D. R. Lehman, Phys. Rev. C 27 (1983) 2085.
  • (20) S. A. Bunyatov, V. V. Lyukov, N. I. Starkov and V. A. Isarev, Sov. J. Part. Nucl. 23 (1992) 253.
  • (21) K. Tsushima and F.C. Khanna Phys. Rev. C 67 (2003) 015211
  • (22) K. Tsushima and F.C. Khanna, J. Phys. G 30 (2004) 1765.
  • (23) V. B. Kopeliovich and A. M. Shunderuk, Eur. Phys. J. A 33 (2007) 277.
  • (24) A. Hosaka, T. Hyodo, K. Sudoh, Y. Yamaguchi and S. Yasui, Prog. Part. Nucl. Phys. 96 (2017) 88.
  • (25) G. Krein, A. W. Thomas and K. Tsushima, arXiv:1706.02688 [hep-ph].
  • (26) F. Frömel, B. Juliá-Díaz and D. O. Riska, Nucl. Phys. A 750 (2005) 337
  • (27) T. Miyamoto [HAL QCD Collaboration], PoS LATTICE 2015 (2016) 090.
  • (28) T. Miyamoto [HAL QCD Collaboration], PoS LATTICE 2016 (2017) 117.
  • (29) T. Miyamoto et al., arXiv:1710.05545 [hep-lat].
  • (30) E. Epelbaum, H. W. Hammer and U.-G. Meißner, Rev. Mod. Phys. 81 (2009) 1773.
  • (31) R. Machleidt and D. R. Entem, Phys. Rep. 503 (2011) 1.
  • (32) J. Haidenbauer and U.-G. Meißner, Phys. Lett. B 706 (2011) 100.
  • (33) J. Haidenbauer and U.-G. Meißner, Nucl. Phys. A 881 (2012) 44.
  • (34) J. Haidenbauer, S. Petschauer, N. Kaiser, U.-G. Meißner and W. Weise, arXiv:1708.08071 [nucl-th], Eur. Phys. J. C, in print.
  • (35) H. Polinder, J. Haidenbauer, and U.-G. Meißner, Nucl. Phys. A 779 (2006) 244.
  • (36) J. Haidenbauer, S. Petschauer, N. Kaiser, U.-G. Meißner, A. Nogga and W. Weise, Nucl. Phys. A 915 (2013) 24.
  • (37) S. R. Beane, M. J. Savage, Nucl. Phys. A 713 (2003) 148.
  • (38) S. R. Beane, M. J. Savage, Nucl. Phys. A 717 (2003) 91.
  • (39) E. Epelbaum, U.-G. Meißner, W. Glöckle, Nucl. Phys. A 714 (2002) 535.
  • (40) E. Epelbaum, U.-G. Meißner, W. Glöckle, [arXiv:nucl-th/0208040].
  • (41) V. Baru, E. Epelbaum, A. A. Filin and J. Gegelia, Phys. Rev. C 92 (2015) 014001.
  • (42) V. Baru, E. Epelbaum and A. A. Filin, Phys. Rev. C 94 (2016) 014001.
  • (43) S. Petschauer and N. Kaiser, Nucl. Phys. A 916 (2013) 1.
  • (44) C. Albertus, E. Hernandez, J. Nieves and J. M. Verde-Velasco, Phys. Rev. D 72 (2005) 094022.
  • (45) K. U. Can, G. Erkol, M. Oka and T. T. Takahashi, Phys. Lett. B 768 (2017) 309.
  • (46) C. Alexandrou, K. Hadjiyiannakou and C. Kallidonis, Phys. Rev. D 94 (2016) 034502.
  • (47) S. Dürr et al. [Budapest-Marseille-Wuppertal Collaboration], Phys. Rev. D 90 (2014) 114504.
  • (48) C. Patrignani et al. (Particle Data Group), Chin. Phys. C 40 (2016) 100001.
  • (49) J.-X. Lu, L.-S. Geng and M. Pavón Valderrama, arXiv:1706.02588 [hep-ph].
  • (50) J. C. Berengut, E. Epelbaum, V. V. Flambaum, C. Hanhart, U.-G. Meißner, J. Nebreda and J. R. Pelaez, Phys. Rev. D 87 (2013) 085018.
  • (51) A. Nogga, H. Kamada and W. Glöckle, Phys. Rev. Lett. 88 (2002) 172501.
  • (52) J. Haidenbauer, U.-G. Meißner, A. Nogga and H. Polinder, Lect. Notes Phys. 724 (2007) 113.
  • (53) A. Nogga, Nucl. Phys. A 914 (2013) 140.
  • (54) B. F. Gibson, I. R. Afnan, J. A. Carlson and D. R. Lehman, Prog. Theor. Phys. Suppl. 117 (1994) 339.
  • (55) K. Miyagawa, H. Kamada, W. Glöckle and V. Stoks, Phys. Rev. C 51 (1995) 2905.
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