Comment on "Λ_c N interaction in leading order covariant chiral effective field theory"
aa r X i v : . [ nu c l - t h ] J a n Comment on “ Λ c N interaction in leading order covariant chiral effective field theory” J. Haidenbauer ∗ Institute for Advanced Simulation, Institut f¨ur Kernphysik,and J¨ulich Center for Hadron Physics, Forschungszentrum J¨ulich, D-52425 J¨ulich, Germany
G. Krein † Instituto de F´ısica Te´orica, Universidade Estadual Paulista,Rua Dr. Bento Teobaldo Ferraz, 271 - Bloco II, 01140-070 S˜ao Paulo, SP, Brazil
Song et al. [Phys. Rev. C 102, 065208 (2020)] presented results for the Λ c N interaction basedon an extrapolation of lattice simulations by the HAL QCD Collaboration at unphysical quarkmasses to the physical point via covariant chiral effective field theory. We point out that theirpredictions for the D partial wave disagree with available lattice results. We discuss the origin ofthat disagreement and present a comparison with predictions from conventional (non-relativistic)chiral effective field theory. Due to the lack of any experimental information, atpresent results from lattice QCD simulations provide theonly model-independent estimate for the strength of theinteraction of charmed baryons with nucleons. Corre-sponding lattice calculations for the Λ c N and Σ c N sys-tems have been published by the HAL QCD Collabo-ration [1–3], though for unphysical quark masses corre-sponding to pion masses of m π = 410 −
700 MeV. Evi-dently, in order to draw conclusions on the physics im-plications and to allow for predictions for future experi-ments an extrapolation of the HAL QCD results to thephysical point is required. Such an extrapolation wasperformed by us in Ref. [4], for Λ c N and in [5] for Σ c N .We used as guideline conventional (non-relativistic) chi-ral effective field theory ( χ EFT) [6] up to next-to-leadingorder (NLO). With regard to the Σ c N interaction, an-other extrapolation of the HAL QCD results was per-formed in Ref. [7] using heavy baryon chiral perturbationtheory and taking into account heavy quark spin symme-try.In this comment we critically discuss results for theΛ c N interaction of yet another and very recent extrap-olation done by Song et al. [8] which utilizes covariant χ EFT. The corresponding work is based on a potential atleading-order (LO) in the chiral expansion. Song et al. [8]found that in the case of the S partial wave the latticeQCD data at unphysical masses ( m π = 410 ,
570 MeV)can be quite well reproduced within covariant χ EFT. Atthe physical point, the predicted phase shifts are rathersimilar to those we obtained in Ref. [4].For the coupled S - D partial waves the situation isquite different. Here the S phase shifts from covariant χ EFT are only in fair agreement with the lattice QCDdata, see Fig. 4 (left side) in Ref. [8]. Specifically for m π = 570 MeV the energy dependence suggested by theHAL QCD results is not properly reproduced. Moreover,the extrapolation to the physical point yields rather dif- ∗ [email protected] † [email protected] ferent results as compared to the predictions in Ref. [4].While conventional χ EFT suggests a moderately attrac-tive S interaction, with phase shifts almost identical tothe one in the S partial wave, the covariant approachleads to a predominantly repulsive result which, in addi-tion, exhibits a strong energy dependence.As a byproduct of their study, Song et al. provided alsopredictions for the D phase shifts and the mixing angle ε – at the physical point and for the pion masses of theHAL QCD simulation – with the intention that thosecan be checked by future lattice QCD calculation. In-terestingly, such results are already available. They havebeen presented in the PhD thesis by Takaya Miyamoto [3]which can be accessed via Inspire. We show the latticeresults for D in Fig. 1(a) together with the predictionsfrom covariant χ EFT. Obviously, there is a strong mis-match. While lattice QCD suggests a weakly attractiveinteraction, the results by Song et al. are strongly repul-sive. Corresponding results of our Λ c N interaction [4],which are likewise predictions, are displayed in Fig. 1(b).Evidently, in case of conventional χ EFT there is a re-markable qualitative agreement with the lattice simula-tion. Specifically, the results for m π = 410 MeV arerather well in line with those by the HAL QCD Collab-oration. This gives us confidence that our extrapolationto the physical point is plausible and reasonable, for D as well as for S .For understanding the origin of the difference let usdiscuss briefly the main features of the two approaches.For further details, specifically how the pion-mass depen-dence is implemented, we refer the reader to Ref. [8] withregard to the extrapolation based on covariant χ EFTand to Ref. [4] for the conventional approach. In eitherframeworks the potential is given in terms of pion ex-changes and a series of contact interactions with an in-creasing number of derivatives. The latter represent theshort-range part of the baryon-baryon force and are pa-rameterized by low-energy constants (LECs), that needto be fixed by a fit to data [9]. The essential differ-ence in the corresponding potentials occurs in the contactterms and arises from the circumstance that in the co-
E (MeV) -30-20-100 δ ( deg )
410 MeV570 MeV (a)
E (MeV) -20246810 δ ( deg )
410 MeV570 MeV (b)
FIG. 1. Predictions for the Λ c N D phase shift. (a) Results from covariant χ EFT taken from Ref. [8]. (b) Results based onthe Λ c N potential from Ref. [4]. Red (black), green (dark grey), and blue (light grey) bands correspond to m π = 138, 410,and 570 MeV, respectively. The width of the bands represent cutoff variations/uncertainties. Lattice results of the HAL QCDCollaboration corresponding to m π = 410 MeV (filled circles) and 570 MeV (open circles) are taken from Ref. [3]. Note thedifferent scales of (a) and (b)! variant formulation the potential is derived with the fullDirac spinor of the baryons included [8]. As a conse-quence, while in conventional χ EFT based on the Wein-berg counting [9], contributions to the contact interac-tion of chiral power ν are proportional to p ν (with p being the modulus of the baryon-baryon center-of-massmomentum) and, thus, are uniquely related to the orderof the chiral expansion, this no longer the case for thecovariant version. Moreover, in the conventional χ EFTa partial wave expansion of the contact interaction withthe involved spin- and momentum-dependent operatorsallows one to rewrite the various contributions in termsof suitably defined LECs that then contribute only tosingle partial waves. This is not possible in the covariantpower counting, in which already the LO contact termcontributes to all J = 0 , D potential from the contact in-teractions already at LO where the corresponding LECs are those that appear likewise in the S and S partialwaves, cf. Eq. (7) in Ref. [8]. Apparently, fixing thoseLECs solely from the S -wave phase shifts of the latticesimulations leads to wrong results for the D . In theWeinberg counting a D contact interaction arises firstat next-to-next-to-next-to-leading order (N LO) [9] andit is independent of the LECs in the S -waves!In summary, conventional χ EFT (up to NLO) em-ployed in Ref. [4] seems to provide a more reliabletool for representing lattice QCD results by the HALQCD Collaboration for the Λ c N interaction at m π =410 ,
570 MeV [3] and for extrapolating them to the phys-ical point. In covariant χ EFT as utilized by Song et al.[8] (at LO) it is obviously more difficult to account forthe results at unphysical pion masses on a quantitativelevel, specifically for the spin triplet case, i.e. in situa-tions where coupled-channel effects could be important.As a consequence, extrapolations are more unstable. Inany case, lattice simulations for quark masses closer tothe physical point would be rather useful to shed furtherlight on the issue of extrapolation and, of course, directexperimental constraints [10] would be helpful too. [1] T. Miyamoto et al. , Nucl. Phys. A , 113 (2018).[2] T. Miyamoto [HAL QCD Collaboration], PoS Hadron , 146 (2018).[3] T. Miyamoto, PhD thesis (Kyoto University, 2019),doi:10.14989/doctor.k21568, https://inspirehep.net/literature/1804661 .[4] J. Haidenbauer and G. Krein, Eur. Phys. J. A , 199(2018).[5] J. Haidenbauer, A. Nogga and I. Vida˜na, Eur. Phys. J.A , 195 (2020). [6] E. Epelbaum, U.-G. Meißner, W. Gl¨ockle, Nucl. Phys. A , 535 (2003).[7] L. Meng, B. Wang and S. L. Zhu, Phys. Rev. C ,064002 (2020).[8] J. Song, Y. Xiao, Z. W. Liu, C. X. Wang, K. W. Li andL. S. Geng, Phys. Rev. C , 065208 (2020).[9] E. Epelbaum, H. W. Hammer and U.-G. Meißner, Rev.Mod. Phys. , 1773 (2009).[10] J. Haidenbauer, G. Krein and T. C. Peixoto, Eur. Phys.J. A56