Pion-mass dependence of the nucleon-nucleon interaction
PPion-mass dependence of the nucleon-nucleon interaction
Qian-Qian Bai and Chun-Xuan Wang
School of Physics, Beihang University, Beijing 102206, China
Yang Xiao
School of Physics, Beihang University, Beijing 102206, China andUniversit´e Paris-Saclay, CNRS / IN2P3, IJCLab, 91405 Orsay, France
Li-Sheng Geng ∗ School of Physics & Beijing Advanced Innovation Center for Big Data-basedPrecision Medicine, Beihang University, Beijing100191, China. andSchool of Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China (Dated: September 30, 2020)Nucleon-nucleon interactions, both bare and e ff ective, play an important role in our understanding of the non-perturbative strong interaction, as well as nuclear structure and reactions. In recent years, tremendous e ff ortshave been seen in the lattice QCD community to derive nucleon-nucleon interactions from first principles.Because of the daunting computing resources needed, most of such simulations were still performed with largerthan physical light quark masses. In the present work, employing the recently proposed covariant chiral e ff ectivefield theory (ChEFT), we study the light quark mass dependence of the nucleon-nucleon interaction extractedby the HALQCD group. It is shown that the pion-full version of the ChEFT can describe the lattice QCDdata with m π =
469 MeV and their experimental counterpart reasonably well, while the pion-less version candescribe the lattice QCD data with m π = , , , S and S - D channels. Theslightly better description of the single channel than the triplet channel indicates that higher order studies arenecessary for the latter. Our results confirmed previous studies that the nucleon-nucleon interaction becomesmore attractive for both the singlet and triplet channels as the pion mass decreases towards its physical value.It is shown that the virtual bound state in the S channel remains virtual down to the chiral limit, while thedeuteron only appears for a pion mass smaller than about 400 MeV. It seems that proper chiral extrapolationsof nucleon-nucleon interaction are possible for pion masses smaller than 500 MeV, similar to the mesonic andone-baryon sectors. PACS numbers:
I. INTRODUCTION
Chiral e ff ective field theories have played an important rolein our understanding of the non-perturbative strong interac-tion [1–4]. In particular, they have provided a modern theo-retical tool to describe nucleon-nucleon interactions, the so-called chiral nuclear forces [5, 6]. Nowadays high-precisionchiral nuclear forces [7–10] are widely used as inputs in abinitio methods to study nuclear structure and reactions [11–13]. More recently, there are increasing interests and e ff ortsin the lattice QCD community to derive nucleon-nucleon (andmore generally baryon-baryon) interactions using quark andgluon degrees of freedom [14]. In addition to providing anon-trivial check on the underlying theory QCD [15], poten-tials derived from lattice QCD simulations can also be usedto constrain chiral nuclear forces as well as provide inputs tonuclear structure studies in the unphysical world [16–18].Lattice QCD simulations have been traditionally performedwith larger than physical light quark masses, finite latticespacing and volume. As a result, multiple extrapolations areneeded to obtain physical results. The extrapolation in quarkmasses are often referred to as chiral extrapolations. It has ∗ Electronic address: [email protected] been well established in the mesonic and one-nucleon sec-tors that chiral extrapolations are reliable at least for rela-tively small pion masses (for chiral extrapolations of baryonmasses and magnetic moments, see, e.g., Refs. [19, 20]). Asfor the baryon-baryon sector, the situation is quite di ff erent.In the case of chiral extrapolations, very few explicit studieshave been performed, and not all of them try to relate simu-lations performed with di ff erent light quark / pion masses [16–18]. Though in the present work we focus on chiral extrap-olations, we note that recently finite volume e ff ects in multi-nucleon systems have been studied [21].In Refs. [22, 23], we have studied the S = − S = − a r X i v : . [ nu c l - t h ] S e p that ChEFT can also be used to test the consistency of latticeQCD simulations (see, e.g., Ref. [29] for a recent applicationfrom such a perspective).This paper is organized as follows. In Sec. II, we briefly in-troduce the covariant ChEFT approach relevant to the presentstudy. Fitting to the lattice QCD data is performed in Sec. III,followed by discussions, and we summarize in Sec. IV. II. THEORETICAL FRAMEWORK
The ChEFT we employ in the present work is described indetail in Refs. [30–33]. Here we only provide a concise intro-duction and relevant formulation needed for chiral extrapola-tions.At leading order, the covariant chiral potential in the pion-full version can be written as V LO = V CT + V OPE (1)where V CT represents contact contributions and V OPE denotesone-pion exchanges. The contact contributions are describedby four covariant four-fermion contact terms without deriva-tives [32], namely. V CT = C S ( u ( p (cid:48) p (cid:48) p (cid:48) , s (cid:48) ) u ( ppp , s ))( u ( − p (cid:48) p (cid:48) p (cid:48) , s (cid:48) ) u ( − ppp , s )) + C V ( u ( p (cid:48) p (cid:48) p (cid:48) , s (cid:48) ) γ µ u ( ppp , s ))( u ( − p (cid:48) p (cid:48) p (cid:48) , s (cid:48) ) γ µ u ( − ppp , s )) (2) + C AV ( u ( p (cid:48) p (cid:48) p (cid:48) , s (cid:48) ) γ γ µ u ( ppp , s ))( u ( − p (cid:48) p (cid:48) p (cid:48) , s (cid:48) ) γ γ µ u ( − ppp , s )) + C T ( u ( p (cid:48) p (cid:48) p (cid:48) , s (cid:48) ) σ µν u ( ppp , s ))( u ( − p (cid:48) p (cid:48) p (cid:48) , s (cid:48) ) σ µν u ( − ppp , s ))where ppp and p (cid:48) p (cid:48) p (cid:48) are initial and final three momentum, s ( s (cid:48) ), s ( s (cid:48) ) are spin projections, C S , V , AV , T are LECs, and u ( u ) areDirac spinors, u ( ppp , s ) = N P (cid:32) σ · p σ · p σ · pE P + M (cid:33) χ S N P = (cid:114) E P + M M (3)with χ S being the Pauli spinor and E P ( M ) being the nucleonenergy (mass). The one-pion-exchange potential in momen-tum space reads V OPE ( p (cid:48) , pp (cid:48) , pp (cid:48) , p ) = − ( g A / f π ) × ( u ( p (cid:48) p (cid:48) p (cid:48) , s (cid:48) ) τ τ τ γ µ γ q µ u ( ppp , s )) × ( u ( − p (cid:48) − p (cid:48) − p (cid:48) , s (cid:48) ) τ τ τ γ ν γ q ν u ( − p − p − p , s )) / (( E p (cid:48) − E p ) − ( p (cid:48) − pp (cid:48) − pp (cid:48) − p ) − m π ) (4)where m π is the pion mass, τ , are the isospin matrcies, g A = .
26, and f π = . ff erent partialwaves in the | LS J (cid:105) basis. In the present work, we are only in-terested in the S and S - D channels. The correspondingpartial wave potentials read V S = ξ N [ C S (1 + R ppp R p (cid:48) p (cid:48) p (cid:48) ) + ˆ C S ( R ppp + R p (cid:48) p (cid:48) p (cid:48) )] , (5) V S = ξ N C S (9 + R ppp R p (cid:48) p (cid:48) p (cid:48) ) + ˆ C S ( R ppp + R p (cid:48) p (cid:48) p (cid:48) )] , (6) V D = ξ N C S R ppp R p (cid:48) p (cid:48) p (cid:48) , (7) V S − D = √ ξ N C S R ppp R p (cid:48) p (cid:48) p (cid:48) + ˆ C S R ppp ] , (8) V D − S = √ ξ N C S R ppp R p (cid:48) p (cid:48) p (cid:48) + ˆ C S R p (cid:48) p (cid:48) p (cid:48) ] , (9)where ξ N = π N p N (cid:48) p , R ppp = | ppp | / ( E p + M ), R p (cid:48) p (cid:48) p (cid:48) = | p (cid:48) p (cid:48) p (cid:48) | / ( E p (cid:48) + M ).A few remarks are in order. First, compared to the leading or-der Weinberg approach, there are two LECs for the S chan-nel, and two for the S channel, instead of only one for the S channel and one for the S channel. Second, the twoLECs for the S channel are also responsible for the D channel and the mixing between S and D . Such a featureenables the rather successful description of the phase shifts ofboth S and S - D up to laboratory energies of about 300MeV [30, 33].The LO contact potentials given above do not contain ex-plicit pion-mass dependent contributions, which are necessaryif one wants to study the pion-mass dependence of latticeQCD nuclear forces. Because m π are counted as of O ( q ),inclusions of such terms require one go to at least the corre-sponding order in the momentum expansion. At such an order,however, the ChEFT has too many LECs [32] which cannot befixed by the limited lattice QCD data of Ref. [28]. As a result,we only add two pion-mass dependent terms in the S and S potentials but keep the momentum expansion at the lead-ing order, similar to, e.g., Ref. [34]. The resulting LECs thenread, explicitly, C S → C S + C π S m π , (10)ˆ C S → ˆ C S + ˆ C π S m π , (11) C S → C S + C π S m π , (12)ˆ C S → ˆ C S + ˆ C π S m π . (13)In principle, we could also take into account the fact that inthe one-pion exchange contributions the LECs, g A and f π , arepion-mass dependent as well, as done, e.g., in Ref. [35]. How-ever, we find that the limited lattice QCD data do not allowone to fully disentangle such contributions from the contactterms. As a result, in our pion-full theory, we use the physicalvalues for g A and f π . In fact, with the pion-mass dependenceof f π of Ref. [36], we have performed alternative studies ofthe physical and lattice QCD data with m π =
469 MeV. Qual-itatively similar but quantitatively slightly worse results wereobtained. In particular, we note that in the S - D coupledchannel, a better fit of the mixing angle and the D phaseshifts can be achieved if the pion mass dependence of f π (and g A ) is taken into account, but in this case the description of the S phase shifts deteriorate a bit.Once the chiral potentials are fixed, we solve the relativisticKadyshevsky scattering equation [37] to obtain the scatteringamplitudes, T ( p (cid:48) p (cid:48) p (cid:48) , ppp ) = V ( p (cid:48) p (cid:48) p (cid:48) , ppp ) + (cid:90) d p (cid:48)(cid:48) p (cid:48)(cid:48) (2 π ) V ( p (cid:48) p (cid:48) p (cid:48) , p (cid:48)(cid:48) p (cid:48)(cid:48) p (cid:48)(cid:48) ) × M N E p (cid:48)(cid:48) E p − E p (cid:48)(cid:48) + i (cid:15) T ( p (cid:48)(cid:48) p (cid:48)(cid:48) p (cid:48)(cid:48) , ppp ) . (14)To avoid ultraviolet divergence, we need to introduce a formfactor of the form f ( ppp , p (cid:48) p (cid:48) p (cid:48) ) = exp [ − ppp n − p (cid:48) p (cid:48) p (cid:48) n Λ n ] with n = Λ the corresponding cuto ff , which is often determined by fittingto data. If the EFT is properly renormalized, physical resultsshould be independent of the corresponding form factor andthe related cuto ff . At this stage, there are still heatly discus-sions about this issue. For a relevant discussion in the co-variant ChEFT, see Ref. [33]. In the present work, we simplytreat the cuto ff Λ as a parameter and let its value determinedby data. III. RESULTS AND DISCUSSIONS
In Ref. [28], nucleon-nucleon interactions are simulatedwith five di ff erent light quark masses. The corresponding pionand nucleon masses are given in Table I. The pion mass rangesfrom 469 MeV to 1171 MeV and the corresponding nucleonmass changes from 1161 MeV to 2274 MeV. Clearly for largelight quark masses, the ChEFT with explicit pion exchangesis not supposed to work, because as the pion mass increases,its range becomes shorter and it is harder to be distinguishedfrom the contributions of contact terms. Though it is di ffi cultto pin down precisely where one should replace the pion-fulltheory with the pion-less theory, we choose a pion mass ofabout 500 MeV. In other words, we will study the latticeQCD data with m π =
469 MeV by the pion-full theory andthose with larger pion masses, m π = ff the one-pion exchange contributions. TABLE I: Pion and nucleon masses (in units of MeV) of lattice QCDsimulations [28] studied in the present work. m π
469 672 837 1015 1171 M N Such a choice is also supported by explicit studies playing with both op-tions. A. S in the pion-full theory First we focus on the S channel.To study the feasibility ofproper chiral extrapolations in the nucleon-nucleon sector, wefirst fit to the experimental phase shifts and the lattice QCDones obtained with m π =
469 MeV. We realized that the ex-perimental data and the lattice QCD data alone are not enoughto fix all the four LECs and the cuto ff . A very good fit canalready be obtained with only three of them, namely, C S , C π S , and the cuto ff . The best fitting results are shown in Fig.1 and the corresponding LECs and cuto ff are given in TableII. To see the sensitivity of the results on the cuto ff , we haveallowed the cuto ff to vary by as large as 50 MeV from the op-timal value of 680 MeV. The corresponding results are shownby the blue and grey bands. Clearly, the cuto ff dependence isrelatively mild as expected. FIG. 1: S phase shifts of the Nijmegen analysis and lattice QCDsimulations with m π =
469 MeV in comparison with the covariantChEFT fits. The blue and grey bands are generated by varying thecuto ff from the optimal value of 680 MeV by ±
50 MeV.TABLE II: Values of the LECs of the best fit to the Nijmegen anal-ysis and lattice QCD simulations with m π =
469 MeV for the S channel. C S [MeV − ] C π S [MeV − ] Λ [MeV]0.046 0.119E-06 680 Clearly, the covariant ChEFT can provide a quite good de-scription of both the experimental data and lattice QCD datasimultaneously. This suggests that if more lattice QCD datawith pion masses between 469 and 139 MeV are available,one should be able to perform reliable extrapolations to thephysical point, similar to the cases of mesonic and one-baryonsectors. Such a finding is indeed very encouraging. In addi-tion, the cuto ff value of the best fit, about 680 MeV, looks veryreasonable. As we will see, in the S - D channel things area bit di ff erent. B. S - D in the pion-full theory Now we turn to the S - D channel. As we alreadybriefly mentioned in Sec. II, the covariant ChEFT can de-scribe the S - D channel already at leading order, thoughat a price, i.e., the same two LECs are responsible for the S , D , and their mixing angle. At the physical point, thesetwo LECs are able to describe the Nijmegen phase shifts rea-sonably well [30, 33]. The situation is a bit di ff erent in thepresent case. First, we notice that the best fit to the physi-cal and lattice QCD phase shifts yields a cuto ff of about 300MeV, which is a bit too small given the fact the lattice QCDpion mass is 469 MeV. Therefore, we decide to fix the cuto ff to some specified values and study to what extent one can de-scribe simultaneously the experimental and lattice QCD phaseshifts. We choose four cuto ff values, 500 MeV, 1000 MeV,1500 MeV, and 2000 MeV for such a purpose. Furthermore,for the triplet channel case, we only fit to those lattice QCDsimulations of T lab . <
40 MeV, di ff erent from the singlet chan-nel case, as it is impossible to obtain a good fit to all the latticeQCD data. This fitting strategy will also be employed for thepion-less case in the following subsection.The results are shown in Fig. 2. It seems that for largercuto ff values, the S and D phase shifts can be describedrelatively well, at least up to T lab . ≈
40 MeV . On the otherhand, the lattice QCD mixing angle (cid:15) stays very close to itsexperimental counterpart, which cannot be fully reproducedby the leading order covariant ChEFT. Though this could bean artifact due to the strong constraint that the same two LECsare responsible for the three observables, the relative insensi-tivity of (cid:15) on the light quark masses needs to be better un-derstood in the future. Another interesting observation is thatas the pion mass increases, the magnitude of both the S andthe D phase shifts decreases, to such a point that for m π =
469 MeV the deuteron becomes unbound, as already noticedin Ref [28].A few remarks are in order concerning the EFT descrip-tion of the singlet and triplet channels. At the physical point,the covariant ChEFT can describe the singlet and triplet chan-nels quite well at least up to T lab . ≈
50 MeV with a singlecuto ff of about 750 MeV [30]. While when the lattice QCDdata with m π =
469 MeV are fitted together with the physicaldata, one cannot describe both channels simultaneously. Thebest fits yield a cuto ff of 680 MeV for the singlet channel andabout 300 MeV for the triplet channel, indicating an incon-sistency between the lattice QCD data and the LO ChEFT. In addition, we note that the rather similar mixing angle (cid:15) at the physical and unphysical pion masses remains to be un-derstood from the EFT perspective. Although one definitelyshould first check these inconsistencies from the EFT perspec-tive by going to higher orders, studies of the same observables As mentioned in the introduction, similar inconsistencies between theNPLQCD determinations of the scattering length and e ff ective range of S and S at m π =
450 MeV using the e ff ective range approximationand those obtained with low energy theorems have been noted in Ref. [29]. at the same unphysical pion masses by lattice QCD groupsother than the HALQCD Collaboration are highly welcome.We note that there are still ongoing discussions on the valid-ity of the HALQCD method and the L¨uscher method in thebaryon-baryon sector, particularly for heavy pion masses, see,e.g., Refs. [38–40]. TABLE III: Values of the LECs of the various fits with di ff erentcuto ff s to the Nijmegen analysis and lattice QCD simulations with m π =
469 MeV for the S - D channel. C S [MeV − ] C π S [MeV − ] Λ [MeV]0.761E-04 -0.189E-09 500-0.815E-04 0.609E-11 1000-0.118E-02 -0.375E-08 1500-0.327E-02 -0.267E-07 2000 C. S and S scattering lengths Once the relevant LECs are determined in the pion-full the-ory, we can study the evolution of physical observables as afunction of the pion mass. In Fig. 3, we show the scatteringlengths for the S and S channels as functions of the pionmass with the LECs tabulated in Tables II and III. First, wenotice that the experimental data are reproduced quite well,though they are not explicitly fitted, reflecting that the de-scriptions of experimental phase shifts close to threshold arereasonably good. Second, for the S channel, though theinteraction becomes more attractive for m π smaller than itsphysical value, it does not become strong enough to generatea bound state. In addition, the interaction strength reaches thelargest around m π ≈
80 MeV and then decreases toward thechiral limit. On the other hand, for the S channel, as ex-pected, a bound state appears somewhere around m π = S and S scattering lengths in termsof light quark mass dependences [35, 42–46], we refrain froma detailed comparison with their results due to the di ff erentinputs used and di ff erent strategies taken to perform the lightquark mass evolution. Nonetheless, it is interesting to notethat our predicted scattering lengths in the chiral limit seem toagree with those of Ref. [46], at least qualitatively. The dependence of the nucleon mass on the pion mass is described us-ing the covariant chiral perturbation theory up to O ( p ) of Ref. [19], withthe two LECs fixed by fitting to the experimental nucleon mass and theHALQCD nucleon mass with m π =
469 MeV.
FIG. 2: S − D phase shifts of the Nijmegen analysis and lattice QCD simulations with m π =
469 MeV in comparison with the covariantChEFT fits. The solid blue / black circles represent experimental / lattice data fitted, while those empty circles are not included in the fits. a[fm] m p [ M e V ] c u t o f f = 6 8 0 p h y s i c a l c u t o f f = 5 0 0 c u t o f f = 1 0 0 0 c u t o f f = 1 5 0 0 c u t o f f = 2 0 0 0 a[fm] m p [ M e V ] p h y s i c a l FIG. 3: S (left) and S (right) scattering lengths as functions of the pion mass in comparison with the experimental data, a S = − . a S = . ff from the optimal value of 680 MeV by ±
50 MeV. D. S and S - D in the pion-less theory FIG. 4: S phase shifts of the lattice QCD simulations with m π = ff fromthe optimal value of 730 MeV by ±
50 MeV.
Next we would like to study the lattice QCD simulationsobtained with m π = , , , ff the one-pion exchangesin solving the Kadyshevsky equation. The results are shownin Figs. 4 and 5, and Tables IV and V. The pion-less versionof the covariant ChEFT can fit the four lattice QCD simula-tions rather well, which is somehow unexpected. Again, thedescription of the S channel is a bit better than that of the S - D coupled channel. We note that the optimal cuto ff for the singlet channel is also di ff erent from that for the tripletchannel, similar to the pion-full case.We have checked that it is impossible to fit all the five latticeQCD data with m π = , , , , ff erent from ours [16–18]. FIG. 5: S - D phase shifts of the lattice QCD simulations with m π = ff from the optimal value of 360 MeV by ±
50 MeV.TABLE IV: Values of the LECs of the best fit to the lattice QCDsimulations with m π = S chan-nel. C S [MeV − ] ˆ C S [MeV − ] C π S [MeV − ] ˆ C π S [MeV − ] Λ [MeV]0.264E-04 -0.114E-02 -0.987E-11 0.491E-09 730TABLE V: Values of the LECs of the best fit to the lattice QCDsimulations with m π = S - D channel. C S [MeV − ] ˆ C S [MeV − ] C π S [MeV − ] ˆ C π S [MeV − ] Λ [MeV]0.295E-04 0.219E-02 -0.534E-11 -0.771E-09 360 IV. SUMMARY AND OUTLOOK
We have studied the lattice QCD simulations of nucleon-nucleon phase shifts together with the experimental data bythe leading order covariant chiral e ff ective field theory. Sup-plemented with pion-mass dependent low-energy constants,we showed that the pion-full ChEFT can describe the exper-imental data and lattice QCD data with m π =
469 MeV rea-sonably well such that chiral extrapolations can be carried outin reasonable confidence, though the description of the S - D coupled channel can still be improved. The relative in-sensitivity of the S - D mixing angle on the pion mass remains to be understood. For lattice QCD simulations with m π = S chan-nel than that in the S - D coupled channel. The di ff erentperformance in the singlet and triplet channels can either beattributed to the fact the leading order covariant ChEFT is in-su ffi cient such that higher order studies are necessary or thatthere is something that we do not fully understand in the lat-tice QCD simulations, i.e., the weak dependence of the mixingangle on the pion mass.Nucleon-nucleon interactions, and more broadly, baryon-baryon interactions play an important role in our understand-ing of the non-perturbative strong interaction. Lattice QCDsimulations are making impressive progress and start to o ff ernew insights on many long-standing issues. To make betteruse of such valuable simulation results, their dependence onlight quark masses need to be studied in more detail. We haveseen tremendous progress in the mesonic and one-baryon sec-tors in this regard, and hope that the present work can inspiremore works in the two-baryon sector. Declaration of competing interest
The authors declare that they have no known competing fi-nancial interests or personal relationships that could have ap-peared to influence the work reported in this paper.
Acknowledgements
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