Low-energy Scattering and Effective Interactions of Two Baryons at m π ∼450 MeV from Lattice Quantum Chromodynamics
Marc Illa, Silas R. Beane, Emmanuel Chang, Zohreh Davoudi, William Detmold, David J. Murphy, Kostas Orginos, Assumpta Parreño, Martin J. Savage, Phiala E. Shanahan, Michael L. Wagman, Frank Winter
IICCUB-20-020, UMD-PP-020-7, MIT-CTP/5238, INT-PUB-20-038FERMILAB-PUB-20-498-T
Low-energy Scattering and Effective Interactions of Two Baryons at m π ∼ MeV from Lattice Quantum Chromodynamics
Marc Illa, Silas R. Beane, Emmanuel Chang, Zohreh Davoudi,
3, 4
William Detmold, David J. Murphy, Kostas Orginos,
6, 7
Assumpta Parreño, Martin J. Savage, Phiala E. Shanahan, Michael L. Wagman, and Frank Winter (NPLQCD Collaboration) Departament de Física Quàntica i Astrofísica and Institut de Ciències del Cosmos,Universitat de Barcelona, Martí Franquès 1, E08028-Spain Department of Physics, University of Washington, Seattle, WA 98195-1560, USA Department of Physics and Maryland Center for Fundamental Physics,University of Maryland, College Park, MD 20742, USA RIKEN Center for Accelerator-based Sciences, Wako 351-0198, Japan Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Department of Physics, College of William and Mary, Williamsburg, VA 23187-8795, USA Jefferson Laboratory, 12000 Jefferson Avenue, Newport News, VA 23606, USA Institute for Nuclear Theory, University of Washington, Seattle, WA 98195-1550, USA Fermi National Accelerator Laboratory, Batavia, IL 60510, USA (Dated: October 24, 2020)The interactions between two octet baryons are studied at low energies using lattice Quan-tum Chromodynamics (LQCD) with larger-than-physical quark masses corresponding to apion mass of m π ∼ MeV and a kaon mass of m K ∼ MeV. The two-baryon systemsthat are analyzed range from strangeness S = 0 to S = − and include the spin-singlet andtriplet N N , Σ N ( I = 3 / ), and ΞΞ states, the spin-singlet ΣΣ ( I = 2 ) and ΞΣ ( I = 3 / )states, and the spin-triplet Ξ N ( I = 0 ) state. The corresponding s -wave scattering phaseshifts, low-energy scattering parameters, and binding energies when applicable, are extractedusing Lüscher’s formalism. While the results are consistent with most of the systems beingbound at this pion mass, the interactions in the spin-triplet Σ N and ΞΞ channels are foundto be repulsive and do not support bound states. Using results from previous studies ofthese systems at a larger pion mass, an extrapolation of the binding energies to the physicalpoint is performed and is compared with available experimental values and phenomenologi-cal predictions. The low-energy coefficients in pionless effective field theory (EFT) relevantfor two-baryon interactions, including those responsible for SU (3) flavor-symmetry break-ing, are constrained. The SU (3) flavor symmetry is observed to hold approximately at thechosen values of the quark masses, as well as the SU (6) spin-flavor symmetry, predicted atlarge N c . A remnant of an accidental SU (16) symmetry found previously at a larger pionmass is further observed. The SU (6) -symmetric EFT constrained by these LQCD calcula-tions is used to make predictions for two-baryon systems for which the low-energy scatteringparameters could not be determined with LQCD directly in this study, and to constrain thecoefficients of all leading SU (3) flavor-symmetric interactions, demonstrating the predictivepower of two-baryon EFTs matched to LQCD. PACS numbers: 11.15.Ha, 12.38.Gc, 12.38.-t, 21.30.Fe, 13.75.Cs, 13.75.Ev.
I. INTRODUCTION
Hyperons ( Y ) are expected to appear in the interior of neutron stars [1], and unless the strong inter-actions between hyperons and nucleons ( N ) are sufficiently repulsive, the equation of state (EoS) ofdense nuclear matter will be softer than for purely non-strange matter, leading to correspondingly a r X i v : . [ h e p - l a t ] O c t lower maximum values for neutron star masses. While experimental data on scattering cross sec-tions in the majority of the Y N channels are scarce, there are reasonably precise constraints on theinteractions in the Λ N channel from scattering and hypernuclear spectroscopy experiments [2, 3],and they indicate that the interactions in this channel are attractive. Given that the Λ baryon islighter than the other hyperons, it is likely the most abundant hyperon in the interior of neutronstars. However, models of the EoS including Λ baryons and attractive Λ N interactions [4] predicta maximum neutron star mass that is below the maximum observed mass at M (cid:12) [5–9]. Severalremedies have been suggested to solve this problem, known in the literature as the “hyperon puz-zle” [11–13]. For example, if hyperons other than the Λ baryon (such as Σ baryons) are present inthe interior of neutron stars and the interactions in the corresponding Y N and
Y Y channels aresufficiently repulsive, the EoS would become more stiff [14, 15]. Another suggestion is that repulsiveinteractions in the
Y N N , Y Y N and
Y Y Y channels may render the EoS stiff enough to producea M (cid:12) neutron star [4, 14, 16–18]. Repulsive density-dependent interactions in systems involvingthe Λ and other hyperons have also been suggested, along with the possibility of a phase transi-tion to quark matter in the interior of neutron stars, see Refs. [11–13] for recent reviews. Giventhe scarcity or complete lack of experimental data on Y N and
Y Y scattering and all three-bodyinteractions involving hyperons, SU (3) flavor symmetry is used to constrain effective field theories(EFTs) and phenomenological meson-exchange models of hypernuclear interactions. In this way,quantities in channels for which experimental data exist can be related via symmetries to thosein channels which lack such phenomenological constraints [19, 20]. For example, the lowest-ordereffective interactions in several channels with strangeness S ∈ {− , − , − } were constrained usingexperimental data on pp phase shifts and the Σ + p cross section in the same SU (3) representation inthe framework of chiral EFT ( χ EFT) in Refs. [21–23]. However, only a few of the SU (3) -breakinglow-energy coefficients (LECs) of the EFT could be constrained [23]. To date, the knowledge ofthese interactions in nature remains unsatisfactory, demanding more direct theoretical approaches. Building upon our previous works [24–31], we present further studies which constrain hyper-nuclear forces in nature by direct calculations starting from the underlying theory of the stronginteractions, Quantum Chromodynamics (QCD). To this end, the numerical technique of latticeQCD (LQCD) is used to obtain information on the low-energy spectra and scattering on two-baryon systems, which can be used to constrain EFTs or phenomenological models of two-baryoninteractions. In recent years, LQCD has allowed a wealth of observables in nuclear physics, fromhadronic spectra and structure to nuclear matrix elements [32–34], to be calculated directly frominteractions of quarks and gluons, albeit with uncertainties that are yet to be fully controlled. Inthe context of constraining hypernuclear interactions, LQCD is a powerful theoretical tool becausethe lowest-lying hyperons are stable when only strong interactions are included in the computation,circumventing the limitations faced by experiments on hyperons and hypernuclei. Nonetheless,LQCD studies in the multi-baryon sector require large computing resources as there is an inherentsignal-to-noise degradation present in the correlation functions of baryons [25, 26, 35–38], amongother issues as discussed in a recent review [34]. Consequently, most studies of two-baryon systemsto date [24, 26–31, 39–51] have used larger-than-physical quark masses to expedite computations,and only recently have results at the physical values of the quark masses emerged [52–55], making itpossible to directly compare with experimental data [56]. The existing studies are primarily basedon two distinct approaches. In one approach, the low-lying spectra of two baryons in finite spatialvolumes are determined from the time dependence of Euclidean correlation functions computed withLQCD, and are then converted to scattering amplitudes at the corresponding energies through the Very recently, the gravitational wave signal GW190814, originated from the merger of a M (cid:12) black hole and a . M (cid:12) compact object, was reported [10], where the nature of the compact object is a subject of discussion. Ifthis compact object was a neutron star, it would have been the most massive one known, imposing a mass-limitconstraint very difficult to fulfill for the majority of existing nuclear EoS models. Note that SU (3) f χ EFT is less convergent than for two flavors. Other observational means to constrain these interactions, such as radius measurements of neutron stars, theirthermal and structural evolution, and the emission of gravitational waves in hot and rapidly rotating newly-born use of Lüscher’s formula [57, 58] or its generalizations [59–75]. In another approach, non-local poten-tials are constructed based on the Bethe-Salpeter wavefunctions determined from LQCD correlationfunctions, and are subsequently used in the Lippmann-Schwinger equation to solve for scatteringphase shifts [55, 76, 77]. Given that Lüscher’s formalism is model-independent below inelasticthresholds, it is this approach that is used in the present study as the basis to constrain scatteringamplitudes and their low-energy parametrizations in a number of two-(octet)baryon channels withstrangeness S ∈ { , − , − , − , − } .While LQCD studies at unphysical values of the quark masses already shed light on the under-standing of (hyper)nuclear and dense-matter physics, a full account of all systematic uncertainties,including precise extrapolations to the physical quark mass, is required to further impact phe-nomenology. Additionally, LQCD results for scattering amplitudes can be used to better constrainthe low-energy interactions within given phenomenological models and applicable EFTs. In thecase of exact SU (3) flavor symmetry and including only the lowest-lying octet baryons, there aresix two-baryon interactions at leading order (LO) in pionless EFT [78, 79] that can be constrainedby the s -wave scattering lengths in two-baryon scattering [80]. LQCD has been used in Ref. [31] toconstrain the corresponding LECs of these interactions by computing the s -wave scattering param-eters of two baryons at an SU (3) flavor-symmetric point with m π ∼ MeV. Strikingly, the firstevidence of a long-predicted SU (6) spin-flavor symmetry in nuclear and hypernuclear interactionsin the limit of a large number of colors ( N c ) [81] was observed in that study, along with an acci-dental SU (16) symmetry. This extended symmetry has been suggested in Ref. [82] to support theconjecture of entanglement suppression in nuclear and hypernuclear forces at low energies, pointingto intriguing aspects of strong interactions in nature.The objective of this paper is to extend our previous study to quark masses that are closer totheir physical values, corresponding to a pion mass of ∼ MeV and a kaon mass of ∼ MeV,and further to study these systems in a setting with broken SU (3) flavor symmetry as is the casein nature. The present study provides new constraints that allow preliminary extrapolations tophysical quark masses to be performed, and complements previous independent LQCD studies atnearby quark masses [24, 26–29, 39, 42, 45, 46, 83, 84]. In particular, predictions for the bind-ing energies of ground states in a number of Y N and
Y Y channels based on the results of thecurrent work and those of Ref. [31] at larger quark masses are consistent with experiments andphenomenological results where they exist. Our LQCD results are used to constrain the leading SU (3) symmetry-breaking coefficients in pionless EFT. This EFT matching enables the explorationof large- N c predictions, pointing to the validity of SU (6) spin-flavor symmetry at this pion mass aswell, and revealing a remnant of an accidental SU (16) symmetry that was observed at a larger pionmass in Ref. [31]. Strategies to make use of the QCD-constrained EFTs to advance the ab initio many-body studies of larger hypernuclear isotopes and dense nuclear matter are beyond the scopeof this work. Nevertheless, the methods applied in Refs. [85–87] to connect the results of LQCDcalculations to higher-mass nuclei can also be applied in the hypernuclear sector using the resultspresented in this work.This paper is organized as follows. Sec. II presents a summary of the computational details(Sec. II A), followed by the results for the lowest-lying energies of two-baryon systems from LQCDcorrelation functions, along with a description of the method used to obtain these spectra (Sec. II B),a determination of the s -wave scattering parameters in the two-baryon channels that are studied,along with the formalism used to extract the scattering amplitude (Sec. II C), and finally the bindingenergies of the bound states that are identified in various channels, including an extrapolation tothe physical point (Sec. II D). Sec. III discusses the constraints that these results impose on thelow-energy coefficients of the next-to-leading order (NLO) pionless EFT Lagrangian, including neutron stars, can be used to indirectly probe the strangeness content of dense matter, and provide complementaryconstraints on models of hypernuclear interactions [11]. some of the SU (3) flavor-symmetry breaking terms (Sec. III A). This is followed by a discussionof the predictions for the values of the coefficients that appear, in the limit of large N c , in the SU (6) spin-flavor Lagrangian at LO (Sec. III B). The main results of this work are summarized inSec. IV. In addition, several appendices are presented to supplement the conclusions of this study.Appendix A contains an analysis of our results in view of the consistency checks of Refs. [88, 89],demonstrating that the checks are unambiguously passed. Appendix B presents an exhaustivecomparison between the results obtained in this work and previous results presented in Ref. [42] forthe two-nucleon channels using the same LQCD correlation functions, as well as with the predictionsof the low-energy theorems analyzed in Ref. [90]. Appendix C includes relations among the LECsof the three-flavor EFT Lagrangian of Ref. [20] and the ones used in the present work, as well asa recipe to access the full set of leading symmetry-breaking coefficients from future studies of amore complete set of two-baryon systems. Lastly, Appendix D contains figures and tables that areomitted from the main body of the paper for clarity of presentation. II. LOWEST-LYING ENERGIES AND LOW-ENERGY SCATTERING PARAMETERSA. Details of the LQCD computation
This work continues, revisits, and expands upon the study of Ref. [42]. In particular, the sameensembles of QCD gauge-field configurations that were used in Ref. [42] to constrain the low-lyingspectra and scattering amplitudes of spin-singlet and spin-triplet two-nucleon systems at a pionmass of ∼ MeV are used here. The same configurations have also been used to study propertiesof baryons and light nuclei at this pion mass, including the rate of the radiative capture process np → dγ [91], the response of two-nucleon systems to large magnetic fields [92], the magneticmoments of octet baryons [93], the gluonic structure of light nuclei [94], and the gluon gravitationalform factors of hadrons [95–97]. For completeness, a short summary of the technical details ispresented here and a more detailed discussion can be found in Ref. [42].The LQCD calculations are performed with n f = 2 + 1 quark flavors, with the Lüscher-Weiszgauge action [98] and a clover-improved quark action [99] with one level of stout smearing ( ρ =0 . ) [100]. The lattice spacing is b = 0 . fm [101]. The strange quark mass is tuned toits physical value, while the degenerate light (up and down)-quark masses produce a pion of mass m π = 450(5) MeV and a kaon of mass m K = 596(6) MeV. Ensembles at these parameters withthree different volumes are used. Using the two smallest volumes with dimensions × and × , two different sets of correlation functions are produced, with sink interpolating operatorsthat are either point-like or smeared with a gauge-invariant Gaussian profile at the quark level.In both cases, the source interpolating operators are smeared. These two types of correlationfunctions are labeled SP and SS, respectively. For the largest ensemble with dimensions × , TABLE I. Parameters of the gauge-field ensembles used in this work. L and T are the spatial and temporaldimensions of the hypercubic lattice, β is related to the strong coupling, b is the lattice spacing, m l ( s ) is thebare light (strange) quark mass, N cfg is the number of configurations used and N src is the total number ofsources computed. For more details, see Ref. [42]. L × T β bm l bm s b [fm] L [fm] T [fm] m π L m π T N cfg N src ×
64 6 . − . − . . . . . . . × ×
96 6 . − . − . . . . . . . × ×
96 6 . − . − . . . . . . . × only SP correlation functions are produced for computational expediency. Table I summarizes theparameters of these ensembles.Correlation functions are constructed by forming baryon blocks at the sink [39]: B ijkB ( p , τ ; x ) = (cid:88) x e i p · x S ( f ) ,i (cid:48) i ( x , τ ; x ) S ( f ) ,j (cid:48) j ( x , τ ; x ) S ( f ) ,k (cid:48) k ( x , τ ; x ) w Bi (cid:48) j (cid:48) k (cid:48) , (1)where S ( f ) ,n (cid:48) n is a quark propagator with flavor f ∈ { u, d, s } and with combined spin-color indices n, n (cid:48) ∈ { , . . . , N s N c } , where N s = 4 is the number of spin components and N c = 3 is the numberof colors. The weights w Bi (cid:48) j (cid:48) k (cid:48) are tensors that antisymmetrize and collect the terms needed tohave the quantum numbers of the baryons B ∈ { N, Λ , Σ , Ξ } . The sum over the sink position x projects the baryon blocks to well-defined three-momentum p . In particular, two-baryon correlationfunctions were generated with total momentum P = p + p , where p i is the three-momentumof the i -th baryon taking the values p i = πL n with n ∈ { (0 , , , (0 , , ± } . Therefore, P = πL d , with d ∈ { (0 , , , (0 , , ± } . Additionally, two baryon correlation functions with back-to-back momenta were generated at the sink, with momenta p = − p = πL n . This latter choiceprovides interpolating operators for the two-baryon system that primarily overlap with states thatare unbound in the infinite-volume limit, providing a convenient means to identify excited statesas well. The construction of the correlation functions continues by forming a fully-antisymmetrizedquark-level wavefunction at the location of the source, with quantum numbers of the two-baryonsystem of interest. Appropriate indices from the baryon blocks at the sink are then contracted withthose at the source, in a way that is dictated by the quark-level wavefunction, see Refs. [30, 102]for more detail. The contraction codes used to produce the correlation functions in this study arethe same as those used to perform the contractions for the larger class of interpolating operatorsused in our previous studies of the SU (3) flavor-symmetric spectra of nuclei and hypernuclei up to A = 5 [30], and two-baryon scattering [31, 41, 42]. .In this study, correlation functions for nine different two-baryon systems have been computed,ranging from strangeness S = 0 to − . Using the notation ( s +1 L J , I ) , where s is the total spin, L is the orbital momentum, J is the total angular momentum, and I is the isospin, the systems are: S = 0 : N N ( S , I = 1) , N N ( S , I = 0) ,S = − N ( S , I = ) , Σ N ( S , I = ) ,S = − S , I = 2) , Ξ N ( S , I = 0) ,S = − S , I = ) ,S = − S , I = 1) , ΞΞ ( S , I = 0) . Under strong interactions, these channels do not mix with other two-baryon channels or otherhadronic states below three-particle inelastic thresholds. In the limit of exact SU (3) flavor symme-try, the states belong to irreducible representations (irreps) of SU (3) : (all the singlet states), (triplet N N ), (triplet Σ N and ΞΞ ), and a (triplet Ξ N ). In the rest of this work, the isospinlabel will be dropped for simplicity. B. Low-lying finite-volume spectra of two baryons
The two-point correlation functions constructed in the previous section have spectral represen-tations in Euclidean spacetime. Explicitly, the correlation function C ˆ O , ˆ O (cid:48) ( τ ; d ) formed using the For the rest of the paper, d = (0 , , ± will be denoted as d = (0 , , for brevity. The same code was generalized to enable studies of np → dγ [91], proton-proton fusion [103], and other electroweakprocesses, as reviewed in Ref. [34]. source (sink) interpolating operators ˆ O ( ˆ O (cid:48) ) can be written as: C ˆ O , ˆ O (cid:48) ( τ ; d ) = (cid:88) x e πi d · x /L (cid:104) ˆ O (cid:48) ( x , τ ) ˆ O † ( , (cid:105) = (cid:88) i Z (cid:48) i Z ∗ i e − E ( i ) τ , (2)where all quantities are expressed in lattice units. E ( i ) is the energy of the i th eigenstate | E ( i ) (cid:105) , Z i ( Z (cid:48) i ) is an overlap factor defined as Z i = √ V (cid:104) | ˆ O ( , | E ( i ) (cid:105) ( Z (cid:48) i = √ V (cid:104) | ˆ O (cid:48) ( , | E ( i ) (cid:105) ), and V = L . The boost-vector dependence of the energies, states, and overlap factors is implicit. Thelowest-lying energies of the one- and two-baryon systems required for the subsequent analyses can beextracted by fitting the correlation functions to this form. To reliably discern the first few exponentsgiven the discrete τ values and the finite statistical precision of the computations is a challengingtask. In particular, a well-known problem in the study of baryons with LQCD is the exponentialdegradation of the signal-to-noise ratio in the correlation function as the source-sink separation timeincreases—an issue that worsens as the masses of the light quarks approach their physical values.First highlighted by Parisi [35] and Lepage [36], and studied in detail for light nuclei in Refs. [25,26], it was later shown that this problem is related to the behavior of the complex phase of thecorrelation functions [38, 104]. Another problem that complicates the study of multi-baryon systemsis the small excitation gaps in the finite-volume spectrum that lead to significant excited-statecontributions to correlation functions. To overcome these issues, sophisticated methods have beendeveloped to analyze the correlation functions, such as Matrix Prony [37] and the generalized pencil-of-function [105] techniques, as well as signal-to-noise optimization techniques [106]. Ultimately, alarge set of single- and multi-baryon interpolating operators with the desired quantum numbers mustbe constructed to provide a reliable variational basis to isolate the lowest-lying energy eigenvaluesvia solving a generalized eigenvalue problem [107], as is done in the mesonic sector [75]. Suchan approach is not yet widely applied to the study of two-baryon correlation functions, given itscomputational-resource requirement, but progress is being made. In Ref. [50], a partial set oftwo-baryon scattering interpolating operators were used to study the two-nucleon and H -dibaryonchannels with results that generally disagreed with previous works [27, 30, 49]. Investigationscontinue to understand and resolve the observed discrepancies [31, 33, 34, 88, 108–110]. For thepresent study, in which only up to two types of interpolating operators were computed, no variationalanalysis could be performed. Instead, we have developed a robust automated fitting methodologyto sample and combine fit range and model selection choices for uncertainty quantification.Given that the correlation functions are only evaluated at a finite number of times and withfinite precision, to fit Eq. (2) the spectral representation is truncated to a relatively small numberof exponentials and fitted in a time range { τ min , τ max } , where τ max is set by a threshold valuedetermined by examining the signal-to-noise ratio, and τ min is chosen to take values in the interval [2 , τ max − τ plateau ] . Here, τ plateau = 5 is chosen to be the minimum length of the fitting window(numbers are expressed in units of the lattice spacing). A scan over all possible fitting windowsis treated as a means to quantify the associated systematic uncertainty. With a fixed window, acorrelated χ -function is minimized to obtain the fit parameters Z (cid:48) i Z ∗ i and E i for i ∈ { , , . . . , e } ,where e +1 is the number of exponentials in a given fit form. Variable projection techniques [111, 112]are used to obtain the value of the overlap factors for a given energy, since they appear linearly in C ˆ O , ˆ O (cid:48) ( τ ; d ) . Furthermore, given the finite statistical sampling of correlation functions, shrinkagetechniques [113] are used to better estimate the covariance matrix. The number of excited statesincluded in the fit is decided via the Akaike information criterion [114]. The confidence intervals ofthe parameters are estimated via the bootstrap resampling method. For a fit to be included in theset of accepted fits (later used to extract the fit parameters and assess the resulting uncertainties),several checks must be passed, including χ /N dof being smaller than 2, and different optimizationalgorithms leading to consistent results for the parameters (within a tolerance)—see Ref. [115] forfurther details on this and other checks. The accepted fits are then combined to give the final resultfor the mean value of the energy, E = (cid:88) f ω f E f , (3)with weights ω f that are chosen to be the following combination of the p -value, p f , and the uncer-tainty of each fit, δE f : ω f = p f /δE f (cid:80) f (cid:48) p f (cid:48) /δE f (cid:48) , (4)see Ref. [116] for a Bayesian framework. Here, the indices f, f (cid:48) run over all the accepted fits. Thestatistical uncertainty is defined as that of the fit with the highest weight, while the systematicuncertainty is defined as the average difference between the weighted mean value and each of theaccepted fits: δE stat = δE f : max [ { w f } ] , δE sys = (cid:115)(cid:88) f w f (cid:0) E f − E (cid:1) . (5)It should be noted that instead of fitting to the correlation function, the effective energy functioncan be employed, derived from the logarithm of the ratio of correlation functions at displaced times, C ˆ O , ˆ O (cid:48) ( τ ; d , τ J ) = 1 τ J log (cid:34) C ˆ O , ˆ O (cid:48) ( τ ; d ) C ˆ O , ˆ O (cid:48) ( τ + τ J ; d ) (cid:35) τ →∞ −−−→ E (0) , (6)where τ J is a non-zero integer that is introduced to improve the extraction of E (0) (for a detailedstudy, see Ref. [37]). Consistent results are obtained when either correlation functions or theeffective energy functions are used as input.In order to identify the shift in the finite-volume energies of two baryons compared with non-interacting baryons, the following ratio of two-baryon and single-baryon correlation functions canbe formed R B B ( τ ; d ) = C ˆ O B B , ˆ O (cid:48) B B ( τ ; d ) C ˆ O B , ˆ O (cid:48) B ( τ ; d ) C ˆ O B , ˆ O (cid:48) B ( τ ; d ) , (7)with an associated effective energy-shift function, R B B ( τ ; d , τ J ) = 1 τ J log (cid:20) R B B ( τ ; d ) R B B ( τ + τ J ; d ) (cid:21) . (8)Constant fits to ratios of correlation functions can be used to obtain energy shifts ∆ E (0) = E (0) − m − m (where m and m are the masses of baryon B and B , respectively), requiring time rangessuch that both the two-baryon and the single-baryon correlation functions are described by a single-state fit. However, if both correlation functions are not in their ground states, cancellations mayoccur between excited states (including the finite-volume states that would correspond to elasticscattering states in the infinite volume), either in correlation function or in ratios of correlationfunctions, producing a “mirage plateau” [88]. Despite this issue, as demonstrated in Ref. [108], ourprevious results, such as those in Refs. [30, 31] are argued to be free of this potential issue (similardiscussions can be found in Ref. [110]). To determine ∆ E (0) in this work, the two-baryon and single-baryon correlation functions are fit to multi-exponential forms (which account for excited states)within the same fitting range, and afterwards the energy shifts are computed at the bootstrap TABLE II. The values of the masses of the octet baryons. The first uncertainty is statistical, while thesecond is systematic. Quantities are expressed in lattice units (l.u.).Ensemble M N [l.u.] M Λ [l.u.] M Σ [l.u.] M Ξ [l.u.] ×
64 0 . . . . ×
96 0 . . . . ×
96 0 . . . . ∞ . . . . level, in such a way that the correlations between the different correlation functions are taken intoaccount. The use of correlated differences of multi-state fit results is convenient in particular forautomated fit range sampling, since the number of excited states can be varied independently forone- and two-baryon correlation functions, unlike fits to the ratio in Eq. (7). Consistent resultswere obtained via fitting the ratio in Eq. (7) in the allowed time regions.The effective mass plots (EMPs) for the single-baryon correlation functions, and for each of theensembles studied in the present work, are displayed in Fig. 22 of Appendix D. The bands shown inthe figures indicate the baryon mass which results from the fitting strategy explained above, withthe statistical and systematic uncertainties included, and the corresponding numerical values listedin Table II. The table also shows the baryon masses extrapolated to infinite volume, obtained byfitting the masses in the three different volumes to the following form: M ( V ) B ( m π L ) = M ( ∞ ) B + c B e − m π L m π L , (9)where M ( ∞ ) B and c B are the two fit parameters. This form incorporates leading-order (LO) volumecorrections to the baryon masses in heavy-baryon chiral perturbation theory (HB χ PT) [117]. Asevident from the m π L values listed in Table I, the volumes used are large enough to ensure smallvolume dependence in the single-baryon masses. This is supported by the observation that thevalue M ( ∞ ) B obtained for each baryon is compatible with all the finite-volume results M ( V ) B . Inphysical units, M N ∼ MeV, M Λ ∼ MeV, M Σ ∼ MeV and M Ξ ∼ MeV. Whilethe Λ baryon is not relevant to subsequent analysis of the two-baryon systems studied in this work,the centroid of the four octet-baryon masses is used to define appropriate units for the EFT LECs,hence M Λ is reported for completeness.The results for the two-baryon energy shifts are shown in Fig. 1. For display purposes, theeffective energy-shift functions, defined in Eq. (8), are shown in Figs. 23-31 of Appendix D, alongwith the corresponding two-baryon effective-energy functions, defined in Eq. (6). The associatednumerical values are listed in Tables XVII-XXV of the same appendix. In each subfigure of Figs. 23-31, two correlation functions are displayed: the one yielding the lowest energy (labeled as n = 1 inTables XVII-XXV) corresponds to having both baryons at rest or, if boosted, with the same valueof the momentum, and the one yielding a higher energy (labeled as n = 2 in the tables) correspondsto the two baryons having different momenta, e.g., having back-to-back momenta or one baryon atrest and the other with non-zero momentum. While the first case ( n = 1 ) couples primarily to the For the smallest volume e − m π L /m π L ∼ − and c B are of O (1) but consistent with zero within uncertainties,e.g., for the nucleon, c B = 3(4)(7) l.u. The channels within the figures/tables are sorted according to the SU (3) irrep they belong to in the limit of exactflavor symmetry, ordered as , , , and a , and within each irrep according to their strangeness, from thelargest to the smallest. ground state, the latter ( n = 2 ) is found to have small overlap onto the ground state, and givesaccess to the first excited state directly.As a final remark, it should be noted that the single-baryon masses and the energies extractedfor the two-nucleon states within the present analysis are consistent within σ with the results ofRef. [42], obtained with the same set of data but using different fitting strategies. Despite thisoverall consistency, the uncertainties of the two-nucleon energies in the present work are generallylarger compared with those reported in Ref. [42] for the channels where results are available inthat work. The reason lies in a slightly more conservative systematic uncertainty analysis employedhere. The comparison between the results of this work and that of Ref. [42] is discussed extensivelyin Appendix B. C. Low-energy scattering phase shifts and effective-range parameters
Below three-particle inelastic thresholds, Lüscher’s quantization condition [57, 58] provides ameans to extract the infinite-volume two-baryon scattering amplitudes from the energy eigenvaluesof two-baryon systems obtained from LQCD calculations, e.g., those presented in Sec. II B. Thiscondition holds if the range of interactions is smaller than (half of) the spatial extent of the cubicvolume, L , and the corrections to this condition scale as e − m π L for the two-baryon systems. Suchcorrections are expected to be small in the present work given the m π L values in Table I. Thequantization conditions are those used in Refs. [31, 42]: in the case of spin-singlet sates, only the s -wave limit of the full quantization condition is considered. For coupled S − D states, in whichthe Blatt-Biedenharn parametrization [118] of the scattering matrix can be used, only the α -waveapproximation of the quantization condition is considered [74]. In both cases, and denoting the( s -wave or α -wave) phase shift by δ , the condition can be written as [63]: k ∗ cot δ = 4 πc d ( k ∗ ; L ) , (10)where k ∗ is the center-of-mass (c.m.) relative momentum of each baryon, d is the total c.m. mo-mentum in units of π/L , and c d lm is a kinematic function related to Lüscher’s Z -function, Z d lm : c d lm ( k ∗ ; L ) = √ πγL (cid:18) πL (cid:19) l − Z d lm [1; ( k ∗ L/ π ) ] , (11)with γ = E/E ∗ being the relativistic gamma factor. Here, E and E ∗ are the energies of the systemin the laboratory and c.m. frames, respectively. The three-dimensional zeta-function is defined as Z d lm [ s ; x ] = (cid:88) n | r | l Y lm ( r )( r − x ) s , (12)where r = ˆ γ − ( n − α d ) and α = (cid:2) m − m ) /E ∗ (cid:3) , with m and m being the masses ofthe two baryons. The factor ˆ γ − acting on a vector u modifies the parallel component with respectto d , while leaving the perpendicular component invariant, i.e., ˆ γ − u = γ − u (cid:107) + u ⊥ . Convenientexpressions have been derived to exponentially accelerate the numerical evaluation of the functionin Eq. (12) [64, 119–121], and the following expression is used in the present analysis: Z d [1; x ] = − γπe x + e x √ π (cid:88) n e −| r | | r | − x + γ π (cid:90) dt e tx t / (cid:88) m (cid:54) = cos(2 πα m · d ) e − π | ˆ γ m | t + 2 tx . (13) Here α should not be confused with α -wave mentioned above.
24 32 480 . . . . . ∆ E [l . u . ] NN ( S )
24 32 48 Σ N ( S )
24 32 48
ΣΣ ( S )
24 32 480 . . . . . ∆ E [l . u . ] ΞΣ ( S )
24 32 48
ΞΞ ( S )
24 32 48 NN ( S )
24 32 48 L [l . u . ] . . . . . ∆ E [l . u . ] Σ N ( S )
24 32 48 L [l . u . ] ΞΞ ( S )
24 32 48 L [l . u . ] Ξ N ( S ) ×
64 : d = (0 , , d = (0 , ,
2) 32 ×
96 : d = (0 , , d = (0 , ,
2) 48 ×
96 : d = (0 , , d = (0 , , FIG. 1. Summary of the energy shifts extracted from LQCD correlation functions for all two-baryon systemsstudied in this work, together with the non-interacting energy shifts defined as ∆ E = (cid:112) m + | p | + (cid:112) m + | p | − m − m , where | p | = | p | = 0 corresponds to systems that are at rest (continuous line), | p | = | p | = ( πL ) corresponds to systems which are either boosted or are unboosted but have back-to-back momenta (dashed line), and | p | = 0 and | p | = ( πL ) corresponds to boosted systems where only onebaryon has non-zero momentum (dashed-dotted line). The points with no boost have been shifted slightlyto the left, and the ones with boosts have been shifted to the right for clarity. Quantities are expressed inlattice units. The values of k ∗ cot δ at given k ∗ values are shown for all two-baryon systems in Fig. 2, andthe associated numerical values are listed in Tables XVII-XXV of Appendix D. The validity ofLüscher’s quantization condition must be verified in each channel, in particular in those that haveexhibited anomalously large ranges, such as Σ N ( S ) , in previous calculations. The consistencybetween solutions to Lüscher’s condition and the finite-volume Hamiltonian eigenvalue equationusing a LO EFT potential was established in Ref. [29] for the Σ N ( S ) channel and at values of thequark masses ( m π ∼ MeV) close to those of the current analysis. The conclusion of Ref. [29],therefore, justifies the use of Lüscher’s quantization condition in the current work for this channel.1 − .
02 0 .
00 0 .
02 0 .
04 0 .
06 0 . − . . . . . . . k ∗ c o t δ [l . u . ] NN ( S ) .
00 0 .
02 0 .
04 0 .
06 0 . − − − Σ N ( S ) .
00 0 .
02 0 .
04 0 . ΣΣ ( S ) .
00 0 .
02 0 .
04 0 . . . . . . . . k ∗ c o t δ [l . u . ] ΞΣ ( S ) .
00 0 .
02 0 .
04 0 . − . . . . . . . . ΞΞ ( S ) .
00 0 .
02 0 .
04 0 . NN ( S ) .
00 0 .
02 0 .
04 0 .
06 0 .
08 0 . k ∗ [l . u . ] − . − . − . − . . . . k ∗ c o t δ [l . u . ] Σ N ( S ) .
00 0 .
02 0 .
04 0 .
06 0 . k ∗ [l . u . ] − . − . − . − . − . − . − . . ΞΞ ( S ) − .
02 0 .
00 0 .
02 0 . k ∗ [l . u . ] − . . . . Ξ N ( S ) ×
64 :32 ×
96 :48 ×
96 : d = (0 , , d = (0 , , d = (0 , , d = (0 , , d = (0 , , d = (0 , , √ πL Z (0 , , : L = 24 L = 32 L = 48 √ πγL Z (0 , , : L = 24 L = 32 L = 48 FIG. 2. k ∗ cot δ values as a function of the squared c.m. momentum k ∗ for all two-baryon systems studiedin this work. The darker uncertainty bands are statistical, while the lighter bands show the statistical andsystematic uncertainties combined in quadrature. The kinematic functions c d lm ( k ∗ ; L ) , given by Eq. (11),are also shown as continuous and dashed lines. Quantities are expressed in lattice units. The energy dependence of k ∗ cot δ can be parametrized by an effective range expansion (ERE)below the t -channel cut [122–124], k ∗ cot δ = − a + 12 rk ∗ + P k ∗ + O ( k ∗ ) , (14)where a is the scattering length, r is the effective range, and P is the leading shape parameter.These parameters can be constrained by fitting k ∗ cot δ values obtained from the use of Lüscher’s Since the pion is the lightest hadron that can be exchanged between any of the two baryons considered in thepresent study, k ∗ t -cut = m π / . k ∗ . To this end, one could use a one-dimensional choice ofthe χ function, minimizing the vertical distance between the fitted point and the function, χ ( a − , r, P ) = (cid:88) i [( k ∗ cot δ ) i − f ( a − , r, P, k ∗ i )] σ i , (15)where f ( a − , r, P, k ∗ ) corresponds to the ERE parametrization given by the right-hand sideof Eq. (14), and the sum runs over all extracted pairs of { k ∗ i , ( k ∗ cot δ ) i } , where the compoundindex i counts data points for different boosts, n values of the level, and different volumes. Eachcontribution is weighted by an effective variance that results from the combination of the uncertaintyin both k ∗ i and ( k ∗ cot δ ) i , σ i = [ δ ( k ∗ cot δ ) i ] + [ δk ∗ i ] , with δx being the mid-68% confidenceinterval of the quantity x . The uncertainty on the { k ∗ i , ( k ∗ cot δ ) i } pair can be understood byrecalling that each pair is a member of a bootstrap ensemble with the distribution obtained inthe previous step of the analysis. To generate the distribution of the scattering parameters, pairsof { k ∗ i , ( k ∗ cot δ ) i } are randomly selected from each bootstrap ensemble and are used in Eq. (15)to obtain a new set of { a − , r, P } parameters. This procedure is repeated N times, where N ischosen to be equal to the number of bootstrap ensembles for { k ∗ i , ( k ∗ cot δ ) i } . This produces anensemble of N values of fit parameters { a − , r, P } , from which the central value and the associateduncertainty in the parameters can be determined (median and mid- intervals are used for thispurpose).Alternatively, one can use a two-dimensional choice of the χ function. Knowing that k ∗ cot δ values must lie along the Z -function, as can be seen from Eq. (10) and Fig. 2, one could take thedistance between the data point and the point where the ERE crosses the Z -function along thisfunction (arc length) in the definition of χ . Explicitly, χ ( a − , r, P ) = (cid:88) i D Z [ { k ∗ i , ( k ∗ cot δ ) i } , { K ∗ i , f ( a − , r, P, K ∗ i ) } ] σ i , (16)where σ i is now defined as σ i = [ δ ( k ∗ cot δ ) i ] + (cid:32) ∂ ( k ∗ cot δ ) i ∂k ∗ (cid:12)(cid:12)(cid:12)(cid:12) k ∗ = k ∗ i (cid:33) [ δk ∗ i ] , (17)and D Z [ { x , y } , { x , y } ] denotes the distance between the two points { x , y } and { x , y } alongthe Z -function. The quantity K ∗ is the point where the ERE ( f in Eq. (16)) crosses the Z -function.To obtain this point, and given the large number of discontinuities present in the Z -function,Householder’s third order method can be used as a reliable root-finding algorithm [125]: πc d ( K ∗ , L ) − f ( a − , r, P, K ∗ ) ≡ F ( K ∗ ) = 0 : K ∗ m +1 = K ∗ m + 3 (1 /F ) (cid:48)(cid:48) (1 /F ) (cid:48)(cid:48)(cid:48) (cid:12)(cid:12)(cid:12)(cid:12) K ∗ m , (18)where the starting point is set to be K ∗ = k ∗ i , and the number of primes over /F indicates theorder of the derivative computed at the point K ∗ m . The stopping criterion is defined as | K ∗ m +1 − K ∗ m | < − , which occurs for m ∼ O (10) . Since the extraction of this point requires knowledge ofscattering parameters, the minimization must be implemented iteratively. This second choice of χ function has been used in the main analysis of this work, however, the use of the one-dimensional The inverse scattering length can be constrained far more precisely compared with the scattering length itself giventhat a − samples can cross zero in the channels considered. As a result, in the following all dependencies on a enter via a − . We thank Sinya Aoki for suggesting that we further explore this choice of χ . − .
01 0 .
00 0 . k ∗ [l . u . ] − . − . . . . k ∗ c o t δ [l . u . ] NN ( S ) − .
01 0 .
00 0 . k ∗ [l . u . ] − . − . . . . Σ N ( S ) − .
01 0 .
00 0 . k ∗ [l . u . ] − . − . . . . k ∗ c o t δ [l . u . ] ΣΣ ( S ) − .
01 0 .
00 0 . k ∗ [l . u . ] − . − . . . . ΞΣ ( S ) − .
01 0 .
00 0 . k ∗ [l . u . ] − . − . . . . ΞΞ ( S ) − .
01 0 .
00 0 . k ∗ [l . u . ] − . − . . . . k ∗ c o t δ [l . u . ] NN ( S ) − .
01 0 .
00 0 . k ∗ [l . u . ] − . − . . . . Ξ N ( S ) ×
64 :32 ×
96 :48 ×
96 : d = (0 , , d = (0 , , d = (0 , , d = (0 , , d = (0 , , d = (0 , ,
2) Two-parameter ERE: stat. / stat.+sys.Three-parameter ERE: stat. / stat.+sys. −√− k ∗ t -channel cut FIG. 3. k ∗ cot δ values as a function of the c.m. momenta k ∗ , along with the band representing the two-(yellow) and three-parameter (red) ERE for the two-baryon channels shown. The bands denote the 68%confidence regions of the fits. Quantities are expressed in lattice units. χ function is shown to yield statistically consistent results (within σ ) for scattering parameters,as demonstrated in Appendix B.For a precise extraction of the ERE parameters, a sufficient number of points below the t -channelcut must be available, for positive or negative k ∗ . In general, for the channels studied throughoutthis work, there are only a few points in the positive k ∗ region below the t -channel cut (startingat k ∗ t -cut ∼ . l.u.). For a non-interacting system, states above scattering threshold have c.m.energies (cid:112) m + k ∗ + (cid:112) m + k ∗ , with the c.m. momenta roughly scaling with the volume as k ∗ ∼ (2 π | n | /L ) . With the minimum value of | n | used in this work ( | n | = 1 ), only the statesfrom the ensemble with L = 48 are expected to lie below the t -channel cut ( π / < k ∗ t -cut ). Thisbehavior is consistent with the data. Comparing with the results of the analysis at m π ∼ MeVin Ref. [31], where lattice configurations of comparable size (in lattice units) were used, the largervalue of the pion mass resulted in the position of the t -channel cut being moved further away fromzero, and the majority of the lowest-lying states extracted in that study remained inside the region4 .
00 0 .
05 0 .
10 0 . a − [l . u . ] − − − r [l . u . ] NN ( S ) .
00 0 .
05 0 .
10 0 . a − [l . u . ] − − − Σ N ( S ) .
00 0 .
05 0 .
10 0 . a − [l . u . ] − − − r [l . u . ] ΣΣ ( S ) .
00 0 .
05 0 .
10 0 . a − [l . u . ] − − − ΞΣ ( S ) .
00 0 .
05 0 .
10 0 . a − [l . u . ] − − − ΞΞ ( S ) .
00 0 .
05 0 .
10 0 . a − [l . u . ] − − − r [l . u . ] NN ( S ) .
00 0 .
05 0 .
10 0 . a − [l . u . ] − − − Ξ N ( S ) Prohibited regions N ERE = 2: L = 24 L = 32 L = 48 L = ∞ Two-parameter ERE 68% C.R.Two-parameter ERE 95% C.R.
FIG. 4. The two-dimensional 68% and 95% confidence regions (C.R.) corresponding to the combined statisti-cal and systematic uncertainty on the scattering parameters a − and r for all two-baryon systems that exhibitbound states, obtained from two-parameter ERE fits. The prohibited regions where the two-parameter EREdoes not cross the Z -function for given volumes (as well as the infinite-volume case) are denoted by hashedareas. Quantities are expressed in lattice units. where the ERE parametrization could be used. Therefore, with only ground-state energies availablefor the analysis of the ERE in the ensembles with L ∈ { , } , the precision in the extraction ofscattering parameters is noticeably reduced compared with the study at m π ∼ MeV in Ref. [31].Inclusion of the shape parameter, P , does not improve the fits, and although the scattering lengthsremain consistent with those obtained with a two-parameter fit, the effective ranges are larger inmagnitude, and the uncertainties in the scattering parameters are increased. Moreover, the centralvalues of the extracted shape parameters are rather large, bringing into question the assumptionthat the contribution of each order in the ERE should be smaller than the previous order. However,uncertainties on the shape parameters are sufficiently large that no conclusive statement can bemade regarding the convergence of EREs. In one case, i.e., the Σ N ( S ) channel, the three-parameter ERE fit is not performed given the large uncertainties. For these reasons, while the5 TABLE III. The values of the inverse scattering length a − , effective range r , and shape parameter P determined from the two- and three-parameter ERE fits to k ∗ cot δ versus k ∗ for various two-baryon channels(see Fig. 3). Quantities are expressed in lattice units.Two-parameter ERE fit Three-parameter ERE fit a − [l.u.] r [l.u.] a − [l.u.] r [l.u.] P [l.u.] N N ( S ) 0 . (+20)(+44)( − − − . (+8 . . − . − . . (+33)(+43)( − − . (+6 . . − . − . (+46)(+510)( − − Σ N ( S ) 0 . (+25)(+14)( − − − . (+6 . . − . − . - - - ΣΣ ( S ) 0 . (+15)(+06)( − − . (+2 . . − . − . . (+17)(+41)( − − . (+3 . . − . − . (+200)(+490)( − − ΞΣ ( S ) 0 . (+16)(+06)( − − . (+3 . . − . − . . (+28)(+21)( − − . (+2 . . − . − . (+310)(+210)( − − ΞΞ ( S ) 0 . (+07)(+07)( − − . (+0 . . − . − . . (+16)(+19)( − − . (+0 . . − . − . (+190)(+200)( − − N N ( S ) 0 . (+18)(+10)( − − . (+5 . . − . − . . (+42)(+18)( − − . (+5 . . − . − . (+570)(+300)( − − Ξ N ( S ) 0 . (+07)(+11)( − − . (+1 . . − . − . . (+14)(+23)( − − . (+3 . . − . − . (+220)(+310)( − − NN ( S ) ⌃ N ( S ) ⌃⌃ ( S ) ⌅ ⌃ ( S ) ⌅⌅ ( S ) NN ( S ) ⌅ N ( S ) . . . . a [l . u . ] NN ( S ) ⌃ N ( S ) ⌃⌃ ( S ) ⌅ ⌃ ( S ) ⌅⌅ ( S ) NN ( S ) ⌅ N ( S ) r [l . u . ] NN ( S ) ⌃ N ( S ) ⌃⌃ ( S ) ⌅ ⌃ ( S ) ⌅⌅ ( S ) NN ( S ) ⌅ N ( S ) . . . . . . r / a FIG. 5. Summary of the inverse scattering length a − (left panel), effective range r (middle panel), and ratio r/a (right panel) determined from the two-parameter ERE fit for the two-baryon systems analyzed. Thebackground color groups the channels by the SU (3) irreps they would belong to if SU (3) flavor symmetrywere exact (orange for , green for , and blue for a ). Quantities are expressed in lattice units. scattering parameters are reported for both the two- and three-parameter fits in this section, onlythose of the two-parameter fits will be used in the EFT study in the next section.Fits to k ∗ cot δ as a function of k ∗ in various two-baryon channels are shown in Fig. 3, along withthe correlation between inverse scattering length and effective range in each channel depicted inFig. 4 using the 68% and 95% confidence regions of the parameters. The areas in the parameter spacethat are prohibited by the constraints imposed by Eq. (10) are also shown in Fig. 4, highlighting thefact that the two-parameter ERE must cross the Z -functions for each volume in the negative- k ∗ region. For fits including higher-order parameters, these constraints are more complicated and arenot shown. For the Σ N ( S ) and ΞΞ ( S ) channels, the ground-state energy is positively shifted,i.e., ∆ E (cid:38) , and only the values of k ∗ associated with the ground states are inside the range ofvalidity of the ERE. As a result, no extraction of the ERE parameters is possible in these channelsgiven the number of data points. Results for the scattering parameters obtained using two- andthree-parameter ERE fits in the other seven channels are summarized in Table III, and are shownin Fig. 5 for better comparison in the case of two-parameter fits.The inverse scattering lengths extracted for all systems are compatible with each other (albeit6 TABLE IV. The values of the ratio of the effective range and scattering length, r/a , determined from thetwo-parameter ERE fit to k ∗ cot δ values in each channel. r/aN N ( S ) Σ N ( S ) ΣΣ ( S ) ΞΣ ( S ) ΞΞ ( S ) N N ( S ) Ξ N ( S ) − . (+0 . . − . − . − . (+43)(+27)( − − . (+06)(+04)( − − . (+13)(+05)( − − . (+02)(+05)( − − . (+22)(+12)( − − . (+10)(+07)( − − within rather large uncertainties), providing a possible signature that SU (3) flavor symmetry isapproximately respected in these two-baryon systems at this pion mass and at low energies, seeSec. III. The effective range in most systems is compatible with zero. Furthermore, the ratio r/a can be used as an indicator of the naturalness of the interactions; for natural interactions, r/a ∼ ,while for unnatural interactions r/a (cid:28) . At the physical point, both N N channels are unnaturaland exhibit large scattering lengths, with r/a being close to . for the spin-singlet channel, and . for the spin-triplet channel. From Table IV, the most constrained ratios are obtained for the ΣΣ ( S ) , ΞΞ ( S ) , and Ξ N ( S ) channels, for which r/a ∼ . − . , indicating unnaturalinteractions at low energies. For other channels, the larger uncertainty in this ratio precludesdrawing conclusions about naturalness. Alternatively, naturalness can be assessed by consideringthe ratio of the binding momentum to the pion mass, as this quantity is better constrained in thisstudy, see Table VI in the next subsection. The values for κ ( ∞ ) /m π in each of the bound two-baryonchannels are between . and . , indicating that the range of interactions mediated by the pionexchange is not the only characteristic scale in the system, suggesting unnaturalness. However,at larger-than-physical quark masses, pion exchange may not be the only significant contributionto the long-range component of the nuclear force, as is discussed in Ref. [41]. For these reasons,both natural and unnatural interactions are considered in the next section when adopting a power-counting scheme in constraining the LECs of the EFT. TABLE V. The values of the parameters ˜ a − , ˜ r, ˜ P from a two- or three-parameter polynomial fit for two-baryon channels that exhibit smooth and monotonic behavior in k ∗ cot δ as a function of k ∗ beyond the t -channel cut. Quantities are expressed in lattice units.Two-parameter polynomial fit Three-parameter polynomial fit ˜ a − [l.u.] ˜ r [l.u.] ˜ a − [l.u.] ˜ r [l.u.] ˜ P [l.u.] ΣΣ ( S ) 0 . (+12)(+09)( − − . (+2 . . − . − . . (+08)(+11)( − − . (+1 . . − . − . (+100)(+98)( − − ΞΣ ( S ) 0 . (+08)(+07)( − − . (+1 . . − . − . . (+11)(+07)( − − . (+1 . . − . − . (+43)(+25)( − − ΞΞ ( S ) 0 . (+03)(+08)( − − . (+0 . . − . − . . (+03)(+06)( − − . (+0 . . − . − . (+23)(+31)( − − N N ( S ) 0 . (+12)(+07)( − − . (+2 . . − . − . . (+12)(+09)( − − . (+2 . . − . − . (+89)(+62)( − − Σ N ( S ) 0 . (+22)(+16)( − − . (+1 . . − . − . . (+23)(+31)( − − . (+2 . . − . − . − (+27)(+16)( − − ΞΞ ( S ) 0 . (+17)(+14)( − − − . (+4 . . − . − . . (+22)(+29)( − − (+14)(+22)( − − − (+87)(+62)( − − Ξ N ( S ) 0 . (+05)(+02)( − − . (+0 . . − . − . . (+02)(+04)( − − . (+0 . . − . − . (+08)(+08)( − − In Appendix B, results for NN channels are compared with the previous scattering parameters obtained in Ref. [42]using the same correlation functions, as well as with the predictions obtained from low-energy theorems in Ref. [90].Through a thorough investigation, the various tensions are discussed and resolved. For a detailed discussion of naturalness in EFTs, see Ref. [126]. .
00 0 .
02 0 .
04 0 . k ∗ [l . u . ] k ∗ c o t δ [l . u . ] ΣΣ ( S ) .
00 0 .
02 0 .
04 0 . k ∗ [l . u . ] . . . . ΞΣ ( S ) .
00 0 .
02 0 .
04 0 . k ∗ [l . u . ] . . . . ΞΞ ( S ) .
00 0 .
02 0 .
04 0 . k ∗ [l . u . ] k ∗ c o t δ [l . u . ] NN ( S ) .
00 0 .
05 0 . k ∗ [l . u . ] − . − . . . Σ N ( S ) .
00 0 .
02 0 .
04 0 .
06 0 . k ∗ [l . u . ] − . − . − . . ΞΞ ( S ) − .
02 0 .
00 0 .
02 0 . k ∗ [l . u . ] − . . . . Ξ N ( S ) ×
64 :32 ×
96 :48 ×
96 : d = (0 , , d = (0 , , d = (0 , , d = (0 , , d = (0 , , d = (0 , ,
2) Two-parameter polynomial: stat. / stat.+sys.Three-parameter polynomial: stat. / stat.+sys.
FIG. 6. k ∗ cot δ values as a function of the c.m. momenta k ∗ , along with the bands representing the two-and three-parameter polynomial fits for two-baryon systems under the assumption that there is a smoothand monotonic behavior in k ∗ cot δ as a function of k ∗ beyond the t -channel cut. Quantities are expressedin lattice units. Although the ERE is only valid below the t -channel cut, one may still fit the k ∗ cot δ valuesbeyond this threshold using a similar polynomial form as the ERE in Eq. (14). To distinguish the“model” fit parameters from those obtained from the ERE, two- and three-parameter polynomialsare characterized by two { ˜ a − , ˜ r } or three { ˜ a − , ˜ r, ˜ P } parameters. Such forms are motivated bythe fact that in most channels, k ∗ cot δ values as a function of k ∗ exhibit smooth and monotonicbehavior beyond the t -channel cut, as is seen in Fig. 2. The only exceptions are the spin-singlet N N and Σ N channels, for which such a polynomial fit will not be performed. The results of thisfit, using the same strategy as described above for ERE fits, are shown in Table V and Fig. 6.In the next section, the EFTs and approximate symmetries of the interactions will be utilized tomake predictions for the inverse scattering length in channels for which ERE fits could not beperformed, i.e., Σ N ( S ) and ΞΞ ( S ) channels, and in those cases, the scattering length is foundconsistent with the ˜ a − values obtained from this model analysis. It should be emphasized thatsuch a polynomial fit beyond the t-channel cut is only one out of many applicable parametrizationsof the amplitude, and a systematic uncertainty associated with multiple model choices and model-selection criteria needs to be assigned to reliably constrain the energy dependence of the amplitudeat higher energies. D. Binding energies
A negative shift in the energy of two baryons in a finite volume compared with that of the non-interacting baryons may signal the presence of a bound state in the infinite-volume limit. However, More precise LQCD results may be required to identify non-polynomial behavior in k ∗ . This is analogous tothe efforts to uniquely identify non-analytic terms in chiral expansions, such as in π - π scattering, where veryhigh-precision calculations are required to reveal the logarithmic dependence on m π , see e.g., Ref. [127]. Explicitly, the infinite-volume binding momentum κ ( ∞ ) can be determined by expanding Eq. (10) in the negative- k ∗ region [63]: | k ∗ | = κ ( ∞ ) + Z L (cid:34)(cid:88) m | ˆ γ m | e i πα m · d e −| ˆ γ m | κ ( ∞ ) L (cid:35) , (19)where Z is the residue of the scattering amplitude at the bound-state pole. In this study, theboost vectors are d = (0 , , and d = (0 , , , and the values of γ deviate from one at the percentlevel. Therefore, all systems considered are non-relativistic to a good approximation. Only thefirst few terms in the sum in Eq. (19), corresponding to | m | ∈ { , , √ } , are considered in thevolume extrapolation performed below, with corrections that scale as O ( e − κ ( ∞ ) L ) .Alternatively, one can compute κ ( ∞ ) by finding the pole location in the s -wave scattering am-plitude: k ∗ cot δ | k ∗ = iκ ( ∞ ) + κ ( ∞ ) = 0 . (20)To obtain κ ( ∞ ) , the scattering amplitude has to first be constrained using Lüscher’s quantizationcondition as discussed in the previous subsection, and then be expressed in terms of an EREexpansion. This approach, therefore, requires an intermediate step compared with the first method,but does not require a truncation of the sum in Eq. (19).Results for the infinite-volume binding momenta κ ( ∞ ) are shown in Table VI. The columnslabeled as d = (0 , , and d = (0 , , correspond, respectively, to fitting separately the valuesof k ∗ with no boost, or with boost d = (0 , , , using Eq. (19). The column labeled as d = { (0 , , , (0 , , } is the result of fitting both sets of k ∗ values simultaneously, i.e., imposing the samevalue for κ ( ∞ ) and Z in both fits. The last column shows the κ ( ∞ ) values obtained using Eq. (20),with the parameters listed in Table III as obtained with a two-parameter ERE fit to k ∗ cot δ . Theresults obtained with the different extractions of κ ∞ are seen to be consistent with each other withinuncertainties. The largest difference observed is in the ΞΞ ( S ) channel, with a difference betweenthe volume-extrapolation and pole-location results of around . σ . The agreement between the twoapproaches suggests that the higher-order terms neglected in the sum in Eq. (19) are not significant.The binding energy, B , is defined in terms of the infinite-volume baryon masses and bindingmomenta as B = M ( ∞ )1 + M ( ∞ )2 − (cid:113) M ( ∞ )21 − κ ( ∞ )2 − (cid:113) M ( ∞ )22 − κ ( ∞ )2 , (21)where M ( ∞ ) i is the infinite-volume mass of baryon i obtained from Eq. (9). This quantity iscomputed for all systems that exhibit a negative c.m. momentum squared in the infinite-volumelimit, i.e., those listed in Table VI. The binding energies in physical units are listed for these systemsin Table VII. The binding energies of the two-nucleon systems computed here are consistent within σ with the values published previously in Ref. [42] using the same LQCD correlation functions.The same two-baryon systems studied here were also studied at m π ∼ MeV in Ref. [31], andwere found to be bound albeit with larger binding energies. While the results at m π ∼ MeVwere inconclusive regarding the presence of bound states in the irrep, the Σ N ( S ) and ΞΞ ( S ) systems are found to be unbound at this pion mass. The results obtained in the present work can be Alternatively, LQCD eigenenergies in a finite volume can be matched to an EFT description of the system in thesame volume to constrain the interactions. The constrained EFT can then be used to obtain the infinite-volumebinding energy, see e.g., Ref. [128]. This approach is more easily applicable to the multi-baryon sector, however itrelies on the validity of the EFT that is used. The largest value of γ is found in the NN ( S ) system with L = 24 , where γ ∼ . . TABLE VI. The infinite-volume binding momenta κ ( ∞ ) for bound states obtained by either using theextrapolation in Eq. (19) or from the pole location of the scattering amplitude as in Eq. (20). Quantitiesare expressed in lattice units. κ ( ∞ ) [l.u.] d = (0 , , d = (0 , , d = { (0 , , , (0 , , } − k ∗ cot δ | k ∗ = iκ ( ∞ ) N N ( S ) 0 . (+08)(+06)( − − . (+10)(+08)( − − . (+05)(+06)( − − . (+06)(+12)( − − Σ N ( S ) 0 . (+13)(+05)( − − . (+09)(+06)( − − . (+08)(+02)( − − . (+12)(+09)( − − ΣΣ ( S ) 0 . (+08)(+08)( − − . (+11)(+07)( − − . (+07)(+06)( − − . (+15)(+07)( − − ΞΣ ( S ) 0 . (+08)(+06)( − − . (+08)(+05)( − − . (+06)(+04)( − − . (+10)(+05)( − − ΞΞ ( S ) 0 . (+05)(+05)( − − . (+06)(+06)( − − . (+04)(+04)( − − . (+05)(+08)( − − N N ( S ) 0 . (+08)(+06)( − − . (+08)(+03)( − − . (+08)(+04)( − − . (+10)(+08)( − − Ξ N ( S ) 0 . (+04)(+06)( − − . (+05)(+06)( − − . (+03)(+05)( − − . (+05)(+06)( − − TABLE VII. Binding energies for bound states in MeV. The values are obtained using κ ( ∞ ) from the volume-extrapolation method with a combined fit to d = (0 , , and d = (0 , , data. The uncertainty from scalesetting is an order of magnitude smaller than the statistical and systematic uncertainties quoted. B [MeV] N N ( S ) Σ N ( S ) ΣΣ ( S ) ΞΣ ( S ) ΞΞ ( S ) N N ( S ) Ξ N ( S )13 . (+2 . . − . − . . (+3 . . − . − . . (+2 . . − . − . . (+2 . . − . − . . (+1 . . − . − . . (+2 . . − . − . . (+1 . . − . − . combined with those of Ref. [31] obtained at m π ∼ MeV to perform a preliminary extrapolationof the binding energies to the physical pion mass. This enables a postdiction of binding energiesin nature in cases where there are experimental data, and a prediction for the presence of boundstates and their binding in cases where no experimental information is available.For systems with non-zero strangeness, experimental knowledge is notably limited in comparisonto the nucleon-nucleon sector, and almost all phenomenological predictions are based on SU (3) flavor-symmetry assumptions as discussed in the introduction. There is a significant body ofwork devoted to building phenomenological models of two-baryon interactions based on one-boson-exchange potentials, such as the Nijmegen hard-core [129–131], soft-core (NSC) [132–134] andextended-soft-core (ESC) [135–143] models, as well as the Jülich [144–146] and Ehime [147, 148]models. EFTs [19, 21–23, 149–154] and quark models [155–157] have also been used to constructtwo-baryon potentials. A short summary of the results in the literature for the relevant channelswith non-zero strangeness is as follows:• The S and S Σ N channels do not exhibit bound states in any of the models listed above.The spin-singlet state behaves in a similar way to N N ( S ) , and the interactions are slightlyattractive, while those in the spin-triplet channel are found to be repulsive. The results in the literature for the binding energies of two-baryon systems obtained at larger-than-physical quarkmasses must be compared with the results of the current work with caution, as the use of different scale settingschemes makes a comparison in physical units meaningless, unless the physical limit of the quantities are taken.In the two-baryon sector, no continuum extrapolation has been performed in any of the previous studies.
0• For the Ξ N ( S ) system, almost all the models find that the interactions are slightly attrac-tive, but only a few exhibit a bound state. Among the most recent results are “ESC08a” [138]which gives B = 0 . MeV ‡ , and “ESC08c1” [139] which gives B = 0 . MeV ‡ . There is oneLQCD calculation of this system near the physical values of the quark masses performed bythe HAL QCD collaboration [158] using a different method than the current work, and nobound state is observed.• The “NSC97” model [134] finds a bound state for the ΣΣ ( S ) channel, with binding energiesranging from . to . MeV. χ EFT at NLO [23] finds a binding energy between and . MeV (no bound state is found with ESC or quark models in this channel).• The
ΞΣ ( S ) system is found to be bound in the “NSC97” model [134], with a binding energybetween . − . MeV, and by χ EFT [22, 23], with a binding energy between . − . MeV at LO and . − . MeV at NLO. With the quark model “fss2” [156], although theinteraction in this system is found to be attractive, no bound state is predicted (similar tothe “ESC08c1” model [139]).• Using one-boson-exchange potentials, with “NSC97” [134] the
ΞΞ ( S ) state is bound witha binding energy between . − . MeV, and with “Ehime” [148] between . − . MeV(no bound state is found with “ESC08c1” [139]). χ EFT [22, 23] also finds this state to bebound with a binding energy of . − . MeV at LO and . − . MeV at NLO. Thequark model “fss2” [156] does not find a bound state. In the
ΞΞ ( S ) channel, no bound stateis found with one-boson-exchange potentials, except for “Ehime” that finds a deeply boundstate with a binding energy of − MeV. “fss2” [156] finds this channel to be repulsive.The quark-mass dependences of multi-baryon spectra have not been studied extensively in theliterature. For the octet-baryon masses, it was found that LQCD calculations performed with 2+1dynamical fermions are consistent with a linear dependence on the pion mass at unphysical valuesof the quark masses, compared to the HB χ PT prediction of quadratic dependence at LO [159–161].Nonetheless, recent precision studies near the physical values of the quark masses appear to be moreconsistent with chiral predictions [162]. In the two-baryon sector the situation is more complicated.On the theoretical side, χ EFT was used in Ref. [163] to extrapolate LQCD results to the physicalpoint, assuming no dependence on the light quark masses for the LECs of the EFT (at a fixedorder). The same premise was taken in Ref. [29] to determine the I = 3 / N interaction at LO,which was used to address the possible appearance of Σ − hyperons in dense nuclear matter. In theabsence of a conclusive form for the quark-mass extrapolation of two-baryon binding energies, twonaive expressions with linear and quadratic m π dependence were used in Ref. [40] to extrapolatethe binding energy of H -dibaryon to its physical value. In Refs. [164–166], under the assumptionthat the H -dibaryon is a compact 6 valence-quark state (and not a two-baryon molecule), χ EFTwas used to extrapolate the binding energies, resulting in an unbound state.Two analytical forms with different m π dependence are used here to obtain the binding energiesat the physical light-quark masses, using the results presented in Ref. [31] at m π ∼ MeV andthose listed in Table VII for m π ∼ MeV: B lin ( m π ) = B (0)lin + B (1)lin m π , (22) B quad ( m π ) = B (0)quad + B (1)quad m π , (23) Since the binding energies are not explicitly computed in these references and only the s -wave scattering parametersare reported, binding energies are computed here using Eqs. (20) and (21), assuming a two-parameter ERE for k ∗ cot δ . These are marked with the symbol ‡ . . . . . m π [GeV ] − B [ M e V ] NN ( S ) . . . . m π [GeV ] − Σ N ( S ) . . . . m π [GeV ] − B [ M e V ] ΣΣ ( S ) . . . . m π [GeV ] − ΞΣ ( S ) . . . . m π [GeV ] − ΞΞ ( S ) . . . . m π [GeV ] − B [ M e V ] NN ( S ) . . . . m π [GeV ] − Ξ N ( S ) NPLQCD n f = 3NPLQCD n f = 2 + 1 Linear extrapolation in m π Quadratic extrapolation in m π NSC97EhimeESC χ EFT LO χ EFT NLOExperimental
FIG. 7. Extrapolation of the binding energies of different two-baryon systems, using the results obtainedin this work and those at m π ∼ MeV from Ref. [31]. For comparison, the results with values obtainedusing one-boson-exchange models or χ EFTs are also shown (and where needed, are shifted slightly in thehorizontal direction for clarity). where B (0)lin , B (1)quad , B (0)lin and B (1)quad are parameters to be constrained by fits to data. These fitsare shown in Fig. 7, along with the experimental value and predictions at the physical point.The binding energies extrapolated to the physical point, i.e., B lin ( m phys π ) and B quad ( m phys π ) , aresummarized in Table VIII. These extrapolations highlight some interesting features. The values obtained at the physicalpoint are consistent with the experimental values for the
N N channels. The rest of the bindingpredictions are at the same level of precision as the phenomenological results. The
ΞΞ ( S ) and Ξ N ( S ) channels are more consistent with being bound than the other channels, using both ex-trapolation functions. Moreover, the Σ N ( S ) channel was found not to support a bound statein this study, a conclusion that is in agreement with phenomenological models. The same conclu- Performing fits to dimensionless ratios of the binding energies to the baryon masses (to minimize the effects ofnon-zero lattice spacing) do not change the qualitative conclusions presented in the text. TABLE VIII. Extrapolated binding energies at the physical quark masses for bound states in MeV usingtwo different forms, linear and quadratic in m π . N N ( S ) Σ N ( S ) ΣΣ ( S ) ΞΣ ( S ) ΞΞ ( S ) N N ( S ) Ξ N ( S ) B lin ( m phys π ) 6 . (+6 . − . . (+7 . − . . (+6 . − . . (+5 . − . . (+4 . − . − . (+6 . − . . (+5 . − . B quad ( m phys π ) 9 . (+4 . − . . (+5 . − . . (+4 . − . . (+3 . − . . (+3 . − . . (+4 . − . . (+3 . − . sion holds for ΞΞ ( S ) , noting that only in one model, namely “Ehime”, a different conclusion isreached [148]. The spread of results and some contradictory conclusions in the models motivate theneed for LQCD studies of these states at near-physical values of the quark masses in the upcomingyears. III. EFFECTIVE LOW-ENERGY INTERACTIONS OF TWO BARYONSA. Leading and next-to-leading order interactions in the EFT
Even though SU (3) flavor symmetry is explicitly broken in this study by the different valuesof the light- and strange-quark masses, it is still useful to classify the different two-(octet)baryonchannels according to the SU (3) irrep that they belong to. In the spin-flavor decomposition of theproduct of two octet baryons with J P =
12 + , the 64 existing channels can be grouped into: ⊗ = ⊕ s ⊕ ⊕ ⊕ ⊕ a . (24)The states belonging to the { , s , } irreps are symmetric with respect to the exchange of twobaryons, and by the Pauli exclusion principle must have total spin J = 0 . The { , , a } irreps,with an antisymmetric flavor wavefunction, have J = 1 . Each of the systems studied in this workbelongs to only one single irrep: all of the singlet states belong to the irrep, N N ( S ) to , the triplet states Σ N and ΞΞ to the irrep, and Ξ N ( S ) to the a irrep. However, since m u = m d (cid:54) = m s , which explicitly breaks SU (3) symmetry, mixing among the irreps will appear.Note that the structure of the LQCD interpolating operators used in this study, i.e., single-pointquark-level wavefunctions at the source, does not allow accessing channels in the s irrep. Moreover,the state in the irrep is a coupled flavor channel, which is excluded from this study given thata large number of kinematic inputs are required to constrain the corresponding coupled-channelscattering amplitudes. The Lagrangian for the low-energy interactions of two octet baryons was first constructed inRef. [80] using the heavy-baryon chiral EFT (HB χ EFT) formalism, and consists of two-baryoncontact operators at LO. These interactions have also been studied in chiral perturbation theory( χ PT) in Refs. [19, 167], where in addition to the momentum-independent operators at LO, thepseudoscalar-meson exchanges are included in the interacting potential. At LO, all terms in bothHB χ EFT and χ PT are SU (3) symmetric. At NLO, there are two types of contributions: the SU (3) -symmetric interactions, obtained by the addition of derivative terms to the LO Lagrangian,and the SU (3) symmetry-breaking interactions, denoted by (cid:24)(cid:24)(cid:24)(cid:24) SU (3) in the following, that arise fromthe inclusion of the quark-mass matrix. The NLO extension of the two-baryon potential within χ PT was first presented in Refs. [20, 151], and includes interactions in higher partial waves. The ground state of the flavor channels belonging to the irrep has been determined in previous LQCD studiesto be bound at larger-than-physical values of the quark masses, corresponding to the long-sought-for H -dibaryonstate, see Refs. [26, 27, 40, 48, 50]. s -wave interactionsare considered. The LO Lagrangian of Ref. [80] is used, and the NLO contributions are formed tofollow the organization of the LO terms. In other words, the same spin-flavor operator structureis preserved in the NLO Lagrangian, up to the inclusion of derivative operators and the quark-mass matrix. The EFT considered is therefore a pionless EFT [78, 79] in the hypernuclear sector.The LO coefficients are known as Savage-Wise coefficients in the literature. This organization isdifferent from that of Petschauer and Kaiser in Ref. [20], and while the notation used here to labelthe NLO LECs is the same as in Ref. [20], their meaning is different. The differences betweenthe two organizations and the relations between both sets of (cid:24)(cid:24)(cid:24)(cid:24) SU (3) coefficients are presented inAppendix C. The full pionless EFT Lagrangian, up to NLO, is written as: L BB = L (0) , SU (3) BB + L (2) , SU (3) BB + L (2) , (cid:24)(cid:24)(cid:24) SU (3) BB , (25)with L (0) , SU (3) BB = − c Tr ( B † i B i B † j B j ) − c Tr ( B † i B j B † j B i ) − c Tr ( B † i B † j B i B j ) − c Tr ( B † i B † j B j B i ) − c Tr ( B † i B i ) Tr ( B † j B j ) − c Tr ( B † i B j ) Tr ( B † j B i ) , (26) L (2) , SU (3) BB = − ˜ c Tr ( B † i ∇ B i B † j B j + h.c. ) − ˜ c Tr ( B † i ∇ B j B † j B i + h.c. ) − ˜ c Tr ( B † i B † j ∇ B i B j + h.c. ) − ˜ c Tr ( B † i B † j ∇ B j B i + h.c. ) − ˜ c [ Tr ( B † i ∇ B i ) Tr ( B † j B j ) + h.c. ] − ˜ c [ Tr ( B † i ∇ B j ) Tr ( B † j B i ) + h.c. ] , (27) L (2) , (cid:24)(cid:24)(cid:24) SU (3) BB = − c χ Tr ( B † i χB i B † j B j ) − c χ Tr ( B † i χB j B † j B i ) − c χ Tr ( B † i B i χB † j B j ) − c χ Tr ( B † i B j χB † j B i ) − c χ Tr ( B † i χB † j B i B j + h.c. ) − c χ Tr ( B † i χB † j B j B i + h.c. ) − c χ Tr ( B † i B † j χB i B j ) − c χ Tr ( B † i B † j χB j B i ) − c χ Tr ( B † i B † j B i B j χ ) − c χ Tr ( B † i B † j B j B i χ ) − c χ Tr ( B † i χB i ) Tr ( B † j B j ) − c χ Tr ( B † i χB j ) Tr ( B † j B i ) , (28)where only terms that contribute to s -wave interactions are included in the NLO Lagrangian L (2) , SU (3) BB . The indices i and j denote spin indices, B is the octet-baryon flavor matrix, B = Σ √ + Λ √ Σ + p Σ − − Σ √ + Λ √ n Ξ − Ξ − (cid:113) Λ , (29)and χ is the quark-mass matrix, which can be written in terms of the meson masses using theGell-Mann–Oakes–Renner relation [168]: χ = 2 B m u m d
00 0 m s ∝ m π m π
00 0 m K − m π , (30)where the constant B is proportional to the quark condensate.In order to constrain the values of the LECs c i , ˜ c i , and c χi , the LO and NLO EREs of the inversescattering amplitudes in the s -wave can be used. It is known that if the interactions between octetbaryons are unnatural, that is r/a (cid:28) , a better justified power-counting scheme in the EFT is4the KSW-vK (Kaplan, Savage and Wise [169, 170] and van Kolck [78]) scheme, where at LO in thescattering amplitude, the contributions from LO momentum-independent operators are summedto all orders. With natural interactions, a power-counting scheme based on naive dimensionalanalysis is used and the expansion of the amplitude remains perturbative in the interaction strength,including for the LO interaction. As mentioned in Section II C, given the large uncertainties in thescattering parameters (in particular in the effective range), the ratio r/a shown in Table IV is notwell constrained, and does not conclusively prove unnaturalness in all channels. Since in at least twochannels the interactions seem unnatural, in the following both the natural and unnatural cases willbe considered in expressing relations between LECs and the scattering parameters. These relationsfor each two-baryon channel can be separated into those which are momentum-independent, withcontributions from LO and NLO (cid:24)(cid:24)(cid:24)(cid:24) SU (3) terms in the Lagrangian, and momentum-dependent, withonly contributions from NLO SU (3) terms: (cid:20) − a B B + µ (cid:21) − = M B B π ( c ( irrep ) + c χ B B ) , (31) r B B (cid:20) − a B B + µ (cid:21) − = M B B π ˜ c ( irrep ) , (32)where c ( irrep ) stands for the appropriate linear combinations of the c i LECs defined in the Lagrangianin Eq. (25). These relations are given in Table IX for each two baryon channel consisting of baryons B and B , where the LECs corresponding to given SU (3) irreps in this table are related to the c i LECs by: c (27) = 2( c − c + c − c ) , c (10) = 2( c + c + c + c ) ,c (8 s ) = 13 ( − c + 4 c − c + 5 c + 6 c − c ) , c (10) = 2( − c − c + c + c ) ,c (1) = 23 ( − c + c − c + 8 c + 3 c − c ) , c (8 a ) = 3 c + 3 c + 2 c + 2 c . (33)The same relations hold for ˜ c ( irrep ) , replacing c i with ˜ c i . Similarly, c χ B B ≡ c χ B B ( m K − m π ) and c χ B B are linear combinations of the c χi LECs as given in Table IX. The variables a B B and r B B are the scattering length and effective range of the channel B B , and M B B is the reduced massof that system. The renormalization scale µ depends on the naturalness of the interactions. For thenatural case µ = 0 , and Eqs. (31) and (32) correspond to a tree-level expansion of the scatteringamplitude. For the unnatural case, the expansion does not converge for momenta larger than a − ,and in the KSW-vK scheme µ is introduced as a renormalization scale for the s -channel two-baryonloops appearing in the all-orders expansion of the amplitude with LO interactions. Since a pionlessEFT is used, a convenient choice is µ = m π (where m π ∼ MeV is the mass of the pion obtainedwith the quark masses used in the LQCD study).Two sets of inputs can be used to constrain the numerical values for the LECs: 1) the scatteringparameters { a − , r } obtained from two-parameter ERE fits in Section II C, tabulated in Table III,can be used to compute LECs of both momentum-independent and momentum-dependent operators(method I), and 2) the binding momenta from Section II D can be used to compute the correspond-ing scattering length, related at LO by − a − + κ ( ∞ ) = 0 , and this single parameter can be usedto constrain the LECs of momentum-independent operators (method II). This second method ismotivated by the fact that κ ( ∞ ) is extracted with higher precision than the parameters from theERE fits, therefore enabling tighter constraints on the LECs of momentum-independent opera-tors. The results for both types of LECs are presented in Table X, and are depicted in Fig. 8. While the relations for the and s irreps are not used here, they will be needed in Sec. III B in connection to the SU (6) spin-flavor symmetry relations. TABLE IX. The LECs of the LO and NLO pionless EFT that contribute to the scattering amplitude of thevarious two-baryon channel. The first three columns are total angular momentum ( J ), strangeness ( S ), andisospin ( I ). J S I
Channel SU (3) LO SU (3) NLO (cid:24)(cid:24)(cid:24) SU (3) NLO
N N c (27) ˜ c (27) c χ − c χ ) − Σ N c (27) ˜ c (27) c χ − c χ ) − c (27) ˜ c (27) − ΞΣ c (27) ˜ c (27) c χ − c χ + c χ − c χ ) − c (27) ˜ c (27) c χ − c χ + c χ − c χ )1 0 0 N N c (10) ˜ c (10) c χ + c χ ) − Σ N c (10) ˜ c (10) − c χ + c χ ) − c (10) ˜ c (10) − c χ + c χ − c χ − c χ ) − N c (8 a ) ˜ c (8 a ) c χ + 2 c χ + 2 c χ + 2 c χ + 2 c χ + 2 c χ + c χ + c χ ) Results are presented in units of π/M B for the momentum-independent operators and π /M B for the momentum-dependent operators, where M B is the centroid of the octet-baryon masses, M B = M N + M Λ + M Σ + M Ξ = 0 . l.u.As can be seen from the values of the LECs that are obtained, the NLO SU (3) coefficientshave large uncertainties, and are mostly consistent with zero, because the effective ranges used toconstrain them have rather large uncertainties. Another feature of the results is that assuming theinteractions to be unnatural leads to better-constrained parameters in general, as a non-zero scale µ in the left-hand side of Eqs. (31) and (32) reduces the effect of uncertainties on the scattering lengths(this was also observed in Ref. [31] for systems at m π ∼ MeV). Furthermore, as is expected,the values obtained with method II have smaller uncertainties than the ones obtained from methodI, given the more precise scattering lengths, although the method is limited to LO predictions.Another anticipated feature is that in the cases where the effective range is resolved from zero
TABLE X. LECs of the momentum-independent and momentum-dependent operators as they appear inTable IX for the two-baryon channels, obtained by solving Eq. (31) in units of [ πM B ] for the momentum-independent operators, and Eq. (31) in units of [ π M B ] for the momentum-dependent operators, where M B isthe centroid of the octet-baryon masses. ˜ c ( irrep ) are only determined using method I.LECs µ Method
N N ( S ) Σ N ( S ) ΣΣ ( S ) ΞΣ ( S ) ΞΞ ( S ) N N ( S ) Ξ N ( S ) c ( irrep ) + c χ B B I − (+9)( − − (+7)( − − (+14)( − − (+7)( − − (+5)( − − (+8)( − − (+3)( − II − (+3)( − − (+3)( − − (+3)( − − (+2)( − − (+1)( − − (+3)( − − (+1)( − m π I . (+4 . − . . (+2 . − . . (+0 . − . . (+0 . − . . (+0 . − . . (+1 . − . . (+0 . − . II . (+0 . − . . (+0 . − . . (+0 . − . . (+0 . − . . (+0 . − . . (+0 . − . . (+0 . − . ˜ c ( irrep ) I − (+1600)( − − (+550)( − (+1800)( − (+390)( − (+160)( − (+570)( − (+86)( − m π I − (+34)( − − (+27)( − (+5)( − (+9)( − (+3)( − (+16)( − (+6)( − c ( i rr e p ) + c B B h ⇡ M B i Natural case ( µ = 0) Unnatural case ( µ = m ⇡ ) NN ( S ) ⌃ N ( S ) ⌃⌃ ( S ) ⌅ ⌃ ( S ) ⌅⌅ ( S ) NN ( S ) ⌅ N ( S ) ˜ c ( i rr e p ) h ⇡ M B i NN ( S ) ⌃ N ( S ) ⌃⌃ ( S ) ⌅ ⌃ ( S ) ⌅⌅ ( S ) NN ( S ) ⌅ N ( S ) FIG. 8. LECs obtained by solving Eqs. (31) (upper panels) and (32) (lower panels) under the assumptionof natural (left panels) and unnatural (right panels) interactions. The LECs of momentum-independentoperators are in units of [ πM B ] and those of the momentum-dependent operators are in units of [ π M B ] , where M B is the centroid of the octet-baryon masses. The gray-circle markers denote quantities that are extractedusing the ERE parameters (method I), while black-square markers are those obtained from scattering lengthsthat are computed from binding momenta (method II). within uncertainties (e.g., in the ΞΞ ( S ) channel), the values from method II are slightly differentfrom those obtained from method I, indicating the non-negligible effect of the NLO effective-rangecontributions that are neglected with this method.It should be noted that the input for scattering parameters is not sufficient to disentangle theLO SU (3) and NLO (cid:24)(cid:24)(cid:24)(cid:24) SU (3) coefficients in general, hence the c ( irrep ) + c χ B B entry in Table Xand Fig. 8. For the systems that belong to the irrep, since the spin-singlet pairs { N N, Σ N } and { ΞΣ , ΞΞ } depend on the same SU (3) LO and (cid:24)(cid:24)(cid:24)(cid:24) SU (3) NLO LECs but with different linearcombinations of the coefficients, a system of equations can be formed to separate each contribution.
TABLE XI. The values of the momentum-independent SU (3) coefficient c (27) and specific linear combinationsof the (cid:24)(cid:24)(cid:24) SU (3) coefficients c χi . Quantities are expressed in units of [ πM B ] , where M B is the centroid of theoctet-baryon masses. µ Method c (27) { N N, Σ N } c (27) { ΣΣ } c (27) { ΞΣ , ΞΞ } c χ − c χ c χ − c χ + c χ − c χ I − (+58)( − − (+14)( − − (+21)( − (+18)( − (+10)( − II − (+9)( − − (+3)( − − (+6)( − − (+4)( − (+2)( − m π I . (+6 . − . . (+0 . − . . (+2 . − . . (+2 . − . − . (+0 . − . II . (+1 . − . . (+0 . − . . (+1 . − . . (+0 . − . . (+0 . − . { NN, ⌃ N } ⌃⌃ { ⌅⌃ , ⌅⌅ } c ( ) h ⇡ M B i Natural case ( µ = 0) { NN, ⌃ N } ⌃⌃ { ⌅⌃ , ⌅⌅ } Unnatural case ( µ = m ⇡ ) { NN, ⌃ N } { ⌅⌃ , ⌅⌅ } c B B h ⇡ M B i c c c c + c c { NN, ⌃ N } { ⌅⌃ , ⌅⌅ } c c c c + c c FIG. 9. The LO SU (3) LEC c (27) (upper panels) and NLO (cid:24)(cid:24)(cid:24) SU (3) LECs c χ B B (lower panels) under theassumption of natural (left panels) and unnatural (right panels) interactions, in units of [ πM B ] , where M B isthe centroid of the octet-baryon masses. The gray-circle markers denote quantities that are extracted usingmethod I, while black-square markers show results obtained from method II. See the text for further details. The results are shown in Table XI and Fig. 9, along with the result for the ΣΣ channel for comparisonpurposes, as there is no contribution from (cid:24)(cid:24)(cid:24)(cid:24) SU (3) interactions for this channel at this order. Fromthese results, it can be seen that the values of the symmetry-breaking coefficients c χ − c χ and c χ − c χ + c χ − c χ are compatible with zero. Together with the observation that the scatteringlengths and binding energies in all of the systems are similar within uncertainties, it appears thatthe SU (3) flavor symmetry remains an approximate symmetry at the quark masses used in thisstudy. These observations in the two-baryon sector are consistent with those in the single-baryonsector as presented in Ref. [42] at the same quark masses. There, it was found that the quantity δ GMO = M B ( M Λ + M Σ − M N − M Ξ ) , which is a measure of SU (3) flavor-symmetry breaking,is an order of magnitude smaller than its experimental value. In Appendix C, the full list of relations needed to independently constrain all 24 different LECsthat appear at LO and NLO are shown, demonstrating that the proper combinations of 18 two-baryon flavor channels are sufficient to extract all these LECs. These channels will be the subjectof upcoming LQCD studies toward the physical values of the quark masses.
B. Compatibility with large- N c predictions In the limit of SU (3) flavor symmetry and large N c , two-baryon interactions are predicted to beinvariant under an SU (6) spin-flavor symmetry, with corrections that generally scale as /N c [81].In the two-nucleon sector, this encompasses the SU (4) spin-flavor Wigner symmetry [173–175],with corrections that scale as /N c . Under SU (6) group transformations, the baryons transformas a three-index symmetric tensor Ψ µνρ , where each SU (6) index is a pair of spin and flavor indices ( iα ) . At LO, only two independent terms contribute to the interacting Lagrangian of two-baryon The violation of the Gell-Mann-Okubo mass relation [171, 172] results from SU (3) breaking transforming in the irrep of SU (3) flavor symmetry, which can only arise from insertions of the light-quark mass matrix or fromnonanalytic meson-mass dependence induced by loops in χ PT. L (0) ,SU (6) BB = − a (Ψ † µνρ Ψ µνρ ) − b Ψ † µνσ Ψ µντ Ψ † ρδτ Ψ ρδσ , (34)where the baryon tensor can be expressed as a function of the octet-baryon matrices, B : Ψ µνρ = Ψ ( iα )( jβ )( kγ ) = 1 √ (cid:16) B αω,i (cid:15) ωβγ (cid:15) jk + B βω,j (cid:15) ωγα (cid:15) ik + B γω,k (cid:15) ωαβ (cid:15) ij (cid:17) . (35)Here, α, β, γ, ω are flavor indices, i, j, k are spin indices, and the Levi-Civita tensor (cid:15) is either inflavor or spin space depending on the type and number of indices. A priori, the relative size of theKaplan-Savage coefficients, a and b , is unknown, and only experimental data or LQCD input mayconstrain these LECs. As is seen in Eqs. (36) below, the contribution from the b coefficient to theLO amplitude is parametrically suppressed compared with that of the coefficient a . As a result,if b in Eq. (34) is comparable or smaller than a , there remains only one type of interaction thatcontributes significantly to the scattering amplitude, a situation that would realize an accidental SU (16) symmetry of the nuclear and hypernuclear forces. The first evidence for SU (16) symmetryin the two-(octet)baryon sector was observed in a LQCD study at a pion mass of ∼ MeV [31],and the goal of the present study is to examine these predictions at smaller values of the light-quarkmasses. Such a symmetry is suggested in Ref. [82] to be consistent with the conjecture of maximumentanglement suppression of the low-energy sector of QCD.As in Sec. III A, the a and b coefficients can be matched to scattering amplitudes in a momentumexpansion at LO. Since at least some of the SU (3) symmetry-breaking LECs c χi were found to beconsistent with zero in this study, one can assume an approximate SU (3) symmetry in general, andrelate the SU (6) LECs a and b directly to the LECs of the LO SU (3) -symmetric Lagrangian for TABLE XII. The leading SU (6) LECs, a and b/ , obtained by solving a given pair of equations in Eqs. (36).The last column shows the results of a constant fit to the LECs obtained in each case as described in Eqs. (37).The spin specifications are dropped from channel labels for brevity, but one clarification is necessary: in thefirst pair of two-baryon channels, N N refers to the spin-singlet case, while in the last pair, it denotes thespin-triplet case. Quantities are expressed in units of [ πM B ] , where M B is the centroid of the octet-baryonmasses.LEC µ Method { N N, Ξ N } { Σ N, Ξ N } { ΣΣ , Ξ N } { ΞΣ , Ξ N } { ΞΞ , Ξ N } { N N, Ξ N } Combined a I − (+3)( − − (+2)( − − (+4)( − − (+2)( − − (+2)( − − (+3)( − − (+4)( − II − . (+0 . − . − . (+0 . − . − . (+1 . − . − . (+0 . − . − . (+0 . − . − . (+0 . − . − . (+1 . − . m π I . (+1 . − . . (+0 . − . . (+0 . − . . (+0 . − . . (+0 . − . . (+0 . − . . (+0 . − . II . (+0 . − . . (+0 . − . . (+0 . − . . (+0 . − . . (+0 . − . . (+0 . − . . (+0 . − . b I (+110)( − (+65)( − (+99)( − (+41)( − (+23)( − (+57)( − (+36)( − II (+11)( − (+12)( − (+14)( − (+9)( − (+6)( − (+12)( − (+9)( − m π I − (+7)( − (+6)( − (+3)( − (+4)( − (+3)( − (+4)( − (+4)( − II (+2)( − (+2)( − (+2)( − (+2)( − (+2)( − (+2)( − (+2)( − Those components of the field Ψ that correspond to decuplet baryons [81] have been neglected as they are notrelevant to the low-energy scattering of two octet baryons. a h ⇡ M B i Natural case ( µ = 0) Unnatural case ( µ = m ⇡ ) { NN , ⌅ N } { ⌃ N , ⌅ N } { ⌃⌃ , ⌅ N } { ⌅ ⌃ , ⌅ N } { ⌅⌅ , ⌅ N } { NN , ⌅ N } b h ⇡ M B i { NN , ⌅ N } { ⌃ N , ⌅ N } { ⌃⌃ , ⌅ N } { ⌅ ⌃ , ⌅ N } { ⌅⌅ , ⌅ N } { NN , ⌅ N } FIG. 10. The leading SU (6) LECs, a (upper panels) and b/ (lower panels), under the assumption ofnatural (left panels) and unnatural (right panels) interactions, in units of [ πM B ] , where M B is the centroidof the octet-baryon masses. The gray-circle markers denote quantities extracted using the ERE parameters(method I), with the light pink band showing the averaged value, while black-square markers show resultsobtained from scattering lengths that are constrained by binding momenta (method II), with the dark pinkband showing the averaged value. given irreps: c (27) = 2 a − b
27 + O (cid:18) N c (cid:19) , c (10) = 2 a − b
27 + O (cid:18) N c (cid:19) ,c (8 s ) = 2 a + 2 b O (cid:18) N c (cid:19) , c (10) = 2 a + 14 b
27 + O (cid:18) N c (cid:19) ,c (1) = 2 a − b O (cid:18) N c (cid:19) , c (8 a ) = 2 a + 2 b
27 + O (cid:18) N c (cid:19) . (36)In order to extract a and b , states in the and irreps can be combined with those in the a irrep,allowing for six possible extractions. The results are shown in Table XII and Fig. 10. As seen inEqs. (36), the contributions from the b coefficient are suppressed by at least a factor of 3 comparedwith those from the a coefficient, thus the rescaled coefficient b/ is considered. Considering thatthe results presented should be valid only up to corrections that scale as /N c , individual values ofthe coefficients a and b/ obtained from different pairs of channels exhibit remarkable agreement,indicating that the SU (6) spin-flavor symmetry is a good approximation at these values of thequark masses. A correlated weighted average of the results is obtained, following the procedure Note that the ERE parameters were obtained in the previous section only for two-baryon channels belonging tothe { , , a } irreps. The average of a series of values { x i } with uncertainties { σ i } is computed as: x average = (cid:88) i x i w i , w i = σ − i (cid:80) j σ − j , σ average = (cid:88) ij w i w j C ij , C ij = σ i σ j , (37)where, since the different values of x i (and their uncertainties) are correlated, a 100% correlation is assumed when TABLE XIII. Predicted SU (3) LECs, c ( irrep ) , as well as the Savage-Wise coefficients, c i , obtained from theKaplan-Savage SU (6) coefficients a and b using the relations in Eqs. (36) and (33). Quantities are expressedin units of [ πM B ] , where M B is the centroid of the octet-baryon masses. µ Method c (27) c (8 s ) c (1) c (10) c (10) c (8 a ) I − (+12)( − (+73)( − − (+73)( − − (+12)( − (+57)( − − (+12)( − II − (+3)( − (+17)( − − (+17)( − − (+3)( − (+14)( − − (+3)( − m π I . (+1 . − . . (+7 . − . . (+7 . − . . (+1 . − . . (+6 . − . . (+1 . − . II . (+0 . − . . (+4 . − . . (+4 . − . . (+0 . − . . (+3 . − . . (+0 . − . µ Method c c c c c c I − (+28)( − (+12)( − (+13)( − − (+19)( − (+24)( − − (+12)( − II − (+6)( − (+3)( − (+3)( − − (+4)( − − (+6)( − − (+3)( − m π I − . (+2 . − . . (+1 . − . . (+1 . − . − . (+1 . − . . (+2 . − . − . (+1 . − . II − . (+1 . − . . (+0 . − . . (+0 . − . − . (+1 . − . . (+1 . − . − . (+0 . − . introduced by Schmelling [176] and used by the FLAG collaboration [177], and is shown as thepink bands in Fig. 10. Given the uncertainty in b/ , no conclusion can be drawn about the relativeimportance of a and b/ . We will return to the question of the presence of an accidental SU (16) symmetry shortly.Given the extracted values of a and b/ , several checks can be performed, and several predictionscan be made. The simplest check is to compute all of the LO SU (3) LECs, c ( irrep ) , using the relationsin Eqs. (36). The results are shown in the first rows of Table XIII and the upper panels of Fig. 11.Columns with hashed backgrounds are the coefficients whose values were used as an input to makepredictions for other coefficients, presented in panels with solid colored backgrounds. These inputcoefficients ( c (27) , c (10) , and c (8 a ) ) can be reevaluated using the average values of a and b/ , whichtherefore gives back consistent values but with different uncertainties (for c (27) , the average of thevalues given in Table XI is computed). The large uncertainties in the c (8 s ) , c (1) , and c (10) coefficientsare due to the fact that b/ , with a larger uncertainty than a , is numerically more important in thesecases, see Eqs. (36). Additionally, the Savage-Wise coefficients c i can be computed by inverting therelations in Eqs. (33), and the resulting values are presented in the last rows of Table XIII and thelower panels of Fig. 11. Due to large uncertainties in the natural case, no conclusions can be maderegarding the relative size of the coefficients. In the unnatural case and at the chosen value of therenormalization scale, the c coefficient has a larger value than the rest of the coefficients. Therelative importance of c is a remnant of an accidental approximate SU (16) symmetry of s -wavetwo-baryon interactions that is more pronounced in the SU (3) -symmetric study with m π ∼ MeV in Ref. [31]. It will be interesting to explore whether the remnant of this symmetry remainsvisible in studies closer to the physical quark masses.The values of the SU (6) coefficients a and b allow predictions to be made for the scatteringlengths of the systems that could not be constrained in this study by an ERE fit, namely the Σ N ( S ) and ΞΞ ( S ) channels. Using the c ( irrep ) coefficients computed previously, the relationsin Eq. (31) can be inverted to obtain a − , assuming that the values of c χi are negligible compared computing σ average . For asymmetric uncertainties in x i , the following procedure is used to symmetrize them: avalue x i = c (+ u )( − l ) is modified to c + ( u − l ) / with uncertainty σ = max [( u + 3 l ) / , (3 u + l ) / . c (27) c (8 S ) c (1) c (10) c (10) c (8 A ) c ( i rr e p ) h ⇡ M B i Natural case ( µ = 0) c (27) c (8 S ) c (1) c (10) c (10) c (8 A ) Unnatural case ( µ = m ⇡ ) c c c c c c c i h ⇡ M B i c c c c c c FIG. 11. The predicted (filled markers) LO SU (3) coefficients c ( irrep ) (upper panels), as well as Savage-Wise coefficients c i (lower panels) reconstructed from the SU (6) relations are compared with the directly-extracted LECs (empty markers) under the assumption of natural (left panels) and unnatural (right panels)interactions, in units of [ πM B ] , where M B is the centroid of the octet-baryon masses. The gray-circle symbolsdenote quantities that have been extracted using the scattering parameters obtained from the ERE fit(method I), while black-square symbols denote those that are obtained from scattering lengths constrainedby binding momenta (method II). The hashed background in the upper panels denotes coefficients whosevalues were used to constrain a and b , and hence are not predictions. with those of c ( irrep ) (an observation that is only confirmed for given linear combinations of theseLECs but is assumed to hold in general given the hints of an approximate SU (3) symmetry in thisstudy). This exercise leads to consistent results for the inverse scattering length for systems forwhich the ERE allowed a direct extraction of this parameter, while it provides predictions for thechannels shown in Table XIV. For the case of natural interactions, the scattering lengths are notconstrained well, although they are consistent within uncertainties with those in the unnatural case,demonstrating the renormalization-scale independence of the scattering length. For the unnaturalcase, both methods are consistent and give rise to inverse scattering lengths that are positive andlarger than those obtained for the rest of the systems studied in this work. This is in agreement with TABLE XIV. Predicted inverse scattering lengths, a − , for the systems where an ERE fit was not possible,using the SU (6) LECs a and b . Quantities are expressed in lattice units. µ Method a − N ( S ) a − S ) I − . (+11)( − − . (+10)( − II . (+33)( − . (+30)( − m π I . (+04)( − . (+03)( − II . (+02)( − . (+02)( − k ∗ cot δ in these channels beyond the t -channelcut, see Table V. IV. CONCLUSIONS
Nuclear and hypernuclear interactions are key inputs into investigations of the properties of matter,and their knowledge continues to be limited in systems with multiple neutrons or when hyperonsare present. In recent years, LQCD has reached the stage where controlled first-principles studiesof nuclei are feasible, and may soon constrain nuclear and hypernuclear few-body interactions innature. The present work demonstrates such a capability in the case of two-baryon interactions,albeit at an unphysically large value of the quark masses corresponding to a pion mass of ∼ MeV. It illustrates how Euclidean two-point correlation functions of systems with the quantumnumbers of two baryons computed with LQCD can be used to constrain a wealth of quantities,from scattering phase shifts to low-energy scattering parameters and binding energies, to EFTsof forces, or precisely the LECs describing the interactions of two baryons. This same approachcan be expected to be followed in upcoming computations with the physical quark masses, and itsoutput, both in form of finite-volume energy spectra and constrained EFT interactions, can serve asinput into quantum many-body studies of larger isotopes, both at unphysical and physical values ofquark masses, see e.g., Refs. [85–87] for previous studies in the nuclear sector. By supplementing themissing experimental input for scattering and spectra of two-baryon systems, such LQCD analysescan constrain phenomenological models and EFTs of hypernuclear forces.In summary, the present paper includes a computation of the lowest-lying spectra of several two-octet baryon systems with strangeness ranging from to − . These results have been computed inthree different volumes, using a single lattice spacing, and with unphysical values of the light-quarkmasses. The finite-volume nature of the energies provides a means to constrain the elastic scatteringamplitudes in these systems through the use of Lüscher’s formalism. Assuming small discretizationartifacts given the improved LQCD action that is employed, our results reveal interesting featuresabout the nature of two-baryon forces with larger-than-physical values of the quark masses. Inparticular, the determination of scattering parameters of two-baryon systems at low energies hasenabled constraints on the LO and NLO interactions of a pionless EFT, both for the SU (3) flavor-symmetric and symmetry-breaking interactions. While the two-baryon channels studied in thiswork only allowed two sets of leading SU (3) symmetry-breaking LECs to be constrained, andthose values are seen to be consistent with zero, the present study is the first such analysis toaccess these interactions, extending the previous EFT matching presented in Ref. [31] at an SU (3) -symmetric point with m π = m K ∼ MeV. Given the limited knowledge of flavor-symmetry-breaking effects in the two-baryon sector in nature, this demonstrates the potential of LQCDto improve the situation. Finally, the observation of an approximate SU (3) symmetry in thetwo-baryon systems of this work led to an investigation of the large- N c predictions of Ref. [81],through matching the LQCD results for scattering amplitudes to the EFT. In particular, the s -wave interactions at LO are found to exhibit an SU (6) spin-flavor symmetry at this pion mass, asalso observed in Ref. [31] at a larger value of the pion mass. Both of the two independent spin-flavor-symmetric interactions at LO are found to contribute to the amplitude. Nonetheless, the extractedvalues of the coefficients of the LO SU (3) -symmetric EFT suggest a remnant of an approximateaccidental SU (16) symmetry observed in the SU (3) flavor-symmetric study at m π ∼ MeV [31].It will be interesting to examine these symmetry considerations in the hypernuclear forces at thephysical values of the quark masses, particularly given the conjectured connections between thenature of forces in nuclear physics and the quantum entanglement in the underlying systems [82].While no attempt is made in the current work to constrain forces within the EFTs at the physical3point, a naive extrapolation is performed using the results of this work and those at m π ∼ MeV, with simple extrapolation functions, to make predictions for the binding energies of severaltwo-baryon channels. The results for ground-state energies of two-nucleon systems are found to becompatible with the experimental values. Furthermore, stronger evidence for the existence of boundstates in the
ΞΞ ( S ) and Ξ N ( S ) channels is observed compared with other two-baryon systems.Such predictions are in agreement with current phenomenological models and EFT predictions, andcan be improved systematically as LQCD studies of multi-baryon systems progress toward physicalvalues of the quark masses in the upcoming years. ACKNOWLEDGMENTS
We would like to thank Joan Soto and Isaac Vidaña for enlightening discussions on the effectivefield theory formalism and on the physics of neutron stars and the hyperon-puzzle, respectively. Wewould like to also thank them, as well as André Walker-Loud, for comments on the first version ofthis manuscript.MI is supported by the Universitat de Barcelona through the scholarship APIF. MI and AP ac-knowledge support from the Spanish Ministerio de Economía y Competitividad (MINECO) underthe project MDM-2014-0369 of ICCUB (Unidad de Excelencia “María de Maeztu”), from the Eu-ropean FEDER funds under the contract FIS2017-87534-P and by the EU STRONG-2020 projectunder the program H2020-INFRAIA-2018-1, grant agreement No. 824093. MI acknowledges theUniversity of Maryland and the Massachusetts Institute of Technology for hospitality and partialsupport during preliminary stages of this work. SRB is supported in part by the U.S. Department ofEnergy, Office of Science, Office of Nuclear Physics under grant Contract No. DE-FG02-97ER-41014.ZD is supported by Alfred P. Sloan fellowship, and by Maryland Center for Fundamental Physicsat the University of Maryland, College Park. WD and PES acknowledge support from the U.S.DOE grant DE-SC0011090. WD is also supported within the framework of the TMD Topical Col-laboration of the U.S. DOE Office of Nuclear Physics, and by the SciDAC4 award DE-SC0018121.PES is additionally supported by the National Science Foundation under CAREER Award 1841699and under EAGER grant 2035015, by the U.S. DOE Early Career Award DE-SC0021006, by aNEC research award, and by the Carl G and Shirley Sontheimer Research Fund. KO and FWare supported by U.S. DOE grant DE-FG02-04ER41302 and by Jefferson Science Associates, LLCunder U.S. DOE Contract DE-AC05-06OR23177. FW is additionally supported by the USQCDScientific Discovery through Advanced Computing (SciDAC) project funded by U.S. Departmentof Energy, Office of Science, Offices of Advanced Scientific Computing Research, Nuclear Physicsand High Energy Physics. MJS is supported by the Institute for Nuclear Theory with DOE grantNo. DE-FG02-00ER41132. This manuscript has been authored by Fermi Research Alliance, LLCunder Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science,Office of High Energy Physics.The results presented in this manuscript were obtained using ensembles of isotropic-clover gauge-field configurations produced several years ago with resources obtained by researchers at the Col-lege of William and Mary and the Thomas Jefferson National Accelerator Facility and by theNPLQCD collaboration. Computations were performed using a College of William and Mary ledXSEDE and NERSC allocation, and NPLQCD PRACE allocations on Curie and MareNostrum,on LLNL machines, on the HYAK computational infrastructure at the UW, and through ALCCallocations. Calculations of propagators and their contractions were performed using computa-tional resources provided by the Extreme Science and Engineering Discovery Environment, whichis supported by National Science Foundation grant No. OCI-1053575, NERSC (supported by U.S.Department of Energy Grant No. DE-AC02-05CH11231), and by the USQCD collaboration. This4research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge Na-tional Laboratory, which is supported by the Office of Science of the U.S. Department of Energyunder Contract No. DE-AC05-00OR22725. The authors thankfully acknowledge the computer re-sources at MareNostrum and the technical support provided by Barcelona Supercomputing Center(RES-FI-2019-2-0032 and RES-FI-2019-3-0024). Parts of the calculations used the Chroma [178]and QUDA [179, 180] software suites. We thank André Walker-Loud and Thomas Luu for contri-butions during initial stages of production prior to 2014.After this manuscript was completed, an additional preprint [181] appeared regarding two-nucleon scattering, using a set of scattering interpolating operators similar to Ref. [50]. [1] V. A. Ambartsumyan and G. S. Saakyan, The Degenerate Superdense Gas of Elementary Particles,Soviet Astronomy , 187 (1960).[2] A. Feliciello and T. Nagae, Experimental review of hypernuclear physics: recent achievements andfuture perspectives, Rept. Prog. Phys. , 096301 (2015).[3] A. Gal, E. Hungerford, and D. Millener, Strangeness in nuclear physics, Rev. Mod. Phys. , 035004(2016), arXiv:1605.00557 [nucl-th].[4] D. 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In Refs. [88, 89], several criteria were presented to validate studies of two-baryon systems that relyon the extraction of finite-volume energies from Euclidean LQCD correlation functions for use inLüscher’s fromalism. The results of the present work are examined and validated with regard tothese criteria. Similar investigations were performed in Refs. [31, 108] for the study at m π ∼ MeV in Ref. [31].-
Interpolating-operator independence : The two different source-sink operator structures, de-noted SP and SS and described in Sec. II A, yield the same energies for both the ground andthe first excited states obtained in this work. This consistency can be verified by examining2 − .
02 0 .
00 0 .
02 0 .
04 0 .
06 0 .
08 0 . NN ( S ) − .
02 0 .
00 0 .
02 0 .
04 0 .
06 0 .
08 0 . Σ N ( S ) − .
02 0 .
00 0 .
02 0 .
04 0 .
06 0 .
08 0 . ΣΣ ( S ) − .
02 0 .
00 0 .
02 0 .
04 0 .
06 0 .
08 0 . ΞΣ ( S ) − .
02 0 .
00 0 .
02 0 .
04 0 .
06 0 .
08 0 . ΞΞ ( S ) − .
02 0 .
00 0 .
02 0 .
04 0 .
06 0 .
08 0 . NN ( S ) − .
02 0 .
00 0 .
02 0 .
04 0 .
06 0 .
08 0 . k ∗ [l . u . ] Σ N ( S ) − .
02 0 .
00 0 .
02 0 .
04 0 .
06 0 .
08 0 . k ∗ [l . u . ] ΞΞ ( S ) − .
02 0 .
00 0 .
02 0 .
04 0 .
06 0 .
08 0 . k ∗ [l . u . ] Ξ N ( S ) ×
64 : d = (0 , , d = (0 , ,
2) 32 ×
96 : d = (0 , , d = (0 , ,
2) 48 ×
96 : d = (0 , , d = (0 , , FIG. 12. The values of k ∗ for all systems analyzed in this work. Quantities are expressed in lattice units. the late-time behavior of the effective-energy and effective-energy-shift functions constructedfrom the SS and SP correlation functions in Figs. 22-31. Moreover, the c.m. momenta k ∗ obtained from the correlation functions with d = (0 , , and d = (0 , , must be consistent,up to negligible relativistic and small O (cid:0) ( m − m ) /E ∗ (cid:1) corrections [63], a feature that isobserved in the results presented here, as shown in Fig. 12. The largest difference is seen inthe N N ( S ) channel for the n = 2 level on the ensemble with L = 24 , for which the c.m.momenta in the unboosted and boosted cases exhibit a ∼ σ difference.- Consistency between ERE parameters for k ∗ < and k ∗ > : In the two-baryon channelsstudied in this work, there are not sufficient data points for k ∗ cot δ below the t -channel cut toextract precise scattering parameters, as pointed out in Sec. II B. Nonetheless, for the casesfor which two sets of data at positive and negative values of k ∗ are available, the ERE fitsobtained by fitting to all k ∗ versus only fitting to k ∗ < values are fully consistent witheach other, as is shown in Fig. 13.- Non-singular scattering parameters : None of the scattering parameters extracted show sin-gular behavior, as can be seen from the values in Table III.-
Requirement on the residue for the scattering amplitude at the bound-state pole : In order tosupport a physical bound state, the slope of the ERE as a function of k ∗ must be smallerthan the slope of the −√− k ∗ at the bound-state pole. The two slopes and associateduncertainty bands are depicted in Fig. 14 for all two-baryon channels and the two-parameterEREs obtained, demonstrating that the needed inequality is satisfied. The values of bindingmomenta used in this analysis are taken from Table VI (the d = { (0 , , , (0 , , } column).- The absence of more than one bound state with an ERE parametrization of amplitudes : None3 − .
01 0 .
00 0 . k ∗ [l . u . ] − . − . . . . k ∗ c o t δ [l . u . ] NN ( S ) − .
01 0 .
00 0 . k ∗ [l . u . ] − . − . . . . Σ N ( S ) − .
01 0 .
00 0 . k ∗ [l . u . ] − . − . . . . k ∗ c o t δ [l . u . ] ΣΣ ( S ) − .
01 0 .
00 0 . k ∗ [l . u . ] − . − . . . . ΞΣ ( S ) − .
01 0 .
00 0 . k ∗ [l . u . ] − . − . . . . ΞΞ ( S ) − .
01 0 .
00 0 . k ∗ [l . u . ] − . − . . . . k ∗ c o t δ [l . u . ] NN ( S ) − .
01 0 .
00 0 . k ∗ [l . u . ] − . − . . . . Ξ N ( S ) ×
64 :32 ×
96 :48 ×
96 : d = (0 , , d = (0 , , d = (0 , , d = (0 , , d = (0 , , d = (0 , ,
2) Two-parameter ERE: n = 1 , n = 1 states −√− k ∗ t -channel cut FIG. 13. k ∗ cot δ values as a function of the c.m. momenta k ∗ , together with bands representing the two-parameter ERE using all the energy levels (ground state n = 1 and excited states n = 2 ) in lighter yellow,or using just the ground state in darker yellow. Quantities are expressed in lattice units. of the systems analyzed exhibit more than one bound state, i.e., the ERE does not cross the −√− k ∗ curve more than once. Therefore, applying the ERE parametrization of the s -wavescattering amplitude in all channels appears to be justified.- Constrained range for ERE parameters in the presence of a bound state : If the system presentsa bound state, the ratio r/a must be smaller than 1/2 for the two-parameter ERE to crossthe −√− k ∗ function once from below, which is the condition for a physical bound state.Moreover, the ERE must cross the Z -functions corresponding to different volumes to satisfyLüscher’s quantization condition, introducing more constraints on scattering parameters.With the use of the two-dimensional χ in this work to fit the k ∗ cot δ values, the confidenceregion of the ERE parameters does not cross these prohibited areas, as was demonstrated inFig. 4.4 − .
01 0 .
00 0 . k ∗ [l . u . ] − . − . . . . k ∗ c o t δ [l . u . ] NN ( S ) − .
01 0 .
00 0 . k ∗ [l . u . ] − . − . . . . Σ N ( S ) − .
01 0 .
00 0 . k ∗ [l . u . ] − . − . . . . k ∗ c o t δ [l . u . ] ΣΣ ( S ) − .
01 0 .
00 0 . k ∗ [l . u . ] − . − . . . . ΞΣ ( S ) − .
01 0 .
00 0 . k ∗ [l . u . ] − . − . . . . ΞΞ ( S ) − .
01 0 .
00 0 . k ∗ [l . u . ] − . − . . . . k ∗ c o t δ [l . u . ] NN ( S ) − .
01 0 .
00 0 . k ∗ [l . u . ] − . − . . . . Ξ N ( S ) ×
64 :32 ×
96 :48 ×
96 : d = (0 , , d = (0 , , d = (0 , , d = (0 , , d = (0 , , d = (0 , ,
2) Two-parameter ERETangent to −√− k ∗ at − κ ( ∞ )2 −√− k ∗ t -channel cut FIG. 14. Comparison between the two-parameter ERE and the slope of −√− k ∗ at k ∗ = − κ ( ∞ )2 , where κ ( ∞ ) is taken from the d = { (0 , , , (0 , , } column of Table VI. Quantities are expressed in lattice units. Appendix B: Comparison with previous LQCD results and those obtained from low-energytheorems
A subset of the correlation functions used in this work has already been analyzed in Ref. [42], wherethe
N N ( S ) and N N ( S ) channels were studied. In the following, we present the outcome ofa careful comparison of the results obtained using both analyses, along with a comparison of theupdated scattering parameters from this work and those obtained from low-energy theorems inRef. [182].
1. Differences in the fitting strategy
The ground-state and first excited-state energies obtained in this work and those from Ref. [42]are shown in Fig. 15. While all numbers are in agreement within uncertainties, it is clear that,5
24 32 48 L [l . u . ] − . . . . . . . ∆ E [l . u . ] NN ( S ) 24 32 48 L [l . u . ] − . . . . . . . NN ( S )This work NPLQCD 15 FIG. 15. Comparison of the ground-state and first excited-state energies obtained in this work (blue circles)and from Ref. [42] (orange diamonds), labeled as NPLQCD 15. The figure shows results with statistical andsystematic uncertainties combined in quadrature. Quantities are expressed in lattice units. in general, the analysis performed in Ref. [42] led to smaller uncertainties (one exception is the
N N ( S ) first excited state with L = 32 ). That analysis consisted of: 1) taking linear combi-nations of the SP and SS correlation functions (except for the L = 48 ensemble, where only SPcorrelation functions were computed), 2) the use of the Hodges-Lehmann (HL) robust estimatorunder bootstrap resampling to estimate the ensemble-averaged correlation functions, and 3) fittingconstants to the effective-(mass) energy functions built from the combinations mentioned above. Inthe present analysis, multi-exponential fits are performed to both SP and SS correlation functionsin a correlated way (when available), using the mean under bootstrap resampling.Taking a closer look at how the statistical and systematic uncertainties are computed, it is worth w f / max( w f ) − . − . − . ∆ E [l . u . ] L = 24 w f / max( w f ) − . − . . L = 32 This work NPLQCD 15
FIG. 16. Ground-state energies for the
N N ( S ) system computed on ensembles with L = 24 (left) and L = 32 (right), sorted by their weight. The weight of each individual fit is indicated by the level oftransparency of each point (darker points have larger weight). The band shows the final result, obtainedby combing the individual points with the corresponding weight according to Eq. (5), with statistical andsystematic uncertainties combined in quadrature. To facilitate the comparison, the orange point in the rightpanel of each figure shows the result of Ref. [42], labeled as NPLQCD 15. One input (SS) Multiple inputs (SS and SP)Effective energy Corr. Effective energy Corr. - - L = M e a n
411 4 11 411 4 11 411411 4 114 11 411411 4 114 11 H L e s t i m a t o r
411 4 11 411 4 11 411411 4 114 11 411411 4 114 11 L = M e a n
411 4 11 411 4 11 411411 4 114 11 411411 4 114 11 - - H L e s t i m a t o r
411 4 11 411 4 11 411411 4 114 11 411411 4 114 11
FIG. 17. Normalized inverse covariance matrices computed for the
N N ( S ) ground state with L = 24 (top)and L = 32 (bottom) for τ ∈ [4 , l.u. using the mean and HL estimators applied to the effective-energyfunction and correlation function. examining the individual fits from all accepted time windows. These are shown in Fig. 16 for the N N ( S ) L = 24 and L = 32 ground states, sorted by their weight, w f , as defined in Eq. (4).As can be seen, there are cases for which the size of the uncertainty is similar to or smaller thanthat presented in Ref. [42]. However, the final combined uncertainty, represented by the band inFig. 16, is larger. This can be understood as using a more conservative procedure for quantifying thesystematic uncertainty, as well as a more thorough one: not only are variations of the fitting rangeconsidered, but also variations in the fitting form, including forms with multiple exponentials, seeSec. II B. Next, the implications of using the HL estimator (instead of the mean) on the individualSP and SS correlation functions are analyzed. When correlations are fully taken into account,the covariance matrix associated with the HL estimator is computed with the Median Absolute7 τ [l . u . ]0 . . . . . . ∆ E [l . u . ] SP , L = 24 4 6 8 10 12 14 16 τ [l . u . ]0 . . . . . . .
14 SP , L = 324 6 8 10 12 14 16 τ [l . u . ]0 . . . . . . ∆ E [l . u . ] SS , L = 24 4 6 8 10 12 14 16 τ [l . u . ]0 . . . . . . .
14 SS , L = 32MeanHodges-Lehmann estimator This workNPLQCD 15
FIG. 18. Comparison of the effective energy-shift plots of the SP and SS correlation functions for the
N N ( S ) L = 24 (left panel) and L = 32 (right panel) first excited states computed using the mean (darkgreen/red circles) and the HL estimator (light green/red squares, shifted horizontally for clarity). The bandsshow the results of this work and of Ref. [42], labeled as NPLQCD 15. Deviation (MAD): C ( τ, τ (cid:48) ) = Median (cid:104) ( ˜ C ( τ ) − Median [ ˜ C ( τ )])( ˜ C ( τ (cid:48) ) − Median [ ˜ C ( τ (cid:48) )]) (cid:105) , (B1)where ˜ C ( τ ) is the bootstrap ensemble computed with the HL estimator of the original correlationfunction C ( τ ) . However, in some cases the resulting covariance matrix is found not to be positivesemi-definite, and it only becomes well behaved when a single type of correlation function is used(or a linear combination of several) in the form of an effective-(mass) energy function. To illustratethis, Fig. 17 shows the normalized inverse covariance matrix, C − ( τ, τ (cid:48) ) / (cid:112) C − ( τ, τ ) C − ( τ (cid:48) , τ (cid:48) ) , forthe N N ( S ) ground state with L = 24 and L = 32 for all possible choices, i.e., HL estimatorversus mean and correlation function versus the effective-energy function.Therefore, in order to incorporate the HL estimator into the fitting strategy used here, only thefully uncorrelated covariance matrix can be used, and this leads to results which are compatiblewith the ones presented here using the mean. In Fig. 18, the effective-energy functions computedwith the mean and HL are compared for the N N ( S ) first excited states, showing agreementwithin uncertainties.To understand the ill-behaved behavior of some of the HL correlation functions, is importantto recall that baryonic correlation functions exhibit distributions that are largely non-Gaussianwith heavy tails, and the mean becomes Gaussian only in the limit of large statistics. However,8 τ [l . u . ]10 − − − − κ ( C ) / C τ [l . u . ] − . − . . . . . κ ( C ) / C × L = 24 L = 32 FIG. 19. The bootstrap estimates of the variance κ ( C ( τ )) /C ( τ ) and skewness κ ( C ( τ )) /C ( τ ) for theSS correlation functions corresponding to the N N ( S ) first excited state with L = 24 (green circles) and L = 32 (red diamonds). The L = 32 points have been shifted slightly along the τ -axis for clarity. at late times, the signal-to-noise degradation worsens, and outliers occur more frequently in thedistribution. For the L = 32 and L = 48 cases, the point at which the HL estimator gives differentresults compared with the usual estimator (mean and standard deviation), which would indicate adeviation from Gaussian behavior, occurs at a much later time compared with the maximum timeincluded in the fits using the automated fitter of this work. For the L = 24 case, the data is morenoisy than on the other two ensembles, showing non-Gaussianity at earlier times. To illustrate thedifferent behavior between the L = 24 and L = 32 ensembles, the second and third cumulants of C ( τ ) , defined as: κ n ( C ( τ )) = (cid:104) C ( τ ) n (cid:105) − n − (cid:88) m =1 (cid:18) n − m − (cid:19) κ m ( C ( τ )) (cid:10) C ( τ ) n − m (cid:11) , (B2)with n ∈ { , } , respectively, are shown in Fig. 19 for the two ensembles in the case of the N N ( S ) first excited state. Looking at the second cumulant (variance), κ , it is clear that L = 24 is morenoisy than L = 32 , and looking at the third cumulant (skewness), κ , it is clear that L = 24 deviatesfrom zero, an indication of the non-Gaussian behavior. The use of robust estimators is, therefore,questionable in this case. This is the main reason for abandoning the use of the HL estimator inthe analysis of correlation functions in the present study.
2. Differences in the scattering parameters
The 68% confidence region of the scattering parameters from a two-parameter ERE extractedin this work and in Ref. [42] are shown in Fig. 21. It can be seen that the values of the parametersobtained in the two analyses do not fully agree at the σ level, although the uncertainties arerather large. There are two significant differences between the two analysis: 1) the use of the newdefinition for the χ function (2D- χ ) in the present work, as opposed to the usual χ function(1D- χ ) used in Ref. [42], and 2) the use of the L -dependent ground-state k ∗ values in the fits toERE in the present work, instead of using only the infinite-volume extrapolated value, κ ( ∞ ) , usedin Ref. [42]. To see the effects of each, a comprehensive analysis has been performed, the resultsof which are shown in Fig. 20. Here, four different possibilities, corresponding to the types of the χ function (1D or 2D) and the use of ground-state k ∗ data ( L -dependent or extrapolated), aretested using the lowest-lying spectra obtained in Ref. [42] and those in the present work.From these tests, several interesting features are observed. First, the use of the 1D- χ , either withthe L -dependent k ∗ or the extrapolated one, is insensitive to the conditions imposed by Lüscher’s9 − .
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08 0 . a − [l . u . ] − r [l . u . ] χ using L -dependent k ∗ N N ( S ) Prohibited regions N ERE = 2: L = 24 L = 32 L = 48 L = ∞ FIG. 20. Comparison of the 68% confidence region of the scattering parameters obtained using the energylevels extracted in this work (yellow area) and from Ref. [42] (gray area) with four different analyses. Theregions include both statistical and systematic uncertainties combined in quadrature. The prohibited regionswhere the two-parameter ERE does not cross the Z -function are the crossed areas. Quantities are expressedin lattice units. − .
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08 0 . a − [l . u . ]010203040 r [l . u . ] NN ( S )Prohibited regions N ERE = 2: L = 24 L = 32 L = 48 L = ∞ This workNPLQCD 15 LO Baru et al.
NLO Baru et al.
FIG. 21. Comparison of the 68% confidence region of the scattering parameters obtained in this work (yellowarea), from Ref. [42] (gray area, labeled as NPLQCD 15), and predictions of low-energy theorems fromRef. [90] (LO and NLO results). The regions include both statistical and systematic uncertainties combinedin quadrature. The prohibited regions where the two-parameter ERE does not cross the Z -functions atgiven volumes or in the infinite-volume limit are denoted as hashed areas. Quantities are expressed in latticeunits. quantization condition, and as a result, the confidence regions of the scattering parameters couldlie on top of the prohibited regions. This is because the distance minimized in the 1D- χ is thevertical one, and not the one along the Z -function, so the ERE is not forced to cross it. Second,when the 2D- χ is used with the extrapolated k ∗ value, κ ( ∞ )2 , the only region that is avoided is theone corresponding to L = ∞ in the figures, which is expected: with the value of the pole positiongiven by Eq. (20), the function k ∗ cot δ | k ∗ = iκ ( ∞ ) equals −√− k ∗ and the ERE crosses the −√− k ∗ function, imposing the r/a < / constraint on the scattering parameters. Third, it is reassuringthat the regions obtained using the two different energy inputs, from this work or from Ref. [42],are always overlapping.Perhaps the most significant observation is that the choice of including the points in the negative k ∗ region in the fit, i.e., the infinite-volume extrapolated value of the momenta versus the L -dependent values, has far more impact on the differences observed than which χ function is used.What the new χ function does is to move the scattering parameters to the allowed region bythe Z -functions. Furthermore, with the new fitting methodology, several questions raised aboutthe validity of the ERE fits are addressed, as was presented in Appendix A. An important one isthat the updated results of this work recover the position of the bound state pole obtained via theinfinite-volume extrapolation of the energies, and do not yield a second pole near threshold, whichwould be incompatible with the use of the ERE. As a final remark, it should be noted that the datafitted to extract these parameters are highly non-Gaussian, as can be seen from the correlationbetween k ∗ and k cot δ in Fig. 2, and exhibit large uncertainties. This can be compared withthe results of Refs. [31, 41] at m π ∼ MeV, where more finite-volume energy eigenvalues, withbetter precision, could be used in the ERE fitting. As a result, it has been verified that either the L -dependent or the infinite-volume extrapolated value of k ∗ in the ERE fitting give compatiblescattering parameters.In Ref. [90], low-energy theorems [182] were used to compute the scattering parameters fromthe binding energies of the N N systems obtained in Ref. [42], and it was pointed out that therewere some tensions with the scattering parameters obtained from the LQCD data using Lüscher’s1method, i.e., those reported in Ref. [42]. Since the binding energies obtained in this work are infull agreement with those obtained in Ref. [42], the results obtained in Ref. [90] can be comparedwith the updated scattering parameters of this work. As is depicted in Fig. 21, the tension hasreduced considerably. For the two-parameter ERE results, the scattering length is now completelyconsistent with the low-energy theorem predictions, both at LO and NLO. For the effective range,since the NLO predictions of the low-energy theorems enter the prohibited region for the two-parameter ERE, the comparison may only be made with the LO results. As is seen, for both the S and S channels, the effective ranges are also in agreement (with the S state having a betteroverlap). Appendix C: On leading flavor-symmetry breaking coefficients in the EFT
Table 10 of Ref. [20] lists the SU (3) flavor-symmetry-breaking LECs c iχ for all of the two-(octet)baryon channels. These coefficients are a combination of different terms in the Lagrangian shownin Table 9 of the same reference (terms 29-40). The relations between the c iχ from Ref. [20] andthe ones in Eq. (28) introduced in the present work are presented in Table XV. Instead of the ( s +1 L J , I ) notation, the channels are labeled as ( I +12 s +1 ) for brevity, as L = 0 in all cases.In Table XVI, a list of the two-baryon channels one needs to study in order to obtain in-dependently all the LECs of this work is provided. There are 6 LO and 12 NLO symmetry-breaking coefficients that are referred to as momentum independent in this paper, as well as 6 NLOmomentum-dependent coefficients, making a total of 24 parameters that need to be constrainedin a more exhaustive study in the future. For the momentum-independent coefficients, the choiceof the systems is not unique, as there are 37 different channels that can be used to constrainonly 18 parameters (assuming SU (2) flavor symmetry and no electromagnetic interaction). Forthe momentum-dependent coefficients, no extra channels are needed besides those used for themomentum-independent coefficients. For simplicity, only channels that do not change the baryoncontent are used (e.g. Σ N → Σ N , denoted as Σ N in short). Appendix D: Supplementary figures and tables
This appendix contains all the figures omitted from the main body of the paper for ease of presenta-tion. These include the effective-mass plots of the single baryons in Fig. 22, and the effective energyand effective energy-shift plots for the two-baryon systems in Figs. 23-31. In Fig. 22, the thin hor-izontal line and the horizontal band surrounding it represent, respectively, the central value of thebaryon mass at each volume, and the associated statistical and systematic uncertainties combinedin quadrature, obtained with the fitting procedure described in Sec. II B. Similarly, in Figs. 23-31the line and the band represent, respectively, the central value of the two-baryon energy shiftscompared to non-interacting baryons at rest (bottom panels) for each volume, and the associatedstatistical and systematic uncertainties combined in quadrature.The appendix also contains the numerical results that were omitted from the main body. Theseinclude the energy shifts, ∆ E , of the two-baryon systems, the c.m. momenta, k ∗ , and the value of k ∗ cot δ for all the systems in Tables XVII-XXV. In these tables, the values in the first and secondparentheses correspond to statistical and systematic uncertainties, respectively, while those in theupper and lower parentheses are, respectively, the right and left uncertainties when the error barsare asymmetric, as is generally the case for the k ∗ cot δ values. When there is a dash sign in thetables, it indicates that the quantity k cot δ diverges due to the singularities in the Z d function.All quantities in the plots and tables are expressed in lattice units.2 TABLE XV. Comparison between the symmetry-breaking LECs of this work and those in Ref. [20] for thetwo-baryon channels for which only one c iχ appears in that reference.Channel ( I +12 s +1 ) Ref. [20] Coefficients in Eq. (28)
N N → N N ( ) c χ c χ − c χ )Λ N → Λ N ( ) c χ (4 c χ − c χ + 9 c χ − c χ − c χ + 4 c χ − c χ + c χ + 4 c χ − c χ )Λ N → Σ N ( ) − c χ c χ − c χ + 2 c χ − c χ + c χ − c χ Σ N → Σ N ( ) c χ − c χ + c χ − c χ + 3 c χ ΛΛ → ΛΛ ( ) c χ (2 c χ − c χ + 2 c χ − c χ − c χ + 4 c χ − c χ + 2 c χ − c χ + 2 c χ + 3 c χ − c χ )Ξ N → Ξ N ( ) c χ − c χ + 2 c χ + c χ − c χ ) N N → N N ( ) c χ c χ + c χ )Λ N → Λ N ( ) c χ (4 c χ + 4 c χ + 7 c χ + 7 c χ + 12 c χ + 12 c χ + 9 c χ + 9 c χ + 4 c χ + 4 c χ )Λ N → Σ N ( ) − c χ − c χ − c χ + 2 c χ + 2 c χ + 3 c χ + 3 c χ Σ N → Σ N ( ) c χ c χ + c χ + 3 c χ + 3 c χ Ξ N → Ξ N ( ) c χ c χ + 2 c χ + 2 c χ + 2 c χ + 2 c χ + 2 c χ + c χ + c χ )Ξ N → Ξ N ( ) c χ c χ + 2 c χ + c χ + c χ ) TABLE XVI. Combinations of two-baryon channels necessary to constrain independently all of the LO+NLOEFT LECs introduced in Sec. III A.Coefficient Channels ( I +12 s +1 ) c (27) ) − ΞΞ( ) c (8 s ) ΛΛ( ) + ΣΣ( ) − ) − Ξ N ( ) + ΞΣ( ) + ΞΣ( ) + ΞΞ( ) c (1) − ) + ΣΣ( ) + 8ΞΛ( ) + Ξ N ( ) − ΞΣ( ) − ΞΣ( ) − ΞΞ( ) c (10) NN ( ) + Σ N ( ) + Σ N ( ) + ΞΛ( ) − Ξ N ( ) − Ξ N ( ) − ΞΣ( ) + ΞΣ( ) − ΞΞ( ) c (10) NN ( ) − Σ N ( ) + Σ N ( ) − ΞΛ( ) + Ξ N ( ) + Ξ N ( ) + ΞΣ( ) − ΞΣ( ) + ΞΞ( ) c (8 a ) − NN ( ) + Σ N ( ) − Σ N ( ) − ΞΛ( ) − Ξ N ( ) + Ξ N ( ) + ΞΣ( ) − ΞΣ( ) − ΞΞ( ) c χ ΛΛ( ) − NN ( ) − Σ N ( ) − Σ N ( ) + Σ N ( ) + ΣΣ( ) − ΞΛ( ) − Ξ N ( )+ Ξ N ( ) − Ξ N ( ) + Ξ N ( ) + ΞΣ( ) + ΞΣ( ) + ΞΣ( ) + ΞΞ( ) − ΞΞ( ) c χ − ΛΛ( ) − NN ( ) − Σ N ( ) + Σ N ( ) + Σ N ( ) − ΣΣ( ) − ΞΛ( ) + Ξ N ( )+ Ξ N ( ) + Ξ N ( ) + Ξ N ( ) + ΞΣ( ) − ΞΣ( ) + ΞΣ( ) + ΞΞ( ) + ΞΞ( ) c χ NN ( ) − Σ N ( ) + Σ N ( ) − Σ N ( ) − ΞΛ( ) + Ξ N ( )+ Ξ N ( ) + ΞΣ( ) − ΞΣ( ) − ΞΣ( ) + ΞΞ( ) + ΞΞ( ) c χ NN ( ) − Σ N ( ) − Σ N ( ) − Σ N ( ) − ΞΛ( ) + Ξ N ( )+ Ξ N ( ) + ΞΣ( ) + ΞΣ( ) − ΞΣ( ) + ΞΞ( ) − ΞΞ( ) c χ NN ( ) − Σ N ( ) + Σ N ( ) + Σ N ( ) − ΞΛ( ) + ΞΛ( ) + Ξ N ( )+ Ξ N ( ) + ΞΣ( ) − ΞΣ( ) − ΞΣ( ) − ΞΣ( ) − ΞΞ( ) + ΞΞ( ) c χ NN ( ) − Σ N ( ) − Σ N ( ) + Σ N ( ) + ΞΛ( ) + ΞΛ( ) + Ξ N ( )+ Ξ N ( ) − ΞΣ( ) − ΞΣ( ) + ΞΣ( ) − ΞΣ( ) − ΞΞ( ) − ΞΞ( ) c χ NN ( ) − Σ N ( ) + Σ N ( ) + Σ N ( ) − ΞΛ( ) + ΞΛ( ) + Ξ N ( ) + Ξ N ( ) − Ξ N ( ) − ΞΣ( ) − ΞΣ( ) − ΞΣ( ) + ΞΣ( ) − ΞΞ( ) + ΞΞ( ) c χ NN ( ) − Σ N ( ) − Σ N ( ) + Σ N ( ) + ΞΛ( ) + ΞΛ( ) + Ξ N ( ) − Ξ N ( ) − Ξ N ( ) + ΞΣ( ) − ΞΣ( ) + ΞΣ( ) + ΞΣ( ) − ΞΞ( ) − ΞΞ( ) c χ − ΛΛ( ) + NN ( ) − Σ N ( ) − Σ N ( ) + Σ N ( ) + ΣΣ( ) + ΞΛ( ) − ΞΛ( ) + Ξ N ( )+ Ξ N ( ) − Ξ N ( ) − Ξ N ( ) − ΞΣ( ) + ΞΣ( ) + ΞΣ( ) + ΞΞ( ) − ΞΞ( ) c χ ΛΛ( ) + NN ( ) − Σ N ( ) + Σ N ( ) + Σ N ( ) − ΣΣ( ) − ΞΛ( ) − ΞΛ( ) − Ξ N ( )+ Ξ N ( ) + Ξ N ( ) − Ξ N ( ) − ΞΣ( ) − ΞΣ( ) + ΞΣ( ) + ΞΞ( ) + ΞΞ( ) c χ − ΛΛ( ) − NN ( ) − Σ N ( ) + Σ N ( ) − Σ N ( ) − ΣΣ( ) − ΞΛ( ) + Ξ N ( )+ Ξ N ( ) + Ξ N ( ) + Ξ N ( ) + ΞΣ( ) − ΞΣ( ) + ΞΣ( ) + ΞΞ( ) + ΞΞ( ) c χ ΛΛ( ) − NN ( ) − Σ N ( ) − Σ N ( ) − Σ N ( ) + ΣΣ( ) − ΞΛ( ) − Ξ N ( )+ Ξ N ( ) − Ξ N ( ) + Ξ N ( ) + ΞΣ( ) + ΞΣ( ) + ΞΣ( ) + ΞΞ( ) − ΞΞ( ) ˜ c (27) ΞΞ( ) ˜ c (10) NN ( ) ˜ c (8 s ) ΛΛ( ) + 2ΣΣ( ) − Ξ N ( ) ˜ c (10) ΞΞ( ) ˜ c (1) − ΛΛ( ) − ΣΣ( ) + Ξ N ( ) ˜ c (8 a ) Ξ N ( ) τ [l . u . ]0 . . . . . C B ( τ ) [l . u . ] N : 24 × τ [l . u . ] N : 32 × τ [l . u . ] N : 48 × τ [l . u . ]0 . . . . C B ( τ ) [l . u . ] Λ : 24 × τ [l . u . ] Λ : 32 × τ [l . u . ] Λ : 48 × τ [l . u . ]0 . . . . C B ( τ ) [l . u . ] Σ : 24 × τ [l . u . ] Σ : 32 × τ [l . u . ] Σ : 48 × τ [l . u . ]0 . . . . . C B ( τ ) [l . u . ] Ξ : 24 × τ [l . u . ] Ξ : 32 × τ [l . u . ] Ξ : 48 × FIG. 22. Single-baryon EMPs for the SP (blue squares) and SS (orange diamonds) source-sink combinations.The SS points have been slightly shifted along the horizontal axis for clarity. . . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . R BB ( τ ) [l . u . ] . . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . R BB ( τ ) [l . u . ] . . . C BB ( τ ) [l . u . ] τ [l . u . ] − . . . . . . R BB ( τ ) [l . u . ] . . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . . . R BB ( τ ) [l . u . ] . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . R BB ( τ ) [l . u . ] . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . . . . R BB ( τ ) [l . u . ] NN ( S ) : d = (0 , , , × NN ( S ) : d = (0 , , , × NN ( S ) : d = (0 , , , × NN ( S ) : d = (0 , , , × NN ( S ) : d = (0 , , , × NN ( S ) : d = (0 , , , × FIG. 23. The effective energy plots (upper panel of each segment) and the effective energy-shift plots (lowerpanel of each segment) for the
N N ( S ) system at rest (left panels) and with boost d = (0 , , (rightpanels) for the SP (blue circles) and SS (orange diamonds) source-sink combinations. . . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . R BB ( τ ) [l . u . ] . . . . C BB ( τ ) [l . u . ] τ [l . u . ] − . . . . R BB ( τ ) [l . u . ] . . . . C BB ( τ ) [l . u . ] τ [l . u . ] − . . . . . R BB ( τ ) [l . u . ] . . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . R BB ( τ ) [l . u . ] . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . . R BB ( τ ) [l . u . ] . . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . . . R BB ( τ ) [l . u . ] Σ N ( S ) : d = (0 , , , × N ( S ) : d = (0 , , , × N ( S ) : d = (0 , , , ×
96 Σ N ( S ) : d = (0 , , , × N ( S ) : d = (0 , , , × N ( S ) : d = (0 , , , × FIG. 24. The effective energy plots (upper panel of each segment) and the effective energy-shift plots (lowerpanel of each segment) for the Σ N ( S ) system at rest (left panels) and with boost d = (0 , , (rightpanels) for the SP (blue circles) and SS (orange diamonds) source-sink combinations. . . . . C BB ( τ ) [l . u . ] τ [l . u . ] − . . . . . R BB ( τ ) [l . u . ] . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . R BB ( τ ) [l . u . ] . . . C BB ( τ ) [l . u . ] τ [l . u . ] − . . . . . R BB ( τ ) [l . u . ] . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . . R BB ( τ ) [l . u . ] . . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . . R BB ( τ ) [l . u . ] . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . . . R BB ( τ ) [l . u . ] ΣΣ ( S ) : d = (0 , , , × S ) : d = (0 , , , × S ) : d = (0 , , , ×
96 ΣΣ ( S ) : d = (0 , , , × S ) : d = (0 , , , × S ) : d = (0 , , , × FIG. 25. The effective energy plots (upper panel of each segment) and the effective energy-shift plots (lowerpanel of each segment) for the
ΣΣ( S ) system at rest (left panels) and with boost d = (0 , , (right panels)for the SP (blue circles) and SS (orange diamonds) source-sink combinations. . . . . C BB ( τ ) [l . u . ] τ [l . u . ] − . . . . . R BB ( τ ) [l . u . ] . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . R BB ( τ ) [l . u . ] . . . C BB ( τ ) [l . u . ] τ [l . u . ] − . . . . . R BB ( τ ) [l . u . ] . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . . R BB ( τ ) [l . u . ] . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . R BB ( τ ) [l . u . ] . . . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . . . R BB ( τ ) [l . u . ] ΞΣ ( S ) : d = (0 , , , × S ) : d = (0 , , , × S ) : d = (0 , , , ×
96 ΞΣ ( S ) : d = (0 , , , × S ) : d = (0 , , , × S ) : d = (0 , , , × FIG. 26. The effective energy plots (upper panel of each segment) and the effective energy-shift plots (lowerpanel of each segment) for the
ΞΣ( S ) system at rest (left panels) and with boost d = (0 , , (right panels)for the SP (blue circles) and SS (orange diamonds) source-sink combinations. . . . C BB ( τ ) [l . u . ] τ [l . u . ] − . . . . . R BB ( τ ) [l . u . ] . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . R BB ( τ ) [l . u . ] . . . . . C BB ( τ ) [l . u . ] τ [l . u . ] − . . . . . R BB ( τ ) [l . u . ] . . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . . R BB ( τ ) [l . u . ] . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . R BB ( τ ) [l . u . ] . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . . . R BB ( τ ) [l . u . ] ΞΞ ( S ) : d = (0 , , , × S ) : d = (0 , , , × S ) : d = (0 , , , ×
96 ΞΞ ( S ) : d = (0 , , , × S ) : d = (0 , , , × S ) : d = (0 , , , × FIG. 27. The effective energy plots (upper panel of each segment) and the effective energy-shift plots (lowerpanel of each segment) for the
ΞΞ( S ) system at rest (left panels) and with boost d = (0 , , (right panels)for the SP (blue circles) and SS (orange diamonds) source-sink combinations. . . . . C BB ( τ ) [l . u . ] τ [l . u . ] − . . . . . R BB ( τ ) [l . u . ] . . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . R BB ( τ ) [l . u . ] . . . . . C BB ( τ ) [l . u . ] τ [l . u . ] − . . . . . . R BB ( τ ) [l . u . ] . . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . . . R BB ( τ ) [l . u . ] . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . R BB ( τ ) [l . u . ] . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . . . . R BB ( τ ) [l . u . ] NN ( S ) : d = (0 , , , × NN ( S ) : d = (0 , , , × NN ( S ) : d = (0 , , , × NN ( S ) : d = (0 , , , × NN ( S ) : d = (0 , , , × NN ( S ) : d = (0 , , , × FIG. 28. The effective energy plots (upper panel of each segment) and the effective energy-shift plots (lowerpanel of each segment) for the
N N ( S ) system at rest (left panels) and with boost d = (0 , , (rightpanels) for the SP (blue circles) and SS (orange diamonds) source-sink combinations. . . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . R BB ( τ ) [l . u . ] . . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . R BB ( τ ) [l . u . ] . . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . . R BB ( τ ) [l . u . ] . . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . R BB ( τ ) [l . u . ] . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . R BB ( τ ) [l . u . ] . . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . . . . R BB ( τ ) [l . u . ] Σ N ( S ) : d = (0 , , , × N ( S ) : d = (0 , , , × N ( S ) : d = (0 , , , ×
96 Σ N ( S ) : d = (0 , , , × N ( S ) : d = (0 , , , × N ( S ) : d = (0 , , , × FIG. 29. The effective energy plots (upper panel of each segment) and the effective energy-shift plots (lowerpanel of each segment) for the Σ N ( S ) system at rest (left panels) and with boost d = (0 , , (rightpanels) for the SP (blue circles) and SS (orange diamonds) source-sink combinations. . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . R BB ( τ ) [l . u . ] . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . R BB ( τ ) [l . u . ] . . C BB ( τ ) [l . u . ] τ [l . u . ] − . . . . . R BB ( τ ) [l . u . ] . . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . . R BB ( τ ) [l . u . ] . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . R BB ( τ ) [l . u . ] . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . . . R BB ( τ ) [l . u . ] ΞΞ ( S ) : d = (0 , , , × S ) : d = (0 , , , × S ) : d = (0 , , , ×
96 ΞΞ ( S ) : d = (0 , , , × S ) : d = (0 , , , × S ) : d = (0 , , , × FIG. 30. The effective energy plots (upper panel of each segment) and the effective energy-shift plots (lowerpanel of each segment) for the
ΞΞ( S ) system at rest (left panels) and with boost d = (0 , , (right panels)for the SP (blue circles) and SS (orange diamonds) source-sink combinations. . . . . C BB ( τ ) [l . u . ] τ [l . u . ] − . . . . R BB ( τ ) [l . u . ] . . . C BB ( τ ) [l . u . ] τ [l . u . ] − . . . . . R BB ( τ ) [l . u . ] . . . . . C BB ( τ ) [l . u . ] τ [l . u . ] − . . . R BB ( τ ) [l . u . ] . . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . . R BB ( τ ) [l . u . ] . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . . R BB ( τ ) [l . u . ] . . . . . C BB ( τ ) [l . u . ] τ [l . u . ] . . . . R BB ( τ ) [l . u . ] Ξ N ( S ) : d = (0 , , , × N ( S ) : d = (0 , , , × N ( S ) : d = (0 , , , ×
96 Ξ N ( S ) : d = (0 , , , × N ( S ) : d = (0 , , , × N ( S ) : d = (0 , , , × FIG. 31. The effective energy plots (upper panel of each segment) and the effective energy-shift plots (lowerpanel of each segment) for the Ξ N ( S ) system at rest (left panels) and with boost d = (0 , , (rightpanels) for the SP (blue circles) and SS (orange diamonds) source-sink combinations. TABLE XVII. The values of the energy shift ∆ E , the c.m. momentum k ∗ , and k ∗ cot δ for the N N ( S ) channel.Ensemble Boost vector State ∆ E [l.u.] k ∗ [l.u.] k ∗ cot δ [l.u.] ×
64 (0 , , n = 1 − . − . − . (+13)(+25)( − − n = 2 0 . . - (0 , , n = 1 0 . − . − . (+23)(+51)( − − n = 2 0 . . − . (+32)(+65)( − − ×
96 (0 , , n = 1 − . − . − . (+29)(+27)( − − n = 2 0 . . . (+3 . . − . − . (0 , , n = 1 0 . − . − . (+18)(+21)( − − n = 2 0 . . - ×
96 (0 , , n = 1 − . − . − . (+13)(+06)( − − n = 2 0 . . . (+0 . . − . − . (0 , , n = 1 0 . − . − . (+38)(+80)( − − n = 2 0 . . -TABLE XVIII. The values of the energy shift ∆ E , the c.m. momentum k ∗ , and k ∗ cot δ for the Σ N ( S ) channel.Ensemble Boost vector State ∆ E [l.u.] k ∗ [l.u.] k ∗ cot δ [l.u.] ×
64 (0 , , n = 1 − . − . − . (+14)(+35)( − − n = 2 0 . . - (0 , , n = 1 0 . − . − . (+24)(+60)( − − n = 2 0 . . − . (+0 . . − . − . ×
96 (0 , , n = 1 − . − . − . (+26)(+25)( − − n = 2 0 . . . (+1 . . − . − . (0 , , n = 1 0 . − . − . (+14)(+16)( − − n = 2 0 . . - ×
96 (0 , , n = 1 − . − . − . (+36)(+43)( − − n = 2 0 . . . (+11)(+11)( − − (0 , , n = 1 0 . − . − . (+42)(+66)( − − n = 2 0 . . - TABLE XIX. The values of the energy shift ∆ E , the c.m. momentum k ∗ , and k ∗ cot δ for the ΣΣ ( S ) channel.Ensemble Boost vector State ∆ E [l.u.] k ∗ [l.u.] k ∗ cot δ [l.u.] ×
64 (0 , , n = 1 − . − . − . (+13)(+17)( − − n = 2 0 . . . (+0 . . − . − . (0 , , n = 1 0 . − . − . (+15)(+39)( − − n = 2 0 . . - ×
96 (0 , , n = 1 − . − . − . (+14)(+20)( − − n = 2 0 . . . (+31)(+71)( − − (0 , , n = 1 0 . − . − . (+17)(+25)( − − n = 2 0 . . - ×
96 (0 , , n = 1 − . − . − . (+17)(+18)( − − n = 2 0 . . . (+66)(+47)( − − (0 , , n = 1 0 . − . − . (+19)(+24)( − − n = 2 0 . . . (+64)(+72)( − −−
96 (0 , , n = 1 − . − . − . (+17)(+18)( − − n = 2 0 . . . (+66)(+47)( − − (0 , , n = 1 0 . − . − . (+19)(+24)( − − n = 2 0 . . . (+64)(+72)( − −− TABLE XX. The values of the energy shift ∆ E , the c.m. momentum k ∗ , and k ∗ cot δ for the ΞΣ ( S ) channel.Ensemble Boost vector State ∆ E [l.u.] k ∗ [l.u.] k ∗ cot δ [l.u.] ×
64 (0 , , n = 1 − . − . − . (+09)(+08)( − − n = 2 0 . . . (+25)(+44)( − − (0 , , n = 1 0 . − . − . (+13)(+19)( − − n = 2 0 . . - ×
96 (0 , , n = 1 − . − . − . (+10)(+12)( − − n = 2 0 . . . (+83)(+98)( − − (0 , , n = 1 0 . − . − . (+09)(+13)( − − n = 2 0 . . . (+16)(+25)( − − ×
96 (0 , , n = 1 − . − . − . (+20)(+08)( − − n = 2 0 . . . (+65)(+41)( − − (0 , , n = 1 0 . − . − . (+22)(+11)( − − n = 2 0 . . . (+11)(+08)( − − TABLE XXI. The values of the energy shift ∆ E , the c.m. momentum k ∗ , and k ∗ cot δ for the ΞΞ ( S ) channel.Ensemble Boost vector State ∆ E [l.u.] k ∗ [l.u.] k ∗ cot δ [l.u.] ×
64 (0 , , n = 1 − . − . − . (+08)(+14)( − − n = 2 0 . . . (+13)(+43)( − − (0 , , n = 1 0 . − . − . (+09)(+13)( − − n = 2 0 . . - ×
96 (0 , , n = 1 − . − . − . (+08)(+11)( − − n = 2 0 . . . (+38)(+58)( − − (0 , , n = 1 0 . − . − . (+07)(+13)( − − n = 2 0 . . . (+04)(+17)( − − ×
96 (0 , , n = 1 − . − . − . (+07)(+08)( − − n = 2 0 . . . (+22)(+27)( − − (0 , , n = 1 0 . − . − . (+08)(+10)( − − n = 2 0 . . . (+27)(+37)( − −−
96 (0 , , n = 1 − . − . − . (+07)(+08)( − − n = 2 0 . . . (+22)(+27)( − − (0 , , n = 1 0 . − . − . (+08)(+10)( − − n = 2 0 . . . (+27)(+37)( − −− TABLE XXII. The values of the energy shift ∆ E , the c.m. momentum k ∗ , and k ∗ cot δ for the N N ( S ) channel.Ensemble Boost vector State ∆ E [l.u.] k ∗ [l.u.] k ∗ cot δ [l.u.] ×
64 (0 , , n = 1 − . − . − . (+17)(+20)( − − n = 2 0 . . . (+1 . . − . − . (0 , , n = 1 0 . − . − . (+31)(+55)( − − n = 2 0 . . - ×
96 (0 , , n = 1 − . − . − . (+15)(+13)( − − n = 2 0 . . . (+97)(+98)( − − (0 , , n = 1 0 . − . − . (+20)(+14)( − − n = 2 0 . . . (+30)(+75)( − − ×
96 (0 , , n = 1 − . − . − . (+18)(+12)( − − n = 2 0 . . . (+13)(+11)( − − (0 , , n = 1 0 . − . − . (+19)(+12)( − − n = 2 0 . . . (+12)(+09)( − − TABLE XXIII. The values of the energy shift ∆ E , the c.m. momentum k ∗ , and k ∗ cot δ for the Σ N ( S ) channel.Ensemble Boost vector State ∆ E [l.u.] k ∗ [l.u.] k ∗ cot δ [l.u.] ×
64 (0 , , n = 1 0 . . - n = 2 0 . . . (+42)(+40)( − − (0 , , n = 1 0 . − . - n = 2 0 . . . (+13)(+09)( − − ×
96 (0 , , n = 1 0 . . − . (+27)(+26)( − − n = 2 0 . . . (+40)(+24)( − − (0 , , n = 1 0 . . − . (+29)(+11)( − − n = 2 0 . . . (+14)(+08)( − − ×
96 (0 , , n = 1 0 . . − . (+37)(+31)( − − n = 2 0 . . . (+14)(+13)( − − (0 , , n = 1 0 . . − . (+39)(+44)( − − n = 2 0 . . − . (+11)(+18)( − −−
96 (0 , , n = 1 0 . . − . (+37)(+31)( − − n = 2 0 . . . (+14)(+13)( − − (0 , , n = 1 0 . . − . (+39)(+44)( − − n = 2 0 . . − . (+11)(+18)( − −− TABLE XXIV. The values of the energy shift ∆ E , the c.m. momentum k ∗ , and k ∗ cot δ for the ΞΞ ( S ) channel.Ensemble Boost vector State ∆ E [l.u.] k ∗ [l.u.] k ∗ cot δ [l.u.] ×
64 (0 , , n = 1 0 . . - n = 2 0 . . − . (+09)(+05)( − − (0 , , n = 1 0 . − . - n = 2 0 . . − . (+05)(+07)( − − ×
96 (0 , , n = 1 0 . . − . (+08)(+06)( − − n = 2 0 . . − . (+57)(+53)( − − (0 , , n = 1 0 . . - n = 2 0 . . − . (+52)(+44)( − − ×
96 (0 , , n = 1 0 . . - n = 2 0 . . - (0 , , n = 1 0 . − . - n = 2 0 . . - TABLE XXV. The values of the energy shift ∆ E , the c.m. momentum k ∗ , and k ∗ cot δ for the Ξ N ( S ) channel.Ensemble Boost vector State ∆ E [l.u.] k ∗ [l.u.] k ∗ cot δ [l.u.] ×
64 (0 , , n = 1 − . − . − . (+09)(+09)( − − n = 2 0 . . . (+29)(+27)( − − (0 , , n = 1 0 . − . − . (+09)(+19)( − − n = 2 0 . . - ×
96 (0 , , n = 1 − . − . − . (+06)(+11)( − − n = 2 0 . . . (+17)(+35)( − − (0 , , n = 1 0 . − . − . (+06)(+08)( − − n = 2 0 . . . (+30)(+45)( − − ×
96 (0 , , n = 1 − . − . − . (+06)(+10)( − − n = 2 0 . . - (0 , , n = 1 0 . − . − . (+06)(+11)( − − n = 2 0 . . − . (+21)(+27)( − −−