PHYSICAL REVIEW D 103, 054508 (2021)
Low-energy scattering and effective interactions of two baryons at mπ ∼ 450 MeV from lattice quantum chromodynamics
Marc Illa ,1,* Silas R. Beane ,2 Emmanuel Chang, Zohreh Davoudi ,3,4 William Detmold ,5 David J. Murphy,5 Kostas Orginos,6,7 Assumpta Parreño ,1 Martin J. Savage ,8 Phiala E. Shanahan ,5 Michael L. Wagman,9 and Frank Winter 7
(NPLQCD Collaboration)
1Departament de Física Qu`antica i Astrofísica and Institut de Ci`encies del Cosmos, Universitat de Barcelona, Martí Franqu`es 1, E08028-Spain 2Department of Physics, University of Washington, Seattle, Washington 98195-1560, USA 3Department of Physics and Maryland Center for Fundamental Physics, University of Maryland, College Park, Maryland 20742, USA 4RIKEN Center for Accelerator-based Sciences, Wako 351-0198, Japan 5Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 6Department of Physics, College of William and Mary, Williamsburg, Virginia 23187-8795, USA 7Jefferson Laboratory, 12000 Jefferson Avenue, Newport News, Virginia 23606, USA 8Institute for Nuclear Theory, University of Washington, Seattle, Washington 98195-1550, USA 9Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA
(Received 17 October 2020; accepted 1 February 2021; published 19 March 2021)
The interactions between two octet baryons are studied at low energies using lattice quantum chromodynamics (LQCD) with larger-than-physical quark masses corresponding to a pion mass of mπ ∼ 450 MeV and a kaon mass of mK ∼ 596 MeV. The two-baryon systems that are analyzed range from strangeness S ¼ 0 to S ¼ −4 and include the spin-singlet and triplet NN, ΣN (I ¼ 3=2), and ΞΞ states, the spin-singlet ΣΣ (I ¼ 2) and ΞΣ (I ¼ 3=2) states, and the spin-triplet ΞN (I ¼ 0) state. The corresponding s-wave scattering phase shifts, low-energy scattering parameters, and binding energies when applicable are extracted using Lüscher’s formalism. While the results are consistent with most of the systems being bound at this pion mass, the interactions in the spin-triplet ΣN and ΞΞ channels are found to be repulsive and do not support bound states. Using results from previous studies of these systems at a larger pion mass, an extrapolation of the binding energies to the physical point is performed and is compared with available experimental values and phenomenological predictions. The low-energy coefficients in pionless effective field theory (EFT) relevant for two-baryon interactions, including those responsible for SUð3Þ flavor- symmetry breaking, are constrained. The SUð3Þ flavor symmetry is observed to hold approximately at the chosen values of the quark masses, as well as the SUð6Þ spin-flavor symmetry, predicted at large Nc.A remnant of an accidental SUð16Þ symmetry found previously at a larger pion mass is further observed. The SUð6Þ-symmetric EFT constrained by these LQCD calculations is used to make predictions for two-baryon systems for which the low-energy scattering parameters could not be determined with LQCD directly in this study, and to constrain the coefficients of all leading SUð3Þ flavor-symmetric interactions, demonstrating the predictive power of two-baryon EFTs matched to LQCD.
DOI: 10.1103/PhysRevD.103.054508
I. INTRODUCTION Hyperons (Y) are expected to appear in the interior of *[email protected] neutron stars [1], and unless the strong interactions between hyperons and nucleons (N) are sufficiently repulsive, the Published by the American Physical Society under the terms of equation of state (EoS) of dense nuclear matter will be the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to softer than for purely nonstrange matter, leading to corre- the author(s) and the published article’s title, journal citation, spondingly lower maximum values for neutron star masses. and DOI. Funded by SCOAP3. While experimental data on scattering cross sections in the
2470-0010=2021=103(5)=054508(56) 054508-1 Published by the American Physical Society MARC ILLA et al. PHYS. REV. D 103, 054508 (2021) majority of the YN channels are scarce, there are reason- Building upon our previous works [24–31], we present ably precise constraints on the interactions in the ΛN further studies which constrain hypernuclear forces in channel from scattering and hypernuclear spectroscopy nature by direct calculations starting from the underlying experiments [2,3], and they indicate that the interactions theory of the strong interactions, quantum chromodynamics Λ in this channel are attractive. Given that the baryon is (QCD). To this end, the numerical technique of lattice QCD lighter than the other hyperons, it is likely the most (LQCD) is used to obtain information on the low-energy abundant hyperon in the interior of neutron stars. spectra and scattering on two-baryon systems, which can be However, models of the EoS including Λ baryons and ΛN used to constrain EFTs or phenomenological models of attractive interactions [4] predict a maximum neutron two-baryon interactions. In recent years, LQCD has allowed star mass that is below the maximum observed mass at 2 M – 1 a wealth of observables in nuclear physics, from hadronic ⊙ [5 9]. Several remedies have been suggested to spectra and structure to nuclear matrix elements [32–34], solve this problem, known in the literature as the “hyperon to be calculated directly from interactions of quarks and puzzle” [11–13]. For example, if hyperons other than the Λ gluons, albeit with uncertainties that are yet to be fully baryon (such as Σ baryons) are present in the interior of neutron stars and the interactions in the corresponding YN controlled. In the context of constraining hypernuclear and YY channels are sufficiently repulsive, the EoS would interactions, LQCD is a powerful theoretical tool because become more stiff [14,15]. Another suggestion is that the lowest-lying hyperons are stable when only strong repulsive interactions in the YNN, YYN, and YYY channels interactions are included in the computation, circumventing the limitations faced by experiments on hyperons and may render the EoS stiff enough to produce a 2 M⊙ neutron star [4,14,16–18]. Repulsive density-dependent inter- hypernuclei. Nonetheless, LQCD studies in the multibaryon actions in systems involving the Λ and other hyperons sector require large computing resources as there is an have also been suggested, along with the possibility of a inherent signal-to-noise degradation present in the correla- phase transition to quark matter in the interior of neutron tion functions of baryons [25,26,35–38], among other issues stars; see Refs. [11–13] for recent reviews. Given the as discussed in a recent review [34]. Consequently, most scarcity or complete lack of experimental data on YN and studies of two-baryon systems to date [24,26–31,39–51] YY scattering and all three-body interactions involving have used larger-than-physical quark masses to expedite hyperons, SUð3Þ flavor symmetry is used to constrain computations, and only recently have results at the physical effective field theories (EFTs) and phenomenological values of the quark masses emerged [52–55], making it meson-exchange models of hypernuclear interactions. In possible to directly compare with experimental data [56]. this way, quantities in channels for which experimental data The existing studies are primarily based on two distinct exist can be related via symmetries to those in channels approaches. In one approach, the low-lying spectra of two which lack such phenomenological constraints [19,20].For baryons in finite spatial volumes are determined from the example, the lowest-order effective interactions in several time dependence of Euclidean correlation functions com- channels with strangeness S ∈ f−2; −3; −4g were con- puted with LQCD, and are then converted to scattering strained using experimental data on pp phase shifts and the amplitudes at the corresponding energies through the use of Σþp cross section in the same SUð3Þ representation in Lüscher’sformula[57,58] or its generalizations [59–75].In the framework of chiral EFT (χEFT) in Refs. [21–23]. another approach, nonlocal potentials are constructed based However, only a few of the SUð3Þ-breaking low-energy 2 on the Bethe-Salpeter wave functions determined from coefficients (LECs) of the EFT could be constrained [23]. LQCD correlation functions, and are subsequently used To date, the knowledge of these interactions in nature in the Lippmann-Schwinger equation to solve for scattering remains unsatisfactory, demanding more direct theoretical ’ 3 phase shifts [55,76,77]. Given that Lüscher s formalism approaches. is model-independent below inelastic thresholds, it is this approach that is used in the present study as the basis to constrain scattering amplitudes and their low-energy para- metrizations in a number of two-(octet)baryon channels 1Very recently, the gravitational wave signal GW190814, with strangeness S ∈ f0; −1; −2; −3; −4g. originated from the merger of a 23 M⊙ black hole and a 2.6 M⊙ compact object, was reported [10], where the nature While LQCD studies at unphysical values of the quark of the compact object is a subject of discussion. If this compact masses already shed light on the understanding of (hyper) object was a neutron star, it would have been the most massive nuclear and dense-matter physics, a full account of all one known, imposing a mass-limit constraint very difficult to systematic uncertainties, including precise extrapolations to fulfill for the majority of existing nuclear EoS models. 2 the physical quark mass, is required to further impact Note that SUð3Þf χEFT is less convergent than for two flavors. phenomenology. Additionally, LQCD results for scattering 3Other observational means to constrain these interactions, amplitudes can be used to better constrain the low-energy such as radius measurements of neutron stars, their thermal and interactions within given phenomenological models and structural evolution, and the emission of gravitational waves in applicable EFTs. In the case of exact SUð3Þ flavor hot and rapidly rotating newly born neutron stars, can be used to indirectly probe the strangeness content of dense matter and symmetry and including only the lowest-lying octet bary- provide complementary constraints on models of hypernuclear ons, there are six two-baryon interactions at leading order interactions [11]. (LO) in pionless EFT [78,79] that can be constrained by the
054508-2 LOW-ENERGY SCATTERING AND EFFECTIVE INTERACTIONS … PHYS. REV. D 103, 054508 (2021) s-wave scattering lengths in two-baryon scattering [80]. predictions for the values of the coefficients that appear, in LQCD has been used in Ref. [31] to constrain the corre- the limit of large Nc, in the SUð6Þ spin-flavor Lagrangian at sponding LECs of these interactions by computing the LO (Sec. III B). The main results of this work are summa- s-wave scattering parameters of two baryons at an SUð3Þ rized in Sec. IV. In addition, several appendixes are presented flavor-symmetric point with mπ ∼ 806 MeV. Strikingly, the to supplement the conclusions of this study. Appendix A first evidence of a long-predicted SUð6Þ spin-flavor sym- contains an analysis of our results in view of the consistency metry in nuclear and hypernuclear interactions in the limit of checks of Refs. [88,89], demonstrating that the checks are unambiguously passed. Appendix B presents an exhaustive a large number of colors (Nc) [81] was observed in that study, along with an accidental SUð16Þ symmetry. This extended comparison between the results obtained in this work and previous results presented in Ref. [42] for the two-nucleon symmetry has been suggested in Ref. [82] to support the channels using the same LQCD correlation functions, as well conjecture of entanglement suppression in nuclear and as with the predictions of the low-energy theorems analyzed hypernuclear forces at low energies, pointing to intriguing in Ref. [90]. Appendix C includes relations among the LECs aspects of strong interactions in nature. of the three-flavor EFT Lagrangian of Ref. [20] and the ones The objective of this paper is to extend our previous used in the present work, as well as a recipe to access the full study to quark masses that are closer to their physical ∼450 set of leading symmetry-breaking coefficients from future values, corresponding to a pion mass of MeV and a studies of a more complete set of two-baryon systems. Last, kaon mass of ∼596 MeV, and further to study these SU 3 Appendix D contains figures and tables that are omitted from systems in a setting with broken ð Þ flavor symmetry the main body of the paper for clarity of presentation. as is the case in nature. The present study provides new constraints that allow preliminary extrapolations to physical quark masses to be performed, and complements II. LOWEST-LYING ENERGIES AND previous independent LQCD studies at nearby quark LOW-ENERGY SCATTERING PARAMETERS – masses [24,26 29,39,42,45,46,83,84]. In particular, pre- A. Details of the LQCD computation dictions for the binding energies of ground states in a number of YN and YY channels based on the results of the This work continues, revisits, and expands upon the current work and those of Ref. [31] at larger quark masses study of Ref. [42]. In particular, the same ensembles of are consistent with experiments and phenomenological QCD gauge-field configurations that were used in Ref. [42] results where they exist. Our LQCD results are used to to constrain the low-lying spectra and scattering amplitudes constrain the leading SUð3Þ symmetry-breaking coeffi- of spin-singlet and spin-triplet two-nucleon systems at a cients in pionless EFT. This EFT matching enables the pion mass of ∼450 MeV are used here. The same con- exploration of large-Nc predictions, pointing to the validity figurations have also been used to study properties of of SUð6Þ spin-flavor symmetry at this pion mass as well, baryons and light nuclei at this pion mass, including the rate and revealing a remnant of an accidental SUð16Þ symmetry of the radiative capture process np → dγ [91], the response that was observed at a larger pion mass in Ref. [31]. of two-nucleon systems to large magnetic fields [92], the Strategies to make use of the QCD-constrained EFTs to magnetic moments of octet baryons [93], the gluonic advance the ab initio many-body studies of larger hyper- structure of light nuclei [94], and the gluon gravitational nuclear isotopes and dense nuclear matter are beyond the form factors of hadrons [95–97]. For completeness, a short scope of this work. Nevertheless, the methods applied in summary of the technical details is presented here and a – Refs. [85 87] to connect the results of LQCD calculations more detailed discussion can be found in Ref. [42]. to higher-mass nuclei can also be applied in the hyper- The LQCD calculations are performed with nf ¼ 2 þ 1 nuclear sector using the results presented in this work. quark flavors, with the Lüscher-Weisz gauge action [98] This paper is organized as follows. Section II presents a and a clover-improved quark action [99] with one level of summary of the computational details (Sec. II A), followed stout smearing (ρ ¼ 0.125) [100]. The lattice spacing is by the results for the lowest-lying energies of two-baryon b ¼ 0.1167ð16Þ fm [101]. The strange quark mass is tuned systems from LQCD correlation functions, along with a to its physical value, while the degenerate light (up and description of the method used to obtain these spectra m s down)-quark masses produce a pion of mass π ¼ (Sec. II B), a determination of the -wave scattering 450 5 m 596 6 parameters in the two-baryon channels that are studied, ð Þ MeV and a kaon of mass K ¼ ð Þ MeV. along with the formalism used to extract the scattering Ensembles at these parameters with three different volumes are used. Using the two smallest volumes with dimensions amplitude (Sec. II C), and finally the binding energies of 3 3 the bound states that are identified in various channels, 24 × 64 and 32 × 96, two different sets of correlation including an extrapolation to the physical point (Sec. II D). functions are produced, with sink interpolating operators Section III discusses the constraints that these results that are either pointlike or smeared with 80 steps of a impose on the low-energy coefficients of the next-to- gauge-invariant Gaussian profile with parameter ρ ¼ 3.5 at leading order (NLO) pionless EFT Lagrangian, including the quark level. In both cases, the source interpolating some of the SUð3Þ flavor-symmetry breaking terms operators are smeared with the same parameters. These two (Sec. III A). This is followed by a discussion of the types of correlation functions are labeled SP and SS,
054508-3 MARC ILLA et al. PHYS. REV. D 103, 054508 (2021)
TABLE I. Parameters of the gauge-field ensembles used in this work. L and T are the spatial and temporal β b m dimensions of the hypercubic lattice, is related to the strong coupling, is the lattice spacing, lðsÞ is the bare light N N (strange) quark mass, cfg is the number of configurations used, and src is the total number of sources computed. For more details, see Ref. [42].
3 L × T β bml bms b [fm] L [fm] T [fm] mπLmπTNcfg Nsrc 243 × 64 6.1 −0.2800 −0.2450 0.1167(16) 2.8 7.5 6.4 17.0 4407 1.16 × 106 323 × 96 6.1 −0.2800 −0.2450 0.1167(16) 3.7 11.2 8.5 25.5 4142 3.95 × 105 483 × 96 6.1 −0.2800 −0.2450 0.1167(16) 5.6 11.2 12.8 25.5 1047 6.8 × 104
2π respectively. For the largest ensemble with dimensions pi ¼ L n with n ∈ fð0; 0; 0Þ; ð0; 0; 1Þg. Therefore, P ¼ 483 96 2π 4 × , only SP correlation functions are produced for L d, with d ∈ fð0; 0; 0Þ; ð0; 0; 2Þg. Additionally, two computational expediency. Table I summarizes the param- baryon correlation functions with back-to-back momenta eters of these ensembles. were generated at the sink, with momenta p1 ¼ −p2 ¼ Correlation functions are constructed by forming baryon 2π L n. This latter choice provides interpolating operators for blocks at the sink [102], the two-baryon system that primarily overlap with states that X are unbound in the infinite-volume limit, providing a con- 0 ijk ip·x ðf1Þ;i BB ðp; τ; x0Þ¼ e Si ðx; τ; x0Þ venient means to identify excited states as well. The con- x struction of the correlation functions continues by forming a 0 0 Sðf2Þ;j x; τ; x Sðf3Þ;k x; τ; x wB ; fully antisymmetrized local quark-level wave function at the × j ð 0Þ k ð 0Þ i0j0k0 ð1Þ location of the source, with quantum numbers of the two- f ;n0 baryon system of interest. Appropriate indices from the Sð Þ f ∈ u; d; s where n is a quark propagator with flavor f g baryon blocks at the sink are then contracted with those at n; n0 ∈ and with combined spin-color indices the source, in a way that is dictated by the quark-level wave 1; …;N N N 4 f s cg, where s ¼ is the number of spin function; see Refs. [30,102] for more detail. The contraction N 3 components and c ¼ is the number of colors. The codes used to produce the correlation functions in this study wB weights i0j0k0 are tensors that antisymmetrize and collect are the same as those used to perform the contractions for the the terms needed to have the quantum numbers of the larger class of interpolating operators used in our previous baryons B ∈ fN;Λ; Σ; Ξg. The interpolating operators for studies of the SUð3Þ flavor-symmetric spectra of nuclei the single-baryon systems studied in this work are local, and hypernuclei up to A ¼ 5 [30], and two-baryon scattering i.e., include no covariant derivatives. Explicitly, [31,41,42].5 In this study, correlation functions for nine different two- 1 baryon systems have been computed, ranging from strange- ˆ abc a b a b c 2s 1 Nμ μ μ ðxÞ¼ϵ pffiffiffi ½uμ ðxÞdμ ðxÞ − dμ ðxÞuμ ðxÞ uμ ðxÞ; S 0 −4 þ L ;I 1 2 3 2 1 2 1 2 3 ness ¼ to . Using the notation ð J Þ, where s is the total spin, L is the orbital momentum, J is the 1 ˆ abc a b a b c I Λμ μ μ ðxÞ¼ϵ pffiffiffi ½uμ ðxÞdμ ðxÞ − dμ ðxÞuμ ðxÞ sμ ðxÞ; total angular momentum, and is the isospin, the 1 2 3 2 1 2 1 2 3 systems are Σˆ x ϵabcua x ub x sc x ; μ1μ2μ3 ð Þ¼ μ1 ð Þ μ2 ð Þ μ3 ð Þ Ξˆ x ϵabcsa x sb x uc x ; 1 3 μ1μ2μ3 ð Þ¼ μ1 ð Þ μ2 ð Þ μ3 ð Þ ð2Þ S ¼ 0∶ NNð S0;I ¼ 1Þ;NNð S1;I ¼ 0Þ; 1 3 3 3 S ¼ −1∶ ΣN S0;I ¼ ; ΣN S1;I ¼ ; where μi denote spin indices and a, b, c denote color indices 2 2 [103]. Only the upper-spin components in the Dirac spinor 1 3 μ p x S ¼ −2∶ ΣΣð S0;I ¼ 2Þ; ΞNð S1;I ¼ 0Þ; basis are used, requiring only specificqffiffi i indices: ↑ð Þ¼ ˆ ˆ þ 2 ˆ ˆ 1 3 N121ðxÞ, Λ↑ðxÞ¼Λ121, Σ ðxÞ¼ ½Σ112ðxÞ − Σ121ðxÞ , S −3∶ ΞΣ S ;I ; qffiffi ↑ 3 ¼ 0 ¼ 2 0 2 ˆ ˆ − and Ξ↑ðxÞ¼ ½Ξ112ðxÞ − Ξ121ðxÞ . The neutron, Σ , 1 3 3 S ¼ −4∶ ΞΞð S0;I ¼ 1Þ; ΞΞð S1;I ¼ 0Þ: and Ξ− operators are obtained by simply interchanging u ↔ d in the expressions above. The sum over the sink position x 4 d 0; 0; 2 in Eq. (1) projects the baryon blocks to well-defined three- For the rest of the paper, ¼ð Þ will be denoted as p d ¼ð0; 0; 2Þ for brevity. momentum . In particular, two-baryon correlation functions 5The same code was generalized to enable studies of np → dγ were generated with total momentum P ¼ p1 þ p2, where pi [91], proton-proton fusion [104], and other electroweak proc- is the three-momentum of the ith baryon taking the values esses, as reviewed in Ref. [34].
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Under strong interactions, these channels do not mix with correlation functions, given its computational-resource other two-baryon channels or other hadronic states below requirement, but progress is being made. In Ref. [50],a three-particle inelastic thresholds. In the limit of exact partial set of two-baryon scattering interpolating operators SUð3Þ flavor symmetry, the states belong to irreducible were used to study the two-nucleon and H-dibaryon channels representations (irreps) of SUð3Þ: 27 (all the singlet states), with results that generally disagreed with previous works 10 (triplet NN), 10 (triplet ΣN and ΞΞ), and 8a (triplet ΞN). [27,30,49]. Investigations continue to understand and resolve In the rest of this work, the isospin label will be dropped for the observed discrepancies [31,33,34,88,109–111]. For the simplicity. present study, in which only up to two types of interpolating operators were computed, no variational analysis could be B. Low-lying finite-volume spectra of two baryons performed. Instead, we have developed a robust automated fitting methodology to sample and combine fit range and The two-point correlation functions constructed in the model selection choices for uncertainty quantification. previous section have spectral representations in Euclidean Given that the correlation functions are only evaluated at spacetime. Explicitly, the correlation function C ˆ ˆ 0 ðτ; dÞ O;O a finite number of times and with finite precision, to fit Oˆ formed using the source (sink) interpolating operators Eq. (3) the spectral representation is truncated to a relatively ˆ 0 (O ) can be written as small number of exponentials and fitted in a time range τ ; τ τ X f min maxg, where max is set by a threshold value deter- 2πid·x=L ˆ 0 ˆ † C ˆ ˆ 0 τ; d e O x; τ O 0; 0 τ O;O ð Þ¼ h ð Þ ð Þi mined by examining the signal-to-noise ratio, and min is x 2; τ − τ X chosen to take values in the interval ½ max plateau . Here, 0 −EðiÞτ τ 5 ¼ ZiZi e ; ð3Þ plateau ¼ is chosen to be the minimum length of the fitting i window (numbers are expressed in units of the lattice spacing). A scan over all possible fitting windows is treated EðiÞ where all quantities are expressed in lattice units. as a means to quantify the associated systematic uncertainty. ðiÞ 0 is the energy of the ith eigenstate jE i, Zi (Zi)isan χ2 pffiffiffiffi With a fixed window, a correlated -function is minimized Z V 0 Oˆ 0; 0 EðiÞ Z0 Z0Z E i ∈ 0; 1; …;e overlapffiffiffiffi factor defined as i ¼ h j ð Þj i ( i ¼ to obtain the fit parameters i i and i for f g, p ˆ i 3 Vh0jO0ð0; 0ÞjEð Þi), and V ¼ L . The boost-vector where e þ 1 is the number of exponentials in a given fit form. dependence of the energies, states, and overlap factors is Variable projection techniques [112,113] are used to obtain implicit. The lowest-lying energies of the one- and two- the value of the overlap factors for a given energy, since they baryon systems required for the subsequent analyses can be appear linearly in COˆ ;Oˆ 0 ðτ; dÞ. Furthermore, given the finite extracted by fitting the correlation functions to this form. To statistical sampling of correlation functions, shrinkage tech- reliably discern the first few exponents given the discrete τ niques [114] are used to better estimate the covariance values and the finite statistical precision of the computations matrix. The number of excited states included in the fit is is a challenging task. In particular, a well-known problem in decided via the Akaike information criterion [115].The the study of baryons with LQCD is the exponential degra- confidence intervals of the parameters are estimated via dation of the signal-to-noise ratio in the correlation function the bootstrap resampling method. For a fit to be included in as the source-sink separation time increases—an issue that the set of accepted fits (later used to extract the fit parameters worsens as the masses of the light quarks approach their and assess the resulting uncertainties), several checks must physical values. First highlighted by Parisi [35] and Lepage χ2=N be passed, including dof being smaller than 2, and [36], and studied in detail for light nuclei in Refs. [25,26],it different optimization algorithms leading to consistent was later shown that this problem is related to the behavior of results for the parameters (within a tolerance)—see the complex phase of the correlation functions [38,105]. Ref. [116] for further details on this and other checks. The Another problem that complicates the study of multibaryon accepted fits are then combined to give the final result for the systems is the small excitation gaps in the finite-volume mean value of the energy, spectrum that lead to significant excited-state contributions to correlation functions. To overcome these issues, sophis- X E ω E ; ticated methods have been developed to analyze the corre- ¼ f f ð4Þ f lation functions, such as Matrix Prony [37] and the generalized pencil-of-function [106] techniques, as well as signal-to-noise optimization techniques [107]. Ultimately, a with weights ωf that are chosen to be the following large set of single- and multibaryon interpolating operators combination of the p-value, pf, and the uncertainty of each with the desired quantum numbers must be constructed to fit, δEf, provide a reliable variational basis to isolate the lowest-lying energy eigenvalues via solving a generalized eigenvalue p =δE2 ω P f f ; problem [108], as is done in the mesonic sector [75]. Such an f ¼ p =δE2 ð5Þ approach is not yet widely applied to the study of two-baryon f0 f0 f0
054508-5 MARC ILLA et al. PHYS. REV. D 103, 054508 (2021) see Ref. [117] for a Bayesian framework. Here, the indices f, differences of multistate fit results is convenient, in f0 run over all the accepted fits. The statistical uncertainty is particular for automated fit range sampling, since the defined as that of the fit with the highest weight, while the number of excited states can be varied independently for systematic uncertainty is defined as the average difference one- and two-baryon correlation functions, unlike fits to the between the weighted mean value and each of the accepted ratio in Eq. (8). Consistent results were obtained via fitting fits: the ratio in Eq. (8) in the allowed time regions. sXffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The effective mass plots (EMPs) for the single-baryon δE δEf∶max½fwfg ; δE w Ef − E 2: correlation functions, and for each of the ensembles stat ¼ sys ¼ fð Þ ð6Þ f studied in the present work, are displayed in Fig. 22 of Appendix D. The bands shown in the figures indicate the It should be noted that instead of fitting to the correlation baryon mass which results from the fitting strategy function, the effective energy function can be employed, explained above, with the statistical and systematic uncer- derived from the logarithm of the ratio of correlation functions tainties included, and the corresponding numerical values at displaced times, listed in Table II. The table also shows the baryon masses extrapolated to infinite volume, obtained by fitting the 1 C ˆ ˆ 0 ðτ; dÞ τ→∞ O;O ð0Þ masses in the three different volumes to the following form: COˆ ;Oˆ 0 ðτ; d; τJÞ¼ log ! E ; ð7Þ τJ COˆ ;Oˆ 0 ðτ þ τJ; dÞ e−mπ L MðVÞ m L Mð∞Þ c ; τJ B ð π Þ¼ B þ B ð10Þ where is a nonzero integer that is introduced to improve the mπL extraction of Eð0Þ (for a detailed study, see Ref. [37]). ð∞Þ Consistent results are obtained when either correlation where MB and cB are the two fit parameters. This form functions or the effective energy functions are used as input. incorporates LO volume corrections to the baryon masses in In order to identify the shift in the finite-volume energies heavy-baryon chiral perturbation theory (HBχPT) [118].As of two baryons compared with noninteracting baryons, the is evident from the mπL values listed in Table I, the volumes following ratio of two-baryon and single-baryon correla- used are large enough to ensure small volume dependence in tion functions can be formed: the single-baryon masses.6 This is supported by the obser- Mð∞Þ C τ; d vation that the value B obtained for each baryon is Oˆ ;Oˆ 0 ð Þ V R τ; d B1B2 B1B2 ; Mð Þ B1B2 ð Þ¼ ð8Þ compatible with all the finite-volume results B .In COˆ ;Oˆ 0 ðτ; dÞCOˆ ;Oˆ 0 ðτ; dÞ M ∼ 1226 M ∼ 1313 M ∼ B1 B1 B2 B2 physical units, N MeV, Λ MeV, Σ 1346 MeV, and MΞ ∼ 1414 MeV. While the Λ baryon is not with an associated effective energy-shift function, relevant to subsequent analysis of the two-baryon systems studied in this work, the centroid of the four octet-baryon 1 RB B ðτ; dÞ R τ; d; τ 1 2 : B1B2 ð JÞ¼ log ð9Þ masses is used to define appropriate units for the EFT LECs; τJ RB B τ τJ; d 1 2 ð þ Þ hence MΛ is reported for completeness. The results for the two-baryon energy shifts are shown in Constant fits to ratios of correlation functions can be used 7 ð0Þ ð0Þ Fig. 1. For display purposes, the effective energy-shift to obtain energy shifts ΔE ¼ E − m1 − m2 (where m1 functions, defined in Eq. (9), are shown in Figs. 23–31 of and m2 are the masses of baryons B1 and B2, respectively), requiring time ranges such that both the two-baryon and the Appendix D, along with the corresponding two-baryon single-baryon correlation functions are described by a effective-energy functions, defined in Eq. (7). The asso- – single-state fit. However, if both correlation functions are ciated numerical values are listed in Tables XVII XXV of 8 – not in their ground states, cancellations may occur between the same appendix. In each subfigure of Figs. 23 31,two excited states (including the finite-volume states that would correlation functions are displayed: the one yielding the correspond to elastic scattering states in the infinite 6 −m L −4 volume), either in correlation function or in ratios of For the smallest volume e π =mπL ∼ 10 and cB are of correlation functions, producing a “mirage plateau” [88]. Oð1Þ but consistent with zero within uncertainties, e.g., for the nucleon, cB ¼ 3ð4Þð7Þ l:u: Despite this issue, as demonstrated in Ref. [109], our 7 previous results, such as those in Refs. [30,31] are argued to The channels within the figures/tables are sorted according to the SUð3Þ irrep they belong to in the limit of exact flavor be free of this potential issue (similar discussions can be 27 10 10 8 0 symmetry, ordered as , , , and a, and within each irrep found in Ref. [111]). To determine ΔEð Þ in this work, the according to their strangeness, from the largest to the smallest. two-baryon and single-baryon correlation functions are fit 8Future studies with a range of values of the lattice spacing will to multiexponential forms (which account for excited be needed to extrapolate the results of two-baryon studies to the states) within the same fitting range, and afterwards the continuum limit. Nonetheless, the use of an improved lattice action in this study suggests that the discretization effects may be energy shifts are computed at the bootstrap level, in such a mild, and the associated systematic uncertainty, which has not way that the correlations between the different correlation been reported in the values in this paper, may not be significant at functions are taken into account. The use of correlated the present level of precision.
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TABLE II. The values of the masses of the octet baryons. The first uncertainty is statistical, while the second is systematic. Quantities are expressed in lattice units (l.u.).
Ensemble MN [l.u.] MΛ [l.u.] MΣ [l.u.] MΞ [l.u.] 243 × 64 0.7261(08)(15) 0.7766(07)(13) 0.7959(07)(10) 0.8364(07)(08) 323 × 96 0.7258(05)(08) 0.7765(05)(06) 0.7963(05)(06) 0.8362(05)(05) 483 × 96 0.7250(06)(12) 0.7761(05)(09) 0.7955(06)(07) 0.8359(08)(08) ∞ 0.7253(04)(08) 0.7763(04)(06) 0.7959(04)(05) 0.8360(05)(05) lowest energy (labeled as n ¼ 1 in Tables XVII–XXV) to the two baryons having different momenta, e.g., having corresponds to having both baryons at rest or, if boosted, back-to-back momenta or one baryon at rest and the other with the same value of the momentum, and the one yielding with nonzero momentum. While the first case (n ¼ 1) a higher energy (labeled as n ¼ 2 in the tables) corresponds couples primarily to the ground state, the latter (n ¼ 2)is
FIG. 1. Summary of the energy shifts extracted from LQCD correlationpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi functionsp forffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi all two-baryon systems studied in this work, 2 2 2 2 2 2 together with the noninteracting energy shifts defined as ΔE ¼ m1 þjp1j þ m2 þjp2j − m1 − m2, where jp1j ¼jp2j ¼ 0 2 2 2π 2 corresponds to systems that are at rest (continuous line), jp1j ¼jp2j ¼ðL Þ corresponds to systems which are either boosted or are 2 2 4π 2 unboosted but have back-to-back momenta (dashed line), and jp1j ¼ 0 and jp2j ¼ðL Þ correspond to boosted systems where only one baryon has nonzero momentum (dash-dotted line). The points with no boost have been shifted slightly to the left, and the ones with boosts have been shifted to the right for clarity. Quantities are expressed in lattice units.
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−1 1 2 2 2 9 found to have a small overlap onto the ground state and where r ¼ γˆ ðn − αdÞ and α ¼ 2 ½1 þðm1 − m2Þ=E , gives access to the first excited state directly. with m1 and m2 being the masses of the two baryons. The As a final remark, it should be noted that the single- factor γˆ−1 acting on a vector u modifies the parallel baryon masses and the energies extracted for the two- component with respect to d, while leaving the perpendicular −1 −1 nucleon states within the present analysis are consistent component invariant, i.e., γˆ u ¼ γ u þ u⊥. Convenient 1σ k within with the results of Ref. [42], obtained with the expressions have been derived to exponentially accelerate the same set of data but using different fitting strategies. numerical evaluation of the function in Eq. (13) [64,120– Despite this overall consistency, the uncertainties of the 122], and the following expression is used in the present two-nucleon energies in the present work are generally analysis: larger compared with those reported in Ref. [42] for the channels where results are available in that work. The 2 2 ex X e−jrj reason lies in a slightly more conservative systematic Zd 1;x2 −γπex2 ffiffiffiffiffiffi 00½ ¼ þp r 2 −x2 uncertainty analysis employed here. The comparison 4π n j j Z 2 between the results of this work and that of Ref. [42] is 1 tx X 2 2 π e −π jγˆmj 2 discussed extensively in Appendix B. γ dt cos 2παm·d e t 2tx : þ 2 t3=2 ð Þ þ 0 m≠0 C. Low-energy scattering phase shifts ð14Þ and effective-range parameters Below three-particle inelastic thresholds, Lüscher’s The values of k cot δ at given k 2 values are shown for quantization condition [57,58] provides a means to extract all two-baryon systems in Fig. 2, and the associated the infinite-volume two-baryon scattering amplitudes from numerical values are listed in Tables XVII–XXV of the energy eigenvalues of two-baryon systems obtained Appendix D. The validity of Lüscher’s quantization con- from LQCD calculations, e.g., those presented in Sec. II B. dition must be verified in each channel, in particular in This condition holds if the range of interactions is smaller those that have exhibited anomalously large ranges, such as L 3 than (half of) the spatial extent of the cubic volume, , and ΣNð S1Þ, in previous calculations. The consistency −m L the corrections to this condition scale as e π for the two- between solutions to Lüscher’s condition and the finite- baryon systems. Such corrections are expected to be small volume Hamiltonian eigenvalue equation using a LO EFT m L 3 in the present work given the π values in Table I. The potential was established in Ref. [29] for the ΣNð S1Þ quantization conditions are those used in Refs. [31,42]:in channel and at values of the quark masses (mπ ∼ 389 MeV) the case of spin-singlet states, only the s-wave limit of the close to those of the current analysis. The conclusion of 3 full quantization condition is considered. For coupled S1 − Ref. [29], therefore, justifies the use of Lüscher’s quanti- 3 D1 states, in which the Blatt-Biedenharn parametrization zation condition in the current work for this channel. [119] of the scattering matrix can be used, only the α-wave The energy dependence of k cot δ can be parametrized approximation of the quantization condition is considered by an effective range expansion (ERE) below the t-channel [74]. In both cases, and with denoting the (s-wave or α- cut [123–125],10 wave) phase shift by δ, the condition can be written as [63] 1 1 2 4 6 d 2 k cot δ ¼ − þ rk þ Pk þ Oðk Þ; ð15Þ k cot δ ¼ 4πc00ðk ; LÞ; ð11Þ a 2 where k is the center-of-mass (c.m.) relative momentum of where a is the scattering length, r is the effective range, and each baryon, d is the total c.m. momentum in units of P is the leading shape parameter. These parameters can be d 2π=L, and clm is a kinematic function related to Lüscher’s constrained by fitting k cot δ values obtained from the use d 2 Z-function, Zlm: of Lüscher’s quantization condition as a function of k .To 2 pffiffiffiffiffiffi this end, one could use a one-dimensional choice of the χ 4π 2π l−2 function, minimizing the vertical distance between the cd ðk 2; LÞ¼ Zd ½1; ðk L=2πÞ2 ; ð12Þ lm γL3 L lm fitted point and the function, with γ ¼ E=E being the relativistic gamma factor. Here, E X k δ − f a−1;r;P;k 2 2 χ2 a−1;r;P ½ð cot Þi ð i Þ ; and E are the energies of the system in the laboratory and ð Þ¼ 2 ð16Þ σi c.m. frames, respectively. The three-dimensional zeta- i function is defined as 9Here α should not be confused with α-wave mentioned above. X r lY r 10Since the pion is the lightest hadron that can be exchanged Zd s; x2 j j lmð Þ ; lm½ ¼ r2 − x2 s ð13Þ between any of the two baryons considered in the present study, n ð Þ k m =2 t-cut ¼ π .
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FIG. 2. k cot δ values as a function of the squared c.m. momentum k 2 for all two-baryon systems studied in this work. The darker uncertainty bands are statistical, while the lighter bands show the statistical and systematic uncertainties combined in quadrature. The d 2 kinematic functions clmðk ; LÞ, given by Eq. (12), are also shown as continuous and dashed lines. Quantities are expressed in lattice units.
11 −1 2 2 2 2 2 2 where fða ;r;P;k Þ corresponds to the ERE para- ki and ðk cot δÞi, σi ¼½δðk cot δÞi þ½δki , with δx metrization given by the right-hand side of Eq. (15), and the being the mid-68% confidence interval of the quantity x. 2 2 sum runs over all extracted pairs of fki ; ðk cot δÞig, where The uncertainty on the fki ; ðk cot δÞig pair can be under- the compound index i counts data points for different stood by recalling that each pair is a member of a bootstrap boosts, n values of the level, and different volumes. Each ensemble with the distribution obtained in the previous step contribution is weighted by an effective variance that of the analysis. To generate the distribution of the scattering 2 results from the combination of the uncertainty in both parameters, pairs of fki ; ðk cot δÞig are randomly selected from each bootstrap ensemble and are used in Eq. (16) to −1 11 obtain a new set of fa ;r;Pg parameters. This procedure The inverse scattering length can be constrained far more N N precisely compared with the scattering length itself given that a−1 is repeated times, where is chosen to be equal to the 2 samples can cross zero in the channels considered. As a result, in number of bootstrap ensembles for fki ; ðk cot δÞig. This the following all dependencies on a enter via a−1. produces an ensemble of N values of fit parameters
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−1 fa ;r;Pg, from which the central value and the associated pstatesffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi above thepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi scattering threshold have c.m. energies 2 2 2 2 uncertainty in the parameters can be determined (median m1 þ k þ m2 þ k , with the c.m. momenta roughly and mid-68% intervals are used for this purpose). scaling with the volume as k 2 ∼ ð2πjnj=LÞ2. With the Alternatively, one can use a two-dimensional choice of minimum value of jnj2 used in this work (jnj2 ¼ 1), only χ2 12 k δ the function. Knowing that cot values must lie the states from the ensemble with L ¼ 48 are expected to along the Z-function, as can be seen from Eq. (11) and t 4π2=482