EEPJ manuscript No. (will be inserted by the editor)
REVIEW OF THE Λ (1405) A CURIOUS CASE OF A STRANGE-NESS RESONANCE
Maxim Mai , a The George Washington University, Washington, DC 20052, USA
Abstract.
During its long-lasting history, the Λ (1405) has become abenchmark for our understanding of the SU(3) hadron dynamics. Start-ing with its theoretical prediction and later experimental verification,until the most recent debates on the existence of the second broad pole,it emerged as a fruitful research area sparking many theoretical and ex-perimental developments.This review intends to provide the reader with the current status ofresearch on the Λ (1405)-resonance, reflecting on historical, experimen-tal and theoretical developments. A common database for experimentalresults and a comparison of most recent theoretical approaches will beprovided in the last two parts of this manuscript. Key words.
Resonances – Strangeness – Unitarity – Baryons
Excited states of strongly interacting particles form a non-trivial and highly popu-lated spectrum. Many features of this hadronic spectrum can be understood studyingthe excitations of three (for baryons) or two (for mesons) constituent quarks [1, 2].This picture is, however, incomplete, leading to, e.g., the so-called missing resonanceproblem or puzzling relative mass ordering between some states. Quantum Chromo-dynamics (QCD) emerged as the field theory of strong interaction successfully passingall tests for nearly half a century. Sophisticated methods have been developed to per-form hadron spectroscopy from QCD leading to many valuable insights, e.g., on thespectrum of baryons [3]. The excited spectrum, however, still contains riddles relatedto the existence of exotic states, gluonic degrees of freedom and interplay of thosewith with multi-hadron dynamics. It is exactly this type of riddles which challengesour understanding of strong interaction and may ultimately lead to a deeper insightinto it. A prominent example of this is the enigmatic Λ (1405) – a I ( J P ) = 0(1 / − )baryonic resonance of strangeness S = −
1. It is the matter of the present review toepitomize the history and current status of our understanding of the nature of thisresonance. Previous reviews on related topics can be found in Refs. [4–8] and in thereview section of Ref. [9]. a e-mail: [email protected] a r X i v : . [ nu c l - t h ] S e p Will be inserted by the editor
This review is organized in three major parts. First, an overview will be given in
Sec-tion 2 , including historical remarks, current values of Λ (1405) resonance parametersas well as the impact of this research on nuclear and other areas of physics. In Sec-tion 3 phenomenological constraints will be summarized.
Section 4 will be devotedto an overview, description and comparison of theoretical approaches. Λ (1405) The history of the Λ (1405) began not long after the initiation of the first large ex-perimental programs on production of kaons in 1950’s, see, e.g., Ref. [4]. In the latterreview on strong interaction, J. J. Sakurai spends long time discussing possible mecha-nisms and controversy related to the resonant structure observed previously by Dalitzand Tuan [10]. In particular, two of four solutions of their K -matrix formulation con-strained by the available experimental data at that time, exhibited resonance-likebehavior below the K − p threshold. Indeed, two years later a resonant structure wasconfirmed in the hydrogen bubble chamber experiments [11, 12] at 1405 MeV in themass ( πΣ ) plots of K − p → Σ .. π .. π .. reactions. In parallel, K − interactions in emul-sion showed peaking behavior in the πΣ mass spectra at the same energy [13].Similar experimental efforts led also to discoveries of further strangeness S = − Σ (1385) I ( J P ) = 1(3 / + ), Λ (1520) I ( J P ) = 0(3 / − ) and others,see Ref. [5] for an overview of early experimental efforts. Thus, the Λ (1405) gained itsname and a permanent position (since 1963 [14]) in the tables of particle data group(PDG). Notably is also that the determination of the spin-parity quantum numberwas precluded for a long time by experimental limitations. This was finally overcomein the studies of photo-induced reactions by the CLAS collaboration [15], directlyconfirming the J P = (1 / − ) hypothesis. Besides discrete quantum numbers such as parity or spin, the universal parameterof a stable hadron is its mass. In the case of unstable states, the latter becomes acomplex valued number quantified by the pole position of the S -matrix analyticallyextrapolated of to the complex energies. The corresponding Riemann surface consistsof one physical and multiple unphysical Riemann sheets . The poles associated witha resonance can only lie on the unphysical sheets, and are typically located on the oneconnected most closely to the physical one. This is because the physical informationfrom experimental measurements or results of Lattice QCD constrain the S -matrixonly along the real energy axis. An example of the scattering amplitude extrapolatedto the second Riemann sheet is presented in Fig. 1 .For very narrow resonances the continuation to the complex energies can be avoidedby approximating the complex-valued pole position by z R ≈ ( M R , − Γ/
2) with thelatter both quantities (resonance mass and width) estimated directly from the exper-imental line-shape. However, in the simple case of 2 → Λ (1405), which is well below the production threshold of the initial¯ KN pair used to conduct scattering experiments. Thus, one is left with two choices: Each Riemann sheet spans over the whole complex plane and is connected analyticallyto the next sheets along the cuts. In the simple case of 2 → N , for N being the number of 2-particle thresholds.ill be inserted by the editor 3 .
25 1 .
30 1 .
35 1 .
40 1 . R e W C M S [ G e V ] . (cid:1)(cid:2)(cid:3)(cid:4) (cid:1)(cid:2)(cid:5) (cid:1)(cid:2)(cid:5)(cid:4) (cid:1)(cid:2)(cid:6) (cid:1)(cid:2)(cid:6)(cid:4)(cid:7)(cid:2)(cid:7)(cid:7) - (cid:7)(cid:2)(cid:7)(cid:4) - (cid:7)(cid:2)(cid:1)(cid:7) (cid:8)(cid:9) (cid:1) (cid:1)(cid:2)(cid:3) [ (cid:10)(cid:9)(cid:11) ] (cid:1) (cid:2) (cid:1) (cid:1) (cid:2) (cid:3) [ (cid:3) (cid:4) (cid:5) ] (cid:1)(cid:2)(cid:3)(cid:4)(cid:1)(cid:2)(cid:3)(cid:5)(cid:1)(cid:2)(cid:3)(cid:6)(cid:1)(cid:2)(cid:3)(cid:7)(cid:1)(cid:2)(cid:3)(cid:3)(cid:7)(cid:8)(cid:1)(cid:8)(cid:8)(cid:1)(cid:7)(cid:8) (cid:1) (cid:1)(cid:2)(cid:3) [ (cid:9)(cid:10)(cid:11) ] (cid:1) [ (cid:1) (cid:2) ] - . - . I m W C M S [ G e V ] | T I =0¯ KN → ¯ KN | Fig. 1.
An example of an analytic continuation of the scattering amplitude (solution 4 fromRef. [16]) to the second Riemann sheet connected to the real energy-axis between the πΣ and ¯ KN thresholds (red shaded area). Gray lines depict the position of the πΣ and ¯ KN thresholds in the isospin symmetric limit. Red line shows the behavior of the amplitude forreal energies, which is constrained by the experimental data as depicted in the bottom rightfigure (c.f. Fig. 3 ). The top right inset shows the contour plot of the same Riemann sheet,which is frequently used to visualize pole structure (c.f.
Figs. 2 and 7 ). (1) Use solely the scattering data above the ¯ KN threshold to constrain the 2 → Λ (1405). Or (2), reduce the invariant mass of thefinal meson-baryon pair (e.g., πΣ ) by introducing one or more additional particlesin the final state. The lineshape with respect to the invariant mass (meson-baryonpair) allows then to extract mass and width of the Λ (1405) approximately as dis-cussed above. Evidently, both these choices have their advantages and are thereforecontained in the most recent PDG tables [9] referred to as “pole positions” and “ex-trapolations below ¯ KN threshold” for (1), and “production experiment” for (2). Adepiction of all quoted results is presented in Fig. 2 . It shall be noted that due tothe intricacy of the theoretical description of the n -body dynamics the latter casegives access only to the approximative quantities ( M R , − Γ/ Λ (1405) would require further development of theoretical many bodytools. For most recent developments of such methods see Refs. [17–19] and referencestherein. Starting with its prediction [10] and later experimental verification, the Λ (1405) ap-peared as a critical test for many theoretical tools. In chronological order three rep-resentative examples are: (1) The early debates about the attractive/repulsive natureof the (anti)kaon-nucleon interaction and the interplay of Yukawa and hyperchargecurrent coupling, as discussed in Ref. [4]; (2) The investigations of radiative decay Will be inserted by the editor � - � � � � π + Σ - π - Σ + π � Σ � Production experimentsExtrapolation below KN Complex pole position ���� ���� ���� ���� ���� ���� ������������������� � [ ��� ] Γ [ � � � ] Fig. 2.
Collection of results for mass vs. width of the Λ (1405) as quoted by the PDG [9],while being separated in three groups. The “pole positions” width and mass have been deter-mined from the complex pole position of the narrow pole via z R = ( M, Γ/
2) for comparison. of Λ (1405) showing the importance of q ¯ q dynamics, see, e.g., Refs. [20, 21]; (3)The surprising observation of the double-pole structure of a dynamically generated Λ (1405), discussed in Section 4 .Besides being a critical test of our understanding of the SU(3) dynamics of QCD, thestrong attraction of the ¯ KN system has a far reaching practical impact on other areasof nuclear physics. One example is the application to Λ b → J/ψΛ (1405) decay [22] andsimilar processes with the finals state interaction dominated by the non-perturbativemeson-baryon dynamics [23]. Another example is the investigation of and search forthe ¯ K -nuclei – ¯ KN N , ¯
KKN , ¯
KN N N , etc., being part of large experimental pro-grams such as, e.g., FINUDA@DAΦNE [24–26], DISTO@Saclay [27–30]. For moredetails on these experimental programs and relation to the theoretical predictions,see reviews [6, 7].In a broader context, the search for ¯ K -nuclei relates to the exploration of the in-medium properties of anti-kaons and strange nuclear matter [31–33]. One naturalapplication of this is the study of equation of state of neutron stars (NS) in relationto the strangeness, see the comprehensive review [7]. In a nutshell, the motivationfor such investigations lies in the fact that compressed to multiples of nuclear matterdensities, the core of neutron stars provides more than enough dense environmentfor appearance of kaon condensates [34–37], hyperons or more extreme scenarios ofstrange quark matter [38]. However, the path between microscopic theory of hadroninteractions (QCD or EFTs thereof) and properties of neutron stars is not an easyone and several challenges need to be overcome. For examples on EFT based cal-culations the reader is referred to Refs. [39–42] and for multikaon systems on thelattice [43, 44]. Currently, it is believed that – if manifested – the strange degreesof freedom (hyperons or kaon condensates) soften the equation of state of neutronstars [35, 36, 45, 46]. While such an effect was preferable to explain earlier astrophys-ical determination of M NS ( R NS ) relation (see, e.g., Ref [35]), it is at odds with morerecent observations [47, 48]. Still, the fact remains that strange degrees of freedom cansizably alter the equation of state of neutron stars and must, thus, be taken seriously. ill be inserted by the editor 5 ��� ��� ��� ����������� ����������������������� � ��� [ ��� ] σ [ � � ] � ��� [ ��� ] ��� ��� ��� ����������� ����������������������� � ��� [ ��� ] σ [ � � ] � ��� [ ��� ] ��� ��� ��� ������������� ����������������������� � ��� [ ��� ] σ [ � � ] � ��� [ ��� ] ��� ��� ��� ������������� ����������������������� � ��� [ ��� ] σ [ � � ] � ��� [ ��� ] ��� ��� ��� ����������� ����������������������� � ��� [ ��� ] σ [ � � ] � ��� [ ��� ] ��� ��� ��� ������������� ����������������������� � ��� [ ��� ] σ [ � � ] � ��� [ ��� ] • [49] (cid:4) [50] (cid:7) [51] (cid:78) [52] (cid:72) [53] ◦ [54] (cid:3) [55] ♦ [56] [57] (cid:79) [58] • [59] (cid:4) [60] (cid:7) [61] Fig. 3.
Summary of experimental data on total cross section for the reaction channels K − p → { K − p , ¯ K n , π / + / − Σ / − / + } from Refs. [49–61]. Horizontal error bars represent thebin size of the corresponding measurement, while the gray vertical line shows the positionof the first inelastic ( ¯ K n ) threshold. Universal parameters of the Λ (1405) can be accessed in a reaction independent wayfrom analytical properties of the scattering amplitude in the complex complex energy-plane as described in the previous section. Several theoretical constraints can be madeon such amplitudes, such as unitarity or low-energy behavior from ChPT. Necessar-ily, this does not fix the amplitude entirely, requiring for a phenomenological inputconstraining the parameters of such models. An overview of experimental data – mostrelevant for the study of Λ (1405) – is the purpose of this section. In addition, sincemany sources of data are old and not well digitalized, the author has collected andsorted it in an open GitHub repository . The largest set of data contains total cross sections for the processes K − p → { K − p ,¯ K n , π / + / − Σ / − / + } measured in 1960’s throughout 1980’s at CERN [50, 57, 58],LBNL [49, 51, 53, 54, 61], BNL [55, 60] and Rutherford Radiation Laboratory [52, 56].In those, a Kaon-beam delivered from, e.g., CERN or Bevatron was followed in a large-volume Hydrogen bubble chamber, placed in a superconducting magnet. In theseimpressive experimental programs, a large number of track photographs (in somecases on the order of 10 ) were taken and evaluated for considered events using eitherdigital techniques or “hand analysis”. The full set of obtained data is depicted in Fig. 3 , where the energy bin sizes are represented by the horizontal error bars. Forconvenience of future studies the data can also be found in a digital form in the openrepository .While rather old and imprecise, these data represents the main bulk of experimentalconstraints on the antikaon-nucleon scattering. Additionally, some data exists on https://github.com/maxim-mai/Experimental-Data/tree/master/Lambda1405 Will be inserted by the editor differential cross sections for elastic and charge-exchange ¯ KN -scattering at somewhathigher P LAB [51, 59, 61, 62]. The latter data is of importance separating the hierarchyof partial waves of the scattering amplitude, see Refs. [63, 64].
A unique opportunity to gain insight into the strong ¯ KN -dynamics is offered by thestudies of the Kaonic hydrogen, with an antikaon taking the role of the electron inthe hydrogen atom. Preparing such a system requires a capture of a K − -meson ona proton, which demands for a very low-momentum Kaon beam. In early years ofthis research this was typically achieved by the emulsion techniques. Using the latter,around 3 million of kaon captures were recorded in an experiment at BNL [65]. Sub-sequently, around 2 (cid:104) decayed to π + Σ − and π − Σ + pairs with the ratio of branchingratios, referred to as γ . Several years later, a follow up experiment [66] was con-ducted at the Rutherford Laboratory measuring the above as well as complimentarythreshold ratios γ = Γ K − p → π + Σ − Γ K − p → π − Σ + , R c = Γ K − p → charged states Γ K − p → all final states , R n = Γ K − p → π Λ Γ K − p → neutral states . (1)The next generation experiments [67–69] on kaonic hydrogen were performed around2000’s. In those, similarly to the earlier experiments [70–73], the low-energy antikaonbeam was stopped in some gaseous target, measuring the X-ray emission (K-series)of the kaonic hydrogen. Compared to the electromagnetic spectrum, the measuredone is shifted due to strong interaction between the K − -meson and the proton. Mostprominently, the energy shift and width of the 1s atomic state can be related to thecomplex-valued K − p (strong) scattering length a K − p . There are some discrepanciesbetween the results of these experiments, which in the past led to many theoreticalcontroverses [74–77]. Eventually, current benchmark refers to the most recent mea-surement by the SIDDHARTA collaboration [69]. Numerical values of all discussedthreshold observables are quoted below γ [65, 66] R c [66] R n [66] ∆E [69] Γ/ . ± .
04 0 . ± .
011 0 . ± .
015 283 ±
42 eV 271 ±
55 eVbut can also be found in the above mentioned GitHub repository .The threshold ratios γ , R c and R n are related to the scattering amplitude by the virtueof ratios of total cross-sections in the corresponding channels. The strong energy shiftand width of kaonic hydrogen is related to the complex-valued K − p scattering length( a K − p ) via the modified Deser-type relation [78] ∆E − iΓ/ − α µ c a K − p (cid:0) − a K − p αµ c (ln α − (cid:1) , (2)where α ’ /
137 is the fine-structure constant, µ c is the reduced mass of the K − p system. For the discussion of higher-order corrections see Ref. [79]. In the contextof Λ (1405), all four threshold kaonic hydrogen data are perhaps the most essentialconstraints on theoretical models for several reasons. First, these precise data liesclosest to the sub-threshold energy region. Secondly, the electro-magnetic part of themeson-baryon interaction becomes important for small P LAB , but is taken care of hereby using branching ratios or by the virtue of Eq. (2), respectively. Thus, the non-trivialimplementation of the Coloumb effects into the antikaon-nucleon amplitudes [80] canbe obviated, allowing one to focus solely on the strong dynamics. ill be inserted by the editor 7
Σ(1660) π − π + π − Σ + K − p ���� ���� ��� ����� π - Σ + [ ��� ] � �� � � � [ �� ] Fig. 4.
Production and sequential decay of Σ (1660) → π − π + π − Σ + asymptotic states [81]. Left:
Reaction mechanism with the double emission of a spectator pion (orange circles) andmeson-baryon interaction of the final π − Σ + -state (blue square). Right:
Invariant mass distribution the final π − Σ + pair as measured in Ref. [81]. Verticalgray lines show the positions of πΣ and ¯ KN thresholds. As discussed in the introduction, an alternative way in accessing the sub-thresholdenergy region of the ¯ KN scattering amplitude is to study multi-particle final decaystates similar to the original experiments of Refs. [11–13]. In such a setup, additional(to the meson-baryon pair) particles carry finite momentum away, such that one isable to probe the meson-baryon system at lower energies than the ¯ KN threshold.An important set of data in this context consists of the invariant-mass distributionof the πΣ sub-system of the K − p → Σ + (1660) π − → π − ( π + ( π − Σ + )) process mea-sured in the bubble chamber experiment at CERN [81]. Here, the meson-baryon statein the innermost parenthesis couples also to the Λ (1405) allowing to scan for thecorresponding “bump” in the line-shape, as depicted in Fig. 4 . The correspondingnumerical data is collected in the open GitHub repository .The apparent advantages of such an experiment are out weighted by the substan-tially increased theoretical complexity in accessing the universal parameters – thecomplex poles position and residuum of the Λ (1405). In particular, analytically un-ambiguous 2 → KN models, see, e.g., Refs. [16, 63, 82] for quantitativeexamples. The most recent experimental progress has been achieved by the CLAS collabora-tion measuring the γp → K + Σπ transition [15, 83] in the dedicated experiment atthe Jefferson Laboratory. As mentioned before, this experiment [15] unambiguouslyconfirmed the spin and parity of the Λ (1405) as J P = 1 / − , in agreement withtheoretical expectations [4, 5]. Equally important is the corresponding high preci-sion measurement [83] of the full Dalitz plot of the final K + Σ + π − , K + Σ − π + and K + Σ π systems and line-shapes of all three πΣ pairs at multiple total energies.The final state of the mentioned reaction consists of three hadrons, which makes theconstruction of data analysis tools cumbersome. Guided by the experience in the two-body sector, many modern methods in constructing such tools rely on unitarity asguiding principle, see e.g., Refs. [17, 18, 84] with some recent applications [85, 86]. Inthe case of γp → K + Σπ the additional complication arises as one needs to include Will be inserted by the editor S = − Kπ → K ∗ → Kπ channel. Nevertheless, a first phenomenological analysis ofthe Dalitz plots has been performed in a framework based on Bonn-Gatchina partial-wave analysis program [63, 87]. Furthermore, constraints on the chiral unitary modelsfrom the πΣ line-shapes [15] have been studied previously in Refs. [16, 88]. There, astructureless ansatz was employed for the initial production vertex γp → K + ( M B )with (
M B ) denoting meson-baryon channels. As demonstrated in these works, theCLAS data indeed can reduce the model space substantially, impacting even the polestructure of the Λ (1405). Two types of near future experiments are expected to become the driving force forthe further progress of the field. First, there is an ongoing effort for an upgrade ofthe so-important SIDDHARTA ¯ KH experiment [69] to measure the spectrum of the¯ Kd -system – the SIDDHARTA-2 experiment [89, 90]. A comparable proposal existsfor an experiment at J-PARC [91, 92]. When measured, the ¯ Kd threshold ampli-tude can be related to the ¯ KN amplitudes directly within a non-relativistic effectivefield theories as derived in Ref. [93–97] or via the non-relativistic three-body Faddeevframework [98–103]. The importance of this complementary measurement lies in thefact that only a combination of both ¯ Kd and ¯ KH measurements can resolve bothIsospin contributions of the ¯ KN scattering amplitude at the threshold unambigu-ously. For example, current theoretical models agree supremely, when projected tothe K − p channel, but exhibit large disagreements in the complimentary K − n chan-nel, see Fig. 6 . This disagreement is concerning, but offers an opportunity to reducethe model space when the new SIDDHARTA-2 data becomes available. Secondly,currently approved Hall D experiment [104] at Jefferson Laboratory intends to usesecondary beam of neutral kaons performing strange hadron spectroscopy. With low-est energies of the beam of around 300 MeV, there is a possibility that the data onantikaon-nucleon cross sections can be improved, taking advantage of the isospin fil-tering [105]. Obviously, this is a highly desirable update of old results from bubblechamber experiments discussed before.
Quantum Chromodynamics is the fundamental theory of the strong interaction, andmust, thus, inevitably grant one an access to the properties of (excited) hadrons.However, Λ (1405) lies at energies where the perturbative approach to QCD is ofno use. Fortunately, there are tools which allow to access this energy regime in asystematic fashion, namely Chiral Perturbation Theory (ChPT) and lattice gaugetheory.ChPT [106, 107] and extensions thereof to the strangeness [108] and baryon sec-tor [109–113] have become a powerful tool and in many cases a benchmark for calcu-lations of different observables in the threshold and subthreshold energy region [114–120]. However, the convergence of perturbative chiral series is impeded by the largeseparations of S = − ill be inserted by the editor 9 a manifestly covariant way leads to the following expansion [121] a I =0¯ KN = (cid:16) (+0 . LO + (+0 . NLO + ( − .
40 + 0 . i ) NNLO + ... (cid:17) fm ,a I =1¯ KN = (cid:16) (+0 . LO + (+0 . NLO + ( − .
26 + 0 . i ) NNLO + ... (cid:17) fm . (3)Thus, once more non-perturbative dynamics prevents one from directly accessing themeson-baryon scattering at energies around the Λ (1405). Of course, perturbative ap-proach is not meaningful when accessing resonances in the first place [122]. Summingup the three leading orders of the chiral expansion yields the antikaon-nucleon scat-tering length, which compares poorly with our best phenomenological knowledge ofthe latter quantityNNLO SU(3) ChPT [121] SIDDHARTA/DTR [69]/[78] a I =0¯ KN = (cid:0) + 1 .
11 + 0 . i (cid:1) fm a I =0¯ KN ≈ (cid:0) − .
53 + 0 . i (cid:1) fmHere the value quoted in the right column refers to the measurement of the energy shiftand width of kaonic hydrogen in the SIDDHARTA experiment at DAΦNE [69], whichis related to the antikaon-nucleon scattering length by the Deser-type formula [78].Extracting the resonance parameters of the Λ (1405) in the twice non-perturbativeregime of QCD is a challenging task, being faced in the past by many theoreticalapproaches. The most representative (pre- and post-QCD) classes of those are: • Potential models [101, 123–134]; • Cloudy bag models [135–137]; • QCD sum rules [138, 139]; • Relativistic Quark Model [1, 21]; • Bound state soliton model [140, 141]; • Chiral Unitary Models [64, 80, 82, 105,142–152]; • Dynamical coupled-channel models [63,87, 153–155] • Heavy Baryon ChPT with explicit reso-nances [156];
The large variety of models is an important asset in estimating the systematic un-certainty in determination of universal parameters of Λ (1405). However, addressingabove approaches in detail would require an extensive discussion of the respectivehistorical context and is beyond the scope of the present review. Thus, further dis-cussion is focused only on currently used approaches, i.e., Chiral Unitary, DynamicalCoupled-Channel and potential models. It is also these types of models which underliethe set of Λ (1405)-parameters quoted in the current PDG tables [9].It shall also be mentioned that a non-perturbative approach to this problem fromQCD is known and already applied to countless cases of hadron spectroscopy, idest the numerical calculations of Lattice QCD. Notably, the results of such calcu-lations are not directly comparable with experimental measurements due to varioustechnicalities, such as finite volume effects, finite lattice spacing or unphysical pionmass. Many systematical methods have been developed to overcome these challenges,see reviews [157, 158]. Especially for the mesonic spectrum great progress has beenachieved, see, e.g., Refs. [159–161], already approaching the physical limit and un-precedented precision. Obviously, the case of Λ (1405) is more complicated due to thepresence of baryons, noisier energy eigenlevels, or larger required operator basis [162].Still, first lattice calculations already exist [163]. Also extraction of infinite volume This differentiation is not unique, due to substantial systematical and historical overlapbetween the potential, Chiral Unitary and dynamical coupled-channel models.0 Will be inserted by the editor quantities is not as simple as in the former case, but some tools have been alreadydeveloped [164–168]. Advancing this progress will certainly lead to the next milestoneof the field either based on finite-volume spectrum of the antikaon-nucleon system(see, e.g., Ref. [169–171] for a recent progress report) or more direct probing of the Λ (1405) structure [172]. Extracting the resonance parameters of the Λ (1405) while still imposing constraintsfrom chiral symmetry of QCD has been the main motivation behind the developmentof the so-called Chiral Unitary models in the late 1990’s. Such models rely on ideasadvocated in Ref. [173–175] and use unitarity as a guiding principle for constructionof the scattering amplitude with chiral amplitude as a driving term including SU(3)coupled-channel dynamics . Recovering two-body unitarity exactly, one pays the priceof giving up the chiral power counting [107] and crossing symmetry. This makes theestimation of the systematic uncertainties intricate, but efforts have been made totackle this problem quantitatively [150, 176].The practical advantage of this type of models is their predictive power, i.e., fittingseveral free parameters a description of very large energy ranges becomes feasibleincluding an extrapolation to the complex energy-plane. In doing so, the pioneeringworks [129, 142] observed clearly a sub-threshold resonance – the dynamically gener-ated Λ (1405), while the next-generation study [143] revealed a surprising presence ofa second, broad pole. Many studies followed these pioneering approaches modifyingthe form of the driving term [80, 82, 144–148], including high energy data [105], higherpartial waves [64, 149], studying various theoretical limits [150–152] or extending tophoto-production channels [16, 88, 177]. While exploring the model space of this classof models, these works have steadily recorded the presence of the second pole. Thus,the double-pole structure as predicted by the Chiral Unitary approaches, seemsto be stable with respect to variations of the models or included data. As we knownow, this phenomenon is common to many hadronic systems as discussed in a recentdedicated review [8].In the modern diagrammatic formulation, the Chiral Unitary approaches begin withthe Chiral Lagrangian L ChPT φB = L (1) φB + L (2) φB + L (3) φB + ... , (4)being an infinite series of terms ordered by their chiral order (powers of meson mo-menta and quark masses), denoted by a subscript. The individual Lagrangians giverise to a finite set of contact terms of the type B → φB , φB → φB , etc., where φ and B denote the 3 × φ ( q ) B ( p ) → φ ( q ) B ( p ), the number of independent structures grows rapidly withthe chiral order, each being accompanied by an unknown low-energy constant (LEC).Thus, for any practical calculation, the above series needs to be truncated. This differ-entiates between various Chiral Unitary approaches, which rely either on the leading(LO) [88, 142, 180] or next-to-leading (NLO) [64, 80, 82, 144–149] order interaction Considering solely ground state mesons and baryons ten meson-baryon channels { K − p ,¯ K n , π Λ , π Σ , π + Σ − , π − Σ + , ηΛ , ηΣ , K + Ξ − , K Ξ } carry the correct ( S = − Q = 0)quantum numbers.ill be inserted by the editor 11 Fig. 5.
Diagrammatic representation of the scattering amplitude T calculated in an infiniteseries of loop diagrams via the Bethe-Salpeter equation (6) with the interaction vertex V off given in Eq. (5). Dashed and full line represent meson and baryon propagators, respectively. term. In the full off-shell form, the latter reads V off ( /q , /q ; P ) = A W T ( /q + /q ) + A Bs /q m − /Ps − m /q + A Bu /q m − /P + /q + /qu − m /q ! LO + A ( q · q ) + A [ /q, /q ] + A M + A (cid:0) /q (cid:0) q · P ) + /q ( q · P (cid:1)(cid:1) ! NLO , (5)where P = p + q = p + q , and s = P , u = ( p − q ) are the usual Mandelstamvariables. The matrices A ... in the 10-dimensional channel-space depend explicitly onthe meson decay constants and axial couplings D , F in the leading, and on the LECs { b , b D , b F , b , ..., b } at the next-to-leading chiral order.The object V off represents the chiral vertex, which can be used directly in constructingFeynman diagrams. Since the number of such diagrams to all chiral orders is infinite,a subset of an infinite cardinality is chosen which: (i) ensures unitarity exactly; (ii)includes the dominant chiral contributions defined in Eq. (5). Technically, this isfulfilled by solving the Bethe-Salpeter equation T off ( /q , /q ; P ) = V off ( /q , /q ; P )+ i Z d k (2 π ) V off ( /q , /k ; P ) 1 /k − m + i(cid:15) k − M + i(cid:15) T off ( /k, /q ; P ) . (6)In the past also a non-relativistic version of a unitary scattering amplitude was ap-proached by using Lippmann-Schwinger equation, see, e.g., Ref. [129]. In the aboveequation a summation in the channel space is performed over the intermediate mesonand baryon states of mass M , and m , respectively. These correspond to their lead-ing chiral order values, which are replaced in practical calculations by the physical(dressed) values, justified by the fact that there is no exact chiral power counting for T off . Regularization of the above integral equation is performed more commonly indimensional regularization with subtraction constants used as free parameters of themodel, but also momentum cutoff techniques have been applied in the past. An analytical solution of the T off ( /q , /q ; P ) for the case of V off consisting only of con-tact interactions was found in Refs. [181, 182]. In that, neglecting the s -channel Borndiagram (last two terms in the LO-parentheses of Eq. (5)) was motivated by the factthat physical masses are used in the propagators of Eq. (6), and that dressing the“Lagrangian” values of the baryon masses already includes an infinite series of iterated s -channel Born diagrams. Furthermore, and given the large variety of independentstructures of the NLO contributions, the expectation – backed by the findings ofRefs. [16, 82] – is that the effects due to the u -channel Born term can be mimickedby the contact terms. The same studies revealed additionally that the off-shell effectsdue to meson-baryon intermediate channels impact the description of the data andprediction of pole-positions of Λ (1405) only slightly. This supports the use of theon-shell condition – applying Dirac equation and, thus, reducing the number of inde-pendent structures in Eq. (5), see Ref. [17]. Note that even after these modifications,the interaction term still contains the full angular structure as encoded in the NLOChiral Lagrangian, allowing for a simultaneous description of S − and P − waves [64].The integral equation can be simplified further by projecting the on-shell potentialto partial waves. For once, this reduces the number of relevant combinations of low-energy constants ( b i → d i ) from, e.g., 14 to 7 for the S -wave. Furthermore, it leadsto a technical advantage that the integral equation (6) transforms an algebraic one.Besides the possibility to incorporate the u -channel Born diagrams approximately,the technical simplicity of this class of Chiral Unitary led to its popularity, see, e.g.,Refs. [80, 129, 144–148, 183]. The price to pay for this simplifications is the loss ofdirect connection to a series of Feynman diagrams. Instead, the formalism approachesthe philosophy underlying the potential models [123–128, 130], but using chiral sym-metry to constrain the form of the potential.In summary, all Chiral Unitary approaches rely on the Chiral Lagrangian, implement-ing driving term of interaction into a unitary formulation of the scattering amplitude.Methods to do so variate in: (i) Using relativistic Bethe-Salpeter or non-relativisticLipmann-Schwinger formulation of the scattering amplitude; (ii) Projecting the driv-ing term on-shell or onto specific partial waves; (iii) Truncating the driving term tothe leading or next-to leading chiral order; (iv) Including all ten channels of groundstate mesons and baryons or only the lightest six into the coupled-channel problem;(v) Regularization of the integral equation. As discussed before the large variety ofdifferent versions of the formalism is crucial to asses the systematic uncertainty inmaking one or another set of assumptions. Obviously, the largest differences occur inthe regions unconstrained by the experimental data, as demonstrated in Ref [150],comparing most recent Chiral Unitary approaches [6, 17, 147]. For example, the pre-diction of the scattering amplitudes overlaps only in the energy region constrained bythe experimental data as demonstrated explicitly in Fig. 6 . The ambiguities of the I = 1 scattering amplitudes, apparent from the right panel of the latter figure empha-size again the need for the kaonic deuterium experiments, such as SIDDHARTA-2.Besides this, the double pole structure is stable across the variations among the ChiralUnitary approaches as visualized in Fig. 7 . Historically, the very broad class of potential models precedes the development of theChiral Unitary approaches. It relies on some form of a potential when solving theLippmann-Schwinger equation T LS ( q , q ; E ) = V ( q , q ) + i Z dkk V ( q , k ) G E ( k ) T LS ( k, q ; E ) , (7) ill be inserted by the editor 13 W CMS [ GeV ] I m f + [f m ] R e f + [f m ] K − p → K − p W CMS [ GeV ] K − n → K − n Fig. 6.
Comparison of Chiral Unitary [6, 17, 147] (colored lines) and potential model [131](black line) predictions for the S -wave amplitude. Left (right) figures shows the elastic K − p ( K − n ) channels with gray vertical line denoting the position of the ¯ KN threshold in theisospin limit. Original figure and further details can be found in Ref. [150]. where E is the total energy of the system, and G E is the Green’s function. The choiceof the potential ( V ) is often motivated by its regulating properties and flexibility whenimplemented into the integral equation, e.g., separable potentials with Yamaguchiform factors [125, 127, 128, 130, 131]. In some cases phenomenological or theoreticalconstraints are implemented, e.g., vector-exchange potentials [123, 126] or matchingto chiral potentials [130, 131].Obviously, the potential models overlap strongly with the Chiral Unitary approaches.The main differences being besides the diagrammatic correspondence of the latter(c.f. Eq. (6) and Fig. 5 ) the relativistic effects. Note, that inclusion of relativistickinematics is only part of such effects, and can be included into potential models [128,130]. These approximations simplify the form of integral equations and still can bejustified in the low energy region, c.f., full black line in
Fig. 6 .There are two main advantages to the use of the potential models in the above sense.First, a very far-reaching impact of the research on antikaon-nucleon scattering is thestudy of properties of strange nuclear matter, see
Section 2.3 . Such calculations [32,186, 187, 187, 188] profit enormously from the use of separable potential forms, whichcover off-shell regions needed for the in-medium applications. Second, as a flexibletool it allows to illuminate the impact of implemented approximations. A recentexample [133], studies the form of the derived potential in relation to the off-shelleffects and chiral symmetry at the leading order, discussing also the double-polestructure of the Λ (1405). This led to a follow-up study [189], which revealed starkconflicts between the latter model and chiral symmetry constraints. In offering animprovement it provides an approach, which leads again to the two-pole solution.Similar observation was made in Ref. [184], but also within the diagrammatic versionof the Chiral Unitary approach [121] the off-shell effects have been studied with noeffect on the double-pole scenario as well. (cid:1)(cid:1) (cid:2)(cid:2) (cid:3)(cid:3) (cid:4)(cid:4) (cid:5)(cid:5) (cid:6)(cid:6)(cid:7)(cid:7) (cid:8)(cid:8) (cid:9)(cid:9) (cid:10) (cid:10) (cid:11)(cid:11) (cid:12)(cid:13) (cid:13)(cid:14)(cid:14) (cid:2)(cid:3)(cid:3) (cid:11)(cid:10) (cid:9)(cid:9) (cid:2)(cid:2) (cid:3)(cid:11) (cid:1)(cid:4)(cid:4) (cid:1)(cid:2)(cid:3)(cid:4) (cid:1)(cid:2)(cid:5)(cid:4) (cid:1)(cid:2)(cid:6)(cid:4) (cid:1)(cid:3)(cid:4)(cid:4) (cid:1)(cid:3)(cid:7)(cid:4) (cid:1)(cid:3)(cid:3)(cid:4) - (cid:1)(cid:8)(cid:4) - (cid:1)(cid:4)(cid:4) - (cid:8)(cid:4)(cid:4) (cid:9)(cid:10) (cid:11) (cid:1)(cid:2)(cid:3) [ (cid:12)(cid:10)(cid:13) ] (cid:1) (cid:2) (cid:3) (cid:1) (cid:2) (cid:3) [ (cid:4) (cid:5) (cid:6) ] (cid:1) Mai:2014xna ( ) (cid:2) Mai:2014xna ( ) (cid:3) Ikeda:2011pi (cid:4)
Sadasivan:2018jig (cid:5)
Guo:2012vv (cid:6)
Feijoo:2018den (cid:7)
Borasoy:2005ie (cid:8)
Oller:2000fj (cid:9)
Jido:2003cb (cid:10)
Morimatsu:2019wvk ( B ) (cid:11) Morimatsu:2019wvk ( C ) (cid:12) Oset:2001cn (cid:13)
Roca:2013av ( ) (cid:14) Roca:2013av ( ) (cid:2) Anisovich:2020lec ( ) (cid:3) Anisovich:2020lec ( ) (cid:11) Fernandez - Ramirez:2015tfa (cid:10)
Zhang:2013sva (cid:9)
Haidenbauer:2010ch (cid:2)
Cieply:2011nq (cid:3)
Revai:2019ipq (cid:11)
Hassanvand:2012dn (cid:1)
Shevchenko:2011ce (cid:4)
Shevchenko:2011ce N a r r o w p o l e B r oad po l e w it h S I DDH AR T A B r oad po l e Chiral Unitary Approaches (cid:4) / • Ref. [97] (cid:13)
Ref. [80] (cid:78)
Ref. [145] Ref. [143] (cid:72)
Ref. [64] (cid:79)
Ref. [151] (cid:7)
Ref. [147] (cid:3) / ♦ Ref. [184] (cid:73)
Ref. [105] ⊗ Ref. [185] (cid:9) / ⊕ Ref. [88]Dynamical coupled-channel models • / (cid:78) Ref. [63] ♦ Ref. [153] (cid:3)
Ref. [154] (cid:79)
Ref. [155]Potential models • Ref. [131] (cid:78)
Ref. [134] ♦ Ref. [132] (cid:4) / (cid:72) Ref. [101]
Fig. 7.
Comparison of pole predictions for poles of Λ (1405) from most recent approaches,id est year ≥ KH datafrom the SIDDHARTA experiment [69]. The gray and dashed areas are drown to guide theeye in differentiating first (narrow) and second (broad) pole of double-pole solutions to the Λ (1405). Vertical lines denote the position of πΣ and ¯ KN thresholds. Re-examining previously accepted approximations is crucial for further developmentof the field. Besides the latter studies, the dynamical coupled-channel models areimportant to independently test our understanding of the antikaon-nucleon scattering.Such models rely on the basic principles of scattering theory in constructing verygeneral parametrization of scattering and (photo-) production amplitudes [190–192].Typically, they have a large number of free parameters fixed in fits to experimentaldata. Thus, the descriptive power of such models is limited to the kinematic regionscovered by the experimental data, and extrapolations to further energy regions (alsocomplex-valued energies) can only be dealt with as consistency checks. An exampleof such a check using techniques from machine learning can be found in a recentstudy [193] of the pole-content of the KΞ channel. In view of the S = − K − p elastic and inelastic scattering was fitted using BnGa [190] model parameterizing theresonant and non-resonant contributions to rather high energies. A series of updateson resonance parameters of hyperons was extracted from such fits, and in a separatework a detailed study was conducted with respect to the Λ (1405). This work includedthe threshold [65, 66, 69] and recent CLAS photo-production data [83]. Two solutionshave been found in this study: a single- and a double-pole one, where the broad polewas fixed to z R = (1380 , − i ) MeV. Thus, at least within this model the broad poledoes not influence the description of the data strongly enough and could neither beexcluded nor precisely determined.The results of all most recent determinations (year ≥ Λ (1405) are collected in Fig. 7 . It demonstrates the relatively small systematic uncer-tainty on the position of the first (narrow) pole. More importantly, the vast majority ill be inserted by the editor 15 of approaches throughout the model classes supports clearly the existence of the sec-ond (broad) pole. The position of the latter is much less restricted, but becomesmuch less volatile when including the most recent kaonic hydrogen [69] and CLASphotoproduction data [15].
The enigma of the Λ (1405) starting from its prediction and experimental verificationto the surprising appearance of the double-pole structure in the complex energy-plane,has become a very fruitful testing ground for approaches to the intermediate (twicenon-perturbative) energy region of QCD.At the current stage, the implementation of constraints due to chiral symmetry ofQCD into unitary form of scattering amplitude seems to demand the existence of thesecond pole at lower energies and deeper in the complex energy-plane. This approachrelies on several well-controlled and -studied assumptions. The obtained double-polehypothesis has been confirmed by a large number of non-redundant studies exploringvarious theoretical limits. Inclusion of modern photoproduction and kaonic hydrogendata led to tighter constraints on the scattering amplitudes and positions of bothpoles.Recent, more data driven approaches are crucial for the critical debate and re-examination of previous assumptions. Without resolving the microscopic dynamicsof the hadronic states such approaches aim simply to ask which of those are de-manded by the data. However, even with the high-precision photo-production databy the CLAS collaboration the two-pole scenario seems to agree with the data.There are two major avenues, which will foster future development of the field. First,new experimental facilities may provide a new complementary data on antikaon-nucleon channel. Most importantly the kaonic deuterium experiments at J-PARCand Frescati will allow to reduce the parameter space of currently available models,which actually disagree strongly in the description of the isovector channel. Secondly,since double-pole hypothesis seems to be tied strongly to the incorporation of QCDsymmetries it is crucial to foster the next-generation of Lattice QCD calculations.Either in obtaining the finite-volume spectrum in the meson-baryon channel or moredirect probing of the Λ (1405)-structure. Acknowledgments
The author would like to thank Jose Antonio Oller for invitation to write this manuscript,and to Ulf-G. Meißner, M. D¨oring, A. Ciepl´y, P. Bruns for their comments and carefulreading of the manuscript. The author thanks R. Brett for help with document preparation,and is grateful to E. Sismanidou for support and motivation.
References
1. S. Capstick and N. Isgur, AIP Conf. Proc. , 267 (1985).2. U. Loring, B. C. Metsch, and H. R. Petry, Eur. Phys. J. A , 395 (2001), arXiv:hep-ph/0103289 .3. S. Durr et al. , Science , 1224 (2008), arXiv:0906.3599 [hep-lat] .4. J. Sakurai, Annals Phys. , 1 (1960).5. R. Dalitz, Ann. Rev. Nucl. Part. Sci. , 339 (1963).6 Will be inserted by the editor6. T. Hyodo and D. Jido, Prog. Part. Nucl. Phys. , 55 (2012), arXiv:1104.4474 [nucl-th].7. A. Gal, E. Hungerford, and D. Millener, Rev. Mod. Phys. , 035004 (2016),arXiv:1605.00557 [nucl-th] .8. U.-G. Meißner, Symmetry , 981 (2020), arXiv:2005.06909 [hep-ph] .9. M. Tanabashi et al. (Particle Data Group), Phys. Rev. D , 030001 (2018).10. R. Dalitz and S. Tuan, Annals Phys. , 100 (1959).11. M. H. Alston, L. W. Alvarez, P. Eberhard, M. L. Good, W. Graziano, H. K. Ticho,and S. G. Wojcicki, Phys. Rev. Lett. , 698 (1961).12. P. Bastien, M. Ferro-Luzzi, and A. Rosenfeld, Phys. Rev. Lett. , 702 (1961).13. Y. Eisenberg, G. Yekutieli, P. Abrahamson, and D. Kessler, “(no further informationis found),” (1961).14. M. Roos, Rev. Mod. Phys. , 314 (1963).15. K. Moriya et al. (CLAS), Phys. Rev. Lett. , 082004 (2014), arXiv:1402.2296 [hep-ex].16. M. Mai and U.-G. Meißner, Eur. Phys. J. A , 30 (2015), arXiv:1411.7884 [hep-ph] .17. M. Mai, B. Hu, M. Doring, A. Pilloni, and A. Szczepaniak, Eur. Phys. J. A , 177(2017), arXiv:1706.06118 [nucl-th] .18. A. Jackura, C. Fern´andez-Ram´ırez, V. Mathieu, M. Mikhasenko, J. Nys, A. Pilloni,K. Salda˜na, N. Sherrill, and A. Szczepaniak (JPAC), Eur. Phys. J. C , 56 (2019),arXiv:1809.10523 [hep-ph] .19. A. Martinez Torres, K. Khemchandani, and E. Oset, Phys. Rev. C , 042203 (2008),arXiv:0706.2330 [nucl-th] .20. H. Burkhardt and J. Lowe, Phys. Rev. C , 607 (1991).21. J. W. Darewych, R. Koniuk, and N. Isgur, Phys. Rev. D , 1765 (1985).22. L. Roca, M. Mai, E. Oset, and U.-G. Meißner, Eur. Phys. J. C , 218 (2015),arXiv:1503.02936 [hep-ph] .23. E. Oset et al. , Int. J. Mod. Phys. E , 1630001 (2016), arXiv:1601.03972 [hep-ph] .24. M. Agnello et al. (FINUDA), Phys. Rev. Lett. , 212303 (2005).25. M. Agnello et al. (FINUDA), Phys. Lett. B , 35 (2005), arXiv:nucl-ex/0506028 .26. V. Filippini et al. (FINUDA), Nucl. Phys. A , 537 (1998).27. M. Maggiora et al. (DISTO), Nucl. Phys. A , 43 (2010), arXiv:0912.5116 [hep-ex] .28. M. Maggiora (DISTO), Nucl. Phys. A , 329 (2001).29. T. Yamazaki et al. , Phys. Rev. Lett. , 132502 (2010), arXiv:1002.3526 [nucl-ex] .30. T. Yamazaki et al. (DISTO), Hyperfine Interact. , 181 (2009), arXiv:0810.5182[nucl-ex] .31. J. Mareˇs, A. Ciepl´y, J. Hrt´ankov´a, M. Sch¨afer, B. Bazak, N. Barnea, E. Friedman, andA. Gal, Acta Phys. Polon. B , 129 (2020).32. A. Gal, E. Friedman, N. Barnea, A. Ciepl´y, J. Mareˇs, and D. Gazda, EPJ Web Conf. , 01018 (2014), arXiv:1411.1241 [nucl-th] .33. J. Hrt´ankov´a, N. Barnea, E. Friedman, A. Gal, J. r. Mareˇs, and M. Sch¨afer, Phys.Lett. B , 90 (2018), arXiv:1805.11368 [nucl-th] .34. D. Kaplan and A. Nelson, Phys. Lett. B , 57 (1986).35. G.-Q. Li, C. Lee, and G. Brown, Nucl. Phys. A , 372 (1997), arXiv:nucl-th/9706057.36. S. Pal, D. Bandyopadhyay, and W. Greiner, Nucl. Phys. A , 553 (2000),arXiv:astro-ph/0001039 .37. C. Lee, Phys. Rept. , 255 (1996).38. G. Baym and S. Chin, Phys. Lett. B , 241 (1976).39. Y. Lim, A. Bhattacharya, J. W. Holt, and D. Pati, (2020), arXiv:2007.06526 [nucl-th].40. K. Hebeler, J. Lattimer, C. Pethick, and A. Schwenk, Phys. Rev. Lett. , 161102(2010), arXiv:1007.1746 [nucl-th] .41. Y. Lim and J. W. Holt, Phys. Rev. Lett. , 062701 (2018), arXiv:1803.02803 [nucl-th].ill be inserted by the editor 1742. A. Ramos, J. Schaffner-Bielich, and J. Wambach, Lect. Notes Phys. , 175 (2001),arXiv:nucl-th/0011003 .43. W. Detmold, K. Orginos, M. J. Savage, and A. Walker-Loud, Phys. Rev. D , 054514(2008), arXiv:0807.1856 [hep-lat] .44. A. Alexandru, R. Brett, C. Culver, M. D¨oring, D. Guo, F. X. Lee, and M. Mai, (2020),arXiv:2009.12358 [hep-lat] .45. D. Lonardoni, A. Lovato, S. Gandolfi, and F. Pederiva, Phys. Rev. Lett. , 092301(2015), arXiv:1407.4448 [nucl-th] .46. T. Hell and W. Weise, Phys. Rev. C , 045801 (2014), arXiv:1402.4098 [nucl-th] .47. P. Demorest, T. Pennucci, S. Ransom, M. Roberts, and J. Hessels, Nature , 1081(2010), arXiv:1010.5788 [astro-ph.HE] .48. J. Antoniadis et al. , Science , 6131 (2013), arXiv:1304.6875 [astro-ph.HE] .49. M. Sakitt, T. Day, R. Glasser, N. Seeman, J. Friedman, W. Humphrey, and R. Ross,Phys. Rev. , B719 (1965).50. M. Csejthey-Barth et al. , Phys. Lett. , 89 (1965).51. T. S. Mast, M. Alston-Garnjost, R. O. Bangerter, A. S. Barbaro-Galtieri, F. T. Solmitz,and R. D. Tripp, Phys. Rev. D , 13 (1976).52. J. Ciborowski et al. , J. Phys. G , 13 (1982).53. M. B. Watson, M. Ferro-Luzzi, and R. D. Tripp, Phys. Rev. , 2248 (1963).54. W. E. Humphrey and R. R. Ross, Phys. Rev. , 1305 (1962).55. J. Kim, Phys. Rev. Lett. , 1074 (1967).56. D. Evans, J. Major, E. Rondio, J. A. Zakrzewski, J. Conboy, D. Miller, and T. Tymie-niecka, J. Phys. G , 885 (1983).57. W. Kittel, G. Otter, and I. Wacek, Phys. Lett. , 349 (1966).58. G. S. Abrams and B. Sechi-Zorn, Phys. Rev. , B454 (1965).59. T. Mast, M. Alston-Garnjost, R. Bangerter, A. Barbaro-Galtieri, F. Solmitz, andR. Tripp, Phys. Rev. D , 3078 (1975).60. J. Kim, Phys. Rev. Lett. , 29 (1965).61. R. Bangerter, M. Alston-Garnjost, A. Barbaro-Galtieri, T. Mast, F. Solmitz, andR. Tripp, Phys. Rev. D , 1484 (1981).62. R. Armenteros et al. , Nucl. Phys. B , 15 (1970).63. A. Anisovich, A. Sarantsev, V. Nikonov, V. Burkert, R. Schumacher, U. Thoma, andE. Klempt, Eur. Phys. J. A , 139 (2020).64. D. Sadasivan, M. Mai, and M. D¨oring, Phys. Lett. B , 329 (2019), arXiv:1805.04534[nucl-th] .65. D. Tovee et al. , Nucl. Phys. B , 493 (1971).66. R. Nowak et al. , Nucl. Phys. B , 61 (1978).67. T. Ito et al. , Phys. Rev. C , 2366 (1998).68. G. Beer et al. (DEAR), Phys. Rev. Lett. , 212302 (2005).69. M. Bazzi et al. (SIDDHARTA), Phys. Lett. B , 113 (2011), arXiv:1105.3090 [nucl-ex] .70. J. Davies, G. Pyle, G. Squier, C. Batty, S. Biagi, S. Hoath, P. Sharman, and A. Clough,Phys. Lett. B , 55 (1979).71. M. Izycki et al. , Z. Phys. A , 11 (1980).72. P. Bird, A. Clough, K. Parker, G. Pyle, G. Squier, S. Baird, C. Batty, A. Kilvington,F. Russell, and P. Sharman, Nucl. Phys. A , 482 (1983).73. M. Iwasaki et al. , Phys. Rev. Lett. , 3067 (1997).74. W. Weise, Nucl. Phys. A , 51 (2010), arXiv:1001.1300 [nucl-th] .75. J. Zmeskal, Prog. Part. Nucl. Phys. , 512 (2008).76. B. Borasoy, R. Nissler, and W. Weise, Phys. Rev. Lett. , 199201 (2006), arXiv:hep-ph/0512279 .77. J. A. Oller, J. Prades, and M. Verbeni, Phys. Rev. Lett. , 172502 (2005), arXiv:hep-ph/0508081 .78. U. G. Meißner, U. Raha, and A. Rusetsky, Eur. Phys. J. C , 349 (2004), arXiv:hep-ph/0402261 .8 Will be inserted by the editor79. A. Cieply and J. Smejkal, Eur. Phys. J. A , 237 (2007), arXiv:0711.4928 [hep-ph] .80. B. Borasoy, R. Nissler, and W. Weise, Eur. Phys. J. A , 79 (2005), arXiv:hep-ph/0505239 .81. R. Hemingway, Nucl. Phys. B , 742 (1985).82. M. Mai and U.-G. Meißner, Nucl. Phys. A , 51 (2013), arXiv:1202.2030 [nucl-th] .83. K. Moriya et al. (CLAS), Phys. Rev. C , 035206 (2013), arXiv:1301.5000 [nucl-ex] .84. H. Kamano, S. Nakamura, T. Lee, and T. Sato, Phys. Rev. D , 114019 (2011),arXiv:1106.4523 [hep-ph] .85. H. Kamano, B. Julia-Diaz, T.-S. Lee, A. Matsuyama, and T. Sato, Phys. Rev. C ,025206 (2009), arXiv:0807.2273 [nucl-th] .86. D. Sadasivan, M. Mai, H. Akdag, and M. D¨oring, Phys. Rev. D , 094018 (2020),arXiv:2002.12431 [nucl-th] .87. A. Anisovich, A. Sarantsev, V. Nikonov, V. Burkert, R. Schumacher, U. Thoma, andE. Klempt, (2019), arXiv:1905.05456 [nucl-ex] .88. L. Roca and E. Oset, Phys. Rev. C , 055201 (2013), arXiv:1301.5741 [nucl-th] .89. C. Curceanu et al. , Acta Phys. Polon. B , 251 (2020).90. J. Zmeskal et al. , JPS Conf. Proc. , 023012 (2019).91. T. Hashimoto et al. (J-PARC E57, E62), JPS Conf. Proc. , 023013 (2019).92. J. Zmeskal et al. , Acta Phys. Polon. B , 101 (2015), arXiv:1501.05548 [nucl-ex] .93. R. Chand and R. Dalitz, Annals Phys. , 1 (1962).94. S. Kamalov, E. Oset, and A. Ramos, Nucl. Phys. A , 494 (2001), arXiv:nucl-th/0010054 .95. U.-G. Meißner, U. Raha, and A. Rusetsky, Eur. Phys. J. C , 473 (2006), arXiv:nucl-th/0603029 .96. M. D¨oring and U.-G. Meißner, Phys. Lett. B , 663 (2011), arXiv:1108.5912 [nucl-th].97. M. Mai, V. Baru, E. Epelbaum, and A. Rusetsky, Phys. Rev. D , 054016 (2015),arXiv:1411.4881 [nucl-th] .98. M. Torres, R. Dalitz, and A. Deloff, Phys. Lett. B , 213 (1986).99. G. Toker, A. Gal, and J. Eisenberg, Nucl. Phys. A , 405 (1981).100. A. Bahaoui, C. Fayard, T. Mizutani, and B. Saghai, Phys. Rev. C , 064001 (2003),arXiv:nucl-th/0307067 .101. N. Shevchenko, Phys. Rev. C , 034001 (2012), arXiv:1103.4974 [nucl-th] .102. N. Shevchenko, Nucl. Phys. A , 50 (2012), arXiv:1201.3173 [nucl-th] .103. N. Shevchenko and J. R´evai, Phys. Rev. C , 034003 (2014), arXiv:1402.3935 [nucl-th].104. M. Amaryan et al. (KLF), (2020), arXiv:2008.08215 [nucl-ex] .105. A. Feijoo, V. Magas, and A. Ramos, Phys. Rev. C , 035211 (2019), arXiv:1810.07600[hep-ph] .106. J. Gasser and H. Leutwyler, Annals Phys. , 142 (1984).107. S. Weinberg, Physica A , 327 (1979).108. J. Gasser and H. Leutwyler, Nucl. Phys. B , 465 (1985).109. J. Gasser, M. Sainio, and A. Svarc, Nucl. Phys. B , 779 (1988).110. V. Bernard, N. Kaiser, J. Kambor, and U. G. Meißner, Nucl. Phys. B , 315 (1992).111. H.-B. Tang, (1996), arXiv:hep-ph/9607436 .112. T. Becher and H. Leutwyler, Eur. Phys. J. C , 643 (1999), arXiv:hep-ph/9901384 .113. P. J. Ellis and H.-B. Tang, Phys. Rev. C , 3356 (1998), arXiv:hep-ph/9709354 .114. V. Bernard and U.-G. Meißner, Ann. Rev. Nucl. Part. Sci. , 33 (2007), arXiv:hep-ph/0611231 .115. V. Bernard, Prog. Part. Nucl. Phys. , 82 (2008), arXiv:0706.0312 [hep-ph] .116. S. Scherer, Adv. Nucl. Phys. , 277 (2003), arXiv:hep-ph/0210398 .117. U. G. Meißner, Rept. Prog. Phys. , 903 (1993), arXiv:hep-ph/9302247 .118. B. Kubis, in Workshop on Physics and Astrophysics of Hadrons and Hadronic Matter (2007) arXiv:hep-ph/0703274 .ill be inserted by the editor 19119. V. Bernard, N. Kaiser, and U.-G. Meißner, Int. J. Mod. Phys. E , 193 (1995),arXiv:hep-ph/9501384 .120. V. Bernard, N. Kaiser, and U. G. Meißner, Phys. Lett. B , 421 (1993), arXiv:hep-ph/9304275 .121. M. Mai, P. C. Bruns, B. Kubis, and U.-G. Meißner, Phys. Rev. D , 094006 (2009),arXiv:0905.2810 [hep-ph] .122. M. J. Savage, Phys. Lett. B , 411 (1994), arXiv:hep-ph/9404285 .123. R. Dalitz, T. Wong, and G. Rajasekaran, Phys. Rev. , 1617 (1967).124. R. H. Landau, Phys. Rev. C , 1324 (1983).125. J. Schnick and R. Landau, Phys. Rev. Lett. , 1719 (1987).126. P. Siegel and W. Weise, Phys. Rev. C , 2221 (1988).127. J. Schnick and R. Landau, (1989).128. J. Fink, P.J., G. He, R. Landau, and J. Schnick, Phys. Rev. C , 2720 (1990).129. N. Kaiser, P. Siegel, and W. Weise, Nucl. Phys. A , 325 (1995), arXiv:nucl-th/9505043 .130. A. Ciepl´y and J. Smejkal, Eur. Phys. J. A , 191 (2010), arXiv:0910.1822 [nucl-th] .131. A. Ciepl´y and J. Smejkal, Nucl. Phys. A , 115 (2012), arXiv:1112.0917 [nucl-th] .132. M. Hassanvand, S. Z. Kalantari, Y. Akaishi, and T. Yamazaki, Phys. Rev. C ,055202 (2013), [Addendum: Phys.Rev.C 88, 019905 (2013)], arXiv:1210.7725 [nucl-th].133. J. R´evai, Few Body Syst. , 49 (2018), arXiv:1711.04098 [nucl-th] .134. J. R´evai, (2019), arXiv:1908.08730 [nucl-th] .135. B. Jennings, Phys. Lett. B , 229 (1986).136. A. W. Thomas, S. Theberge, and G. A. Miller, Phys. Rev. D , 216 (1981).137. Y. Umino and F. Myhrer, Phys. Rev. D , 3391 (1989).138. D. B. Leinweber, Annals Phys. , 203 (1990).139. L. S. Kisslinger and E. M. Henley, Eur. Phys. J. A , 8 (2011), arXiv:0911.1179[hep-ph] .140. C. L. Schat, N. N. Scoccola, and C. Gobbi, Nucl. Phys. A , 627 (1995), arXiv:hep-ph/9408360 .141. T. Ezoe and A. Hosaka, Phys. Rev. D , 014046 (2020), arXiv:2006.03788 [hep-ph] .142. E. Oset and A. Ramos, Nucl. Phys. A , 99 (1998), arXiv:nucl-th/9711022 .143. J. Oller and U. G. Meißner, Phys. Lett. B , 263 (2001), arXiv:hep-ph/0011146 .144. J. A. Oller, Eur. Phys. J. A , 63 (2006), arXiv:hep-ph/0603134 .145. Y. Ikeda, T. Hyodo, and W. Weise, Phys. Lett. B , 63 (2011), arXiv:1109.3005[nucl-th] .146. Y. Ikeda, T. Hyodo, and W. Weise, Nucl. Phys. A , 98 (2012), arXiv:1201.6549[nucl-th] .147. Z.-H. Guo and J. Oller, Phys. Rev. C , 035202 (2013), arXiv:1210.3485 [hep-ph] .148. B. Borasoy, U.-G. Meißner, and R. Nissler, Phys. Rev. C , 055201 (2006), arXiv:hep-ph/0606108 .149. D. Jido, E. Oset, and A. Ramos, Phys. Rev. C , 055203 (2002), arXiv:nucl-th/0208010 .150. A. Ciepl´y, M. Mai, U.-G. Meißner, and J. Smejkal, Nucl. Phys. A , 17 (2016),arXiv:1603.02531 [hep-ph] .151. D. Jido, J. Oller, E. Oset, A. Ramos, and U.-G. Meißner, Nucl. Phys. A , 181(2003), arXiv:nucl-th/0303062 .152. M. Lutz and E. Kolomeitsev, (2001), 10.1023/A:1012655116247, arXiv:nucl-th/0105068 .153. C. Fernandez-Ramirez, I. Danilkin, D. Manley, V. Mathieu, and A. Szczepaniak, Phys.Rev. D , 034029 (2016), arXiv:1510.07065 [hep-ph] .154. H. Zhang, J. Tulpan, M. Shrestha, and D. Manley, Phys. Rev. C , 035205 (2013),arXiv:1305.4575 [hep-ph] .155. J. Haidenbauer, G. Krein, U.-G. Meißner, and L. Tolos, Eur. Phys. J. A , 18 (2011),arXiv:1008.3794 [nucl-th] .0 Will be inserted by the editor156. C.-H. Lee, D.-P. Min, and M. Rho, Nucl. Phys. A , 334 (1996), arXiv:hep-ph/9505283 .157. S. Aoki et al. (Flavour Lattice Averaging Group), Eur. Phys. J. C , 113 (2020),arXiv:1902.08191 [hep-lat] .158. R. A. Briceno, J. J. Dudek, and R. D. Young, Rev. Mod. Phys. , 025001 (2018),arXiv:1706.06223 [hep-lat] .159. R. A. Briceno, J. J. Dudek, R. G. Edwards, and D. J. Wilson, Phys. Rev. Lett. ,022002 (2017), arXiv:1607.05900 [hep-ph] .160. M. Mai, C. Culver, A. Alexandru, M. D¨oring, and F. X. Lee, Phys. Rev. D , 114514(2019), arXiv:1908.01847 [hep-lat] .161. M. Fischer, B. Kostrzewa, M. Mai, M. Petschlies, F. Pittler, M. Ueding, C. Urbach,and M. Werner (ETM), (2020), arXiv:2006.13805 [hep-lat] .162. J. Fallica, “Lambda baryon spectroscopy and pion-pion scattering with partial wavemixing in lattice qcd,” (2018).163. B. J. Menadue, W. Kamleh, D. B. Leinweber, and M. Mahbub, Phys. Rev. Lett. ,112001 (2012), arXiv:1109.6716 [hep-lat] .164. M. Doring, M. Mai, and U.-G. Meißner, Phys. Lett. B , 185 (2013), arXiv:1302.4065[hep-lat] .165. R. Molina and M. D¨oring, Phys. Rev. D , 056010 (2016), [Addendum: Phys.Rev.D94, 079901 (2016)], arXiv:1512.05831 [hep-lat] .166. M. Doring, J. Haidenbauer, U.-G. Meißner, and A. Rusetsky, Eur. Phys. J. A , 163(2011), arXiv:1108.0676 [hep-lat] .167. Z.-W. Liu, J. M. Hall, D. B. Leinweber, A. W. Thomas, and J.-J. Wu, Phys. Rev. D , 014506 (2017), arXiv:1607.05856 [nucl-th] .168. M. Lage, U.-G. Meißner, and A. Rusetsky, Phys. Lett. B , 439 (2009),arXiv:0905.0069 [hep-lat] .169. J. Bulava, in (2019) arXiv:1909.13097 [hep-lat] .170. S. Paul et al. , PoS LATTICE2018 , 089 (2018), arXiv:1812.01059 [hep-lat] .171. C. Morningstar, J. Bulava, B. Singha, R. Brett, J. Fallica, A. Hanlon, and B. H¨orz,Nucl. Phys. B , 477 (2017), arXiv:1707.05817 [hep-lat] .172. J. M. M. Hall, W. Kamleh, D. B. Leinweber, B. J. Menadue, B. J. Owen, A. W. Thomas,and R. D. Young, Phys. Rev. Lett. , 132002 (2015), arXiv:1411.3402 [hep-lat] .173. A. Dobado, M. J. Herrero, and T. N. Truong, Phys. Lett. B , 134 (1990).174. S. Weinberg, Phys. Lett. B , 288 (1990).175. J. Gasser and U. G. Meißner, Nucl. Phys. B , 90 (1991).176. E. Epelbaum, H. Krebs, and U. Meißner, Eur. Phys. J. A , 53 (2015),arXiv:1412.0142 [nucl-th] .177. L. Geng and E. Oset, Eur. Phys. J. A , 405 (2007), arXiv:0707.3343 [hep-ph] .178. M. Frink and U.-G. Meißner, Eur. Phys. J. A , 255 (2006), arXiv:hep-ph/0609256 .179. A. Krause, Helv. Phys. Acta , 3 (1990).180. E. Oset, L. Roca, M. Mai, and J. Nieves, AIP Conf. Proc. , 040004 (2016).181. P. C. Bruns, M. Mai, and U. G. Meißner, Phys. Lett. B , 254 (2011),arXiv:1012.2233 [nucl-th] .182. M. Mai, From meson-baryon scattering to meson photoproduction , Ph.D. thesis, U.Bonn (main) (2013).183. B. Borasoy, R. Nissler, and W. Weise, Phys. Rev. Lett. , 213401 (2005), arXiv:hep-ph/0410305 .184. O. Morimatsu and K. Yamada, Phys. Rev. C , 025201 (2019), arXiv:1903.12380[hep-ph] .185. E. Oset, A. Ramos, and C. Bennhold, Phys. Lett. B , 99 (2002), [Erratum:Phys.Lett.B 530, 260–260 (2002)], arXiv:nucl-th/0109006 .186. A. Ciepl´y, E. Friedman, A. Gal, D. Gazda, and J. Mareˇs, Phys. Rev. C , 045206(2011), arXiv:1108.1745 [nucl-th] .ill be inserted by the editor 21187. A. Ciepl´y, E. Friedman, A. Gal, D. Gazda, and J. Mareˇs, Phys. Lett. B , 402(2011), arXiv:1102.4515 [nucl-th] .188. A. Ciepl´y and V. Krejˇciˇr´ık, Nucl. Phys. A , 311 (2015), arXiv:1501.06415 [nucl-th].189. P. Bruns and A. Ciepl´y, Nucl. Phys. A , 121702 (2020), arXiv:1911.09593 [nucl-th].190. A. Anisovich, E. Klempt, V. Nikonov, A. Sarantsev, and U. Thoma, Eur. Phys. J. A , 153 (2011), arXiv:1109.0970 [hep-ph] .191. D. R¨onchen, M. D¨oring, F. Huang, H. Haberzettl, J. Haidenbauer, C. Hanhart, S. Kre-wald, U.-G. Meißner, and K. Nakayama, Eur. Phys. J. A , 101 (2014), [Erratum:Eur.Phys.J.A 51, 63 (2015)], arXiv:1401.0634 [nucl-th] .192. H. Kamano, S. Nakamura, T. S. H. Lee, and T. Sato, Phys. Rev. C , 035209 (2013),arXiv:1305.4351 [nucl-th] .193. J. Landay, M. Mai, M. D¨oring, H. Haberzettl, and K. Nakayama, Phys. Rev. D ,016001 (2019), arXiv:1810.00075 [nucl-th] .194. K. S. Myint, Y. Akaishi, M. Hassanvand, and T. Yamazaki, PTEP , 073D01(2018), arXiv:1804.08240 [nucl-th] .195. M. Matveev, A. Sarantsev, V. Nikonov, A. Anisovich, U. Thoma, and E. Klempt, Eur.Phys. J. A , 179 (2019), arXiv:1907.03645 [nucl-ex] .196. A. Sarantsev, M. Matveev, V. Nikonov, A. Anisovich, U. Thoma, and E. Klempt, Eur.Phys. J. A55