183. Pole Structure of the Λ(1405) Region

83. Pole Structure of the Λ(1405) Region
Revised June 2021 by T. Hyodo (Tokyo Metropolitan U.) and U.-G. Meißner (Bonn U.; FZ Jülich).
The Λ(1405) resonance emerges in the meson-baryon scattering amplitude with the strangenessS = −1 and isospin I = 0. It is the archetype of what is called a dynamically generated resonance,as pioneered by Dalitz and Tuan [1]. The most powerful and systematic approach for the low-energy
regime of the strong interactions is chiral perturbation theory (ChPT), see e.g. Ref. [2]. A perturbative calculation is, however, not applicable to this sector because of the existence of the Λ(1405)
just below the K̄N threshold. In this case, ChPT has to be combined with a non-perturbative resummation technique, just as in the case of the nuclear forces. By solving the Lippmann-Schwinger
equation with the interaction kernel determined by ChPT and using a particular regularization,
in Ref. [3] a successful description of the low-energy K − p scattering data as well as the mass
distribution of the Λ(1405) was achieved (for further developments, see Ref. [4–7] and referencestherein).
The study of the pole structure was initiated by Ref. [8], which finds two poles of the scattering
amplitude in the complex energy plane between the K̄N and πΣ thresholds. The spectrum inexperiments exhibits one effective resonance shape, while the existence of two poles results in thereaction-dependent lineshape [9]. The origin of this two-pole structure is attributed to the two
attractive channels of the leading order interaction in the SU(3) basis (singlet and octet) [9] and in
the isospin basis (K̄N and πΣ) [10]. It is remarkable that the sign and the strength of the leading
order interaction is determined by a low-energy theorem of chiral symmetry, i.e. the so-called
Weinberg-Tomozawa term. The two-pole nature of the Λ(1405) is qualitatively different from thecase of the N(1440) resonance. Two poles of the N(1440) appear on different Riemann sheets of thecomplex energy plane separated by the π∆ branch point. These poles reflect a single state, witha nearby pole and a more distant shadow pole. In contrast, the two poles in the Λ(1405) region
on the same Riemann sheet (where πΣ channels are unphysical and all other channels physical,
correspondingly to the one, connected to the real axis beween the πΣ and K̄N thresholds) are
generated from two attractive forces mentioned above [9, 10].
Recently, various new experimental results on the Λ(1405) have become available [4]. Amongthese, the most striking measurement is the precise determination of the energy shift and widthof kaonic hydrogen by the SIDDHARTA collaboration [11, 12], which provides a quantitative andstringent constraint on the K − p amplitude at threshold through the improved Deser formula [13].
Systematic studies with error analyses based on the next-to-leading order ChPT interaction including the SIDDHARTA constraint have been performed by various groups [14–18] All these studies
confirm that the new kaonic hydrogen data are compatible with the scattering data above threshold.
The results of the pole positions of Λ(1405) in the various approaches are summarized in Table
83.1. We may regard the difference among the calculations as a systematic error, which stems fromthe various approximations of the Bethe-Salpeter equation, the fitting procedure, and also the inclusion of SU(3) breaking effects such as the choice of the various meson decay constants, and so on.
A detailed comparison of the various approaches that enter the table is given in Ref. [19]. A recent
analysis including also the J P = 1/2+ P-wave contribution (and also an explicit Σ(1385) 3/2+ state)
gives results consistent with the findings reported above, with the pole positions at (1364−i43) MeVand (1430 − i15) MeV, respectively [20].
The main component for the Λ(1405) is the pole 1, whose position converges within a relatively
small region near the K̄N threshold. On the other hand, the position of the pole 2 shows a
sizeable scatter. Detailed studies of the πΣ spectrum in various reaction processes, together withthe precise experimental lineshape (see e.g. the recent precise photoproduction data from theS. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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283. Pole Structure of the Λ(1405) Region

LEPS collaboration [21] and from the CLAS collaboration [22, 23], electroproduction data fromthe CLAS collaboration [24], and proton-proton collision data from COSY [25] and the HADEScollaboration [26]), will shed light on the position of the second pole. The πΣ spectra from theCLAS data are analyzed in Ref. [27] and Ref. [18]. It was shown in Ref. [18] that several solutions,which agree with the scattering data, are ruled out if confronted with the recent CLAS data. The
remaining solutions are collected as solution #2 and solution #4 in Table 83.1. The HADES data
are analyzed in Ref. [28] and Ref [29]. Although the result of the pole found in Ref. [28] is notcompatible with other results, the authors of Ref. [29] invoke the anomalous triangle singularity
mechanism to argue that the invariant mass distribution of the πΣ system is found at lower masses
than in other reactions. It is thus desirable to perform more comprehensive analyses of πΣ spectra
together with the systematic error analysis of the scattering data.
Table 83.1: Comparison of the pole positions of Λ(1405) in the complex
energy plane from next-to-leading order chiral unitary coupled-channel
approaches including the SIDDHARTA constraint. The lower two results
also include the CLAS photoproduction data.
approachRefs. [14, 15], NLO
Ref. [17], Fit II
Ref. [18], solution #2
Ref. [18], solution #4

pole 1 [MeV]+31424+7−23 − i 26−14+3+81421−2 − i 19−5+21434+2−2 − i 10−1+8+21429−7 − i 12−3pole 2 [MeV]+191381+18−6 − i 81−8+91388−9 − i 114+24−25+171330+4−i56−5−11+121325+15−15 − i 90−18References[1] R. Dalitz and S. Tuan, Phys. Rev. Lett. 2, 425 (1959).
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383. Pole Structure of the Λ(1405) Region[19] A. Cieplý et al., Nucl. Phys. A 954, 17 (2016), [arXiv:1603.02531].
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