Silica-based nanocarriers of anionic Ru(II) nitrosyl complexes for “dark” anticancer cytotoxicity

Phys. Lett. B 855 (2024) 138802 Contents lists available at ScienceDirect Physics Letters B journal homepage: www.elsevier.com/locate/physletb Letter Light-quark mass dependence of the Λ(1405) resonance Xiu-Lei Ren Helmholtz Institute Mainz, Staudingerweg 18, Mainz, 55128, Germany A R T I C L E I N F O Editor: A. Schwenk Keywords: Kaon-nucleon interaction Chiral symmetry Λ(1405) A B S T R A C T We present the light-quark mass dependence of the Λ(1405) resonance at leading order in a renormalizable framework of covariant chiral effective field theory. The meson-baryon scattering amplitudes, which are obtained by solving the scattering equation within time-ordered perturbation theory, follow the quark mass trajectory of the Coordinated Lattice Simulations consortium. At 𝑀𝜋 ≈ 200 MeV and 𝑀𝐾 ≈ 487 MeV, our parameter-free prediction of Λ(1405) poles is consistent with the recent lattice results of BaSc Collaboration [Phys. Rev. Lett. 132, 051901 (2024)]. Varying the pion mass from 135 MeV to 400 MeV, we present the evolution of double-pole ̄ threshold; whereas the lower pole positions of Λ(1405): the higher pole remains a resonance around the 𝐾𝑁 undergoes a transition from resonance to a virtual state, and ultimately to a bound state of the 𝜋Σ system, which could be verified by the forthcoming lattice QCD simulations. 1. Introduction Studying the meson-baryon scattering in the SU(3) sector can deepen our understanding of nonperturbative QCD. In particular, the attractive interaction between anti-kaon (𝐾̄ ) and nucleon (𝑁 ), which triggers the existence of the Λ(1405) resonance [1], the antikaonic- and multi-antikaonic nuclei/atoms e.g., in Refs. [2,3], and the possible antikaonic condensation in neutron star [4,5], plays an important role in the strangeness nuclear physics [6,7]. The Λ(1405) state, which is a 𝐼(𝐽 𝑃 ) = 0(1∕2− ) baryonic resonance of strangeness 𝑆 = −1, was first predicted by Dalitz and Tuan [8,9] and then confirmed by hydrogen bubble chamber experiments in the 𝜋Σ spectrum [10]. Λ(1405) does not adhere to the prediction of the constituent quark model [11], thus categorizing it as an exotic candidate. With the development of chiral perturbation theory (ChPT) [12] and its unitization technique — chiral unitary approach (CUA) [13], Oller and Meißner [14] found that the Λ(1405) state could be composed of two dynamically generated 𝐼 = 0 poles, which are located ̄ thresholds. This further arouses tremendous between the 𝜋Σ and 𝐾𝑁 efforts in experimental and theoretical research to uncover the nature of the Λ(1405) resonance. The detailed discussion and the references can be found in the recent reviews [15–17] and the section 83 of Particle Data Group (PDG) review [1]. Besides, the most up-to-date experimental researches have reported, e.g., the photoproduction cross sections of 𝐾 + Λ(1405) and Λ(1405) lineshape at the BGOOD experiment [18]; the 𝐾 − 𝑝 → 𝜋 0 Σ0 , 𝜋 0 Λ cross sections near threshold at DAΦNE [19]; the 𝜋Σ invariant mass spectra in 𝐾 − -induced reactions on deuteron by J-PARC E31 Collaboration [20]; the 𝐾 − 𝑝 femtoscopic correlations by ALICE Collaboration [21]. On the theoretical side, the most recent ̄ interacstudies can be found e.g., in Ref. [22] to investigate the 𝐾𝑁 tion up to next-to-next-to-leading order of CUA; in Ref. [23] to perform the uncertainty analysis by including the correlations between the pole parameters within the next-to-leading order CUA; in Ref. [24] to constrain the chirally motivated models with the 𝜋Σ photoproduction mass spectra. Note that a consensus has yet to be reached on the one-/twopole structure of Λ(1405). Although the latest PDG review [1] has listed Λ(1405) as a four-star state and the second pole of Λ(1405) as Λ(1380) with 2-star rating. With the rapid progress of computing techniques, lattice QCD simulation is going to be a promising approach to identifying the Λ(1405) state and clarifying the controversy of the one-/two-pole structure. The earlier efforts can be found in e.g. Refs. [25–29]. Most recently, Baryon Scattering (BaSc) Collaboration has performed a first lattice calculation ̄ scattering amplitudes [30,31]. Usof the coupled-channel 𝜋Σ and 𝐾𝑁 ing a single 𝑁𝑓 = 2 + 1 ensemble generated by the Coordinated Lattice Simulations (CLS) consortium, they found two poles of the Λ(1405) resonance with the pseudoscalar masses 𝑀𝜋 ≈ 200 MeV and 𝑀𝐾 ≈ 487 MeV: ̄ threshold, and the the higher pole is a resonance just below the 𝐾𝑁 lower pole is a virtual bound state below the 𝜋Σ threshold. Although the finite-volume correction and the finite-lattice spacing effect are not yet implemented, the BaSc results provide an ideal playground for checking/verifying the predictive power of existing phenomenological models and chiral unitary approaches. E-mail address: xiulei.ren@uni-mainz.de. https://doi.org/10.1016/j.physletb.2024.138802 Received 5 April 2024; Received in revised form 29 May 2024; Accepted 9 June 2024 Available online 13 June 2024 0370-2693/© 2024 The Author(s). Published by Elsevier B.V. Funded by SCOAP³. (http://creativecommons.org/licenses/by/4.0/). This is an open access article under the CC BY license Physics Letters B 855 (2024) 138802 X.-L. Ren That is one purpose of the current work: to examine our renormalizable approach for the meson-baryon scattering, which was proposed in Ref. [32]. Starting with the covariant chiral effective Lagrangians and employing the time-ordered perturbation theory (TOPT), we define an effective potential as the sum of two-particle irreducible contributions of time-ordered diagrams. The corresponding (coupled-channel) scattering equation 𝑇 = 𝑉 + 𝑉 𝐺𝑇 can be derived self-consistently in TOPT. At leading order (LO), we take into account the full off-shell dependence in solving the scattering equation. The renormalized scattering amplitude is obtained by utilizing the subtractive renormalization scheme [33]. Note that the subtracted terms are fixed by the chiral symmetry, which is different from the often-used strategies in CUA, where the finite-cutoff or the subtraction constant varies to match the data. In this sense, our framework at LO provides a model-independent prediction for the meson-baryon scattering. As shown in Ref. [32], our approach has been successfully applied to the pion-nucleon system. Then, we extend this framework to the strangeness 𝑆 = −1 sector with isospin 𝐼 = 0 and study the Λ(1405) resonance at the physical point [34]. We solve the scattering equation with ̄ , 𝜂Λ, and 𝐾Ξ, and obtain the renorthe four coupled channels 𝜋Σ, 𝐾𝑁 malized 𝑇 -matrix in the isospin limit. Two Λ(1405) poles are found on the second Riemann sheet: the lower pole (1337.7 − 𝑖79.1 MeV) is close ̄ to the 𝜋Σ threshold, the higher pole (1430.9 − 𝑖8.0 MeV) just below 𝐾𝑁 threshold. Our LO results are consistent with the ones of the next-toleading order (NLO) study in CUA [35]. Coming to the unphysical quark mass region, it is interesting to extrapolate our parameter-free prediction of Λ(1405) poles and compare it with the BaSc lattice data. If such an agreement is reached, it indicates that our approach exhibits the predictive capability to some extent. Therefore, it enables us to investigate the quark mass dependence of the Λ(1405) state. Similar studies along this line can be found in Refs. [36–41]. Here we focus on the results along with the quark-mass trajectory of the CLS ensembles. Our model variables, the pseudoscalar meson masses, octet baryon masses, vector meson masses, and meson decay constants, have been calculated in Refs. [42–44] based on the CLS configuration. It provides a good opportunity to investigate the light-quark mass dependence of Λ(1405), which could be verified by the forthcoming results of the BaSc Collaboration. Note that this LO study can be thought as a first step and the higher order corrections are planned in the future to refine our conclusion. This article is organized as follows. In Sect. 2, we briefly present our renormalizable approach to study the 𝑆 = −1 meson-baryon scattering, followed by the description of the CLS lattice data. The comparison with the BaSc results and the quark mass dependence of Λ(1405) poles are given in Sect. 3. Finally, we summarize our study in Sect. 4. that we employ the vector-meson exchange (VME) to saturate the oftenused Weinberg-Tomozawa (WT) term since the former has a better ultraviolet behavior without changing the low-energy physics. For the strangeness 𝑆 = −1 sector with isospin 𝐼 = 0, the LO interaction for the process 𝑀𝑖 (𝑞1 ) + 𝐵𝑖 (𝑞2 ) → 𝑀𝑗 (𝑞2 ) + 𝐵𝑗 (𝑝2 ) is given as in TOPT ) ( 𝑀𝑉2 1 ∑ 𝑉 𝐶𝑀 𝐵 ,𝑀 𝐵 𝜔𝑀𝑖 (𝒒 1 ) + 𝜔𝑀𝑗 (𝒒 2 ) 2 𝑗 𝑗 𝑖 𝑖 𝜔𝑉 (𝒒 1 − 𝒒 2 ) 32𝐹0 𝑉 [ 1 × 𝐸 − 𝜔𝐵𝑖 (𝒑1 ) − 𝜔𝑉 (𝒒 1 − 𝒒 2 ) − 𝜔𝑀𝑗 (𝒒 2 ) ] 1 + 𝐸 − 𝜔𝐵𝑗 (𝒑2 ) − 𝜔𝑉 (𝒒 1 − 𝒒 2 ) − 𝜔𝑀𝑖 (𝒒 1 ) 𝑉LO = − 𝑚𝐵 (𝝈 ⋅ 𝒒 2 )(𝝈 ⋅ 𝒒 1 ) 1 ∑ 𝐵 𝐶𝑀 𝐵 ,𝑀 𝐵 2 𝑗 𝑗 𝑖 𝑖 𝜔 4𝐹0 𝐵 𝐵 (𝑷 ) 𝐸 − 𝜔𝐵 (𝑷 ) (𝝈 ⋅ 𝒒 1 )(𝝈 ⋅ 𝒒 2 ) 𝑚𝐵 1 ∑ ̃𝐵 , 𝐶𝑀 𝐵 ,𝑀 𝐵 + 2 𝑗 𝑗 𝑖 𝑖 𝜔 (𝑲) 𝐸 − 𝜔 (𝒒 4𝐹0 𝐵 𝐵 𝑀𝑖 1 ) − 𝜔𝑀𝑗 (𝒒 2 ) − 𝜔𝐵 (𝑲) + (1) where 𝐹0 and 𝑀𝑉 are the pseudoscalar decay constant and the vector meson mass, respectively, in the chiral limit. Both are related via the Kawarabayashi-Suzuki-Riazuddin-Fayyazuddin relation [47–49]: 𝑀𝑉2 = 2𝑔 2 𝐹02 with the coupling 𝑔 of the vector-field self-interaction. The constant coefficients 𝐶 𝑉 of the VME contribution are given in Table 1 of Ref. [34], and the coefficients of the Born (𝐶 𝐵 ) and crossed-Born (𝐶̃ 𝐵 ) terms, as the functions of the axial vector couplings 𝐷 and 𝐹 , are given in Tables 2 and 3 of Ref. [34]. In the center-of-mass (c.m.) frame, the four momenta of initial and final states are ( ) ( ) 𝑞1𝜇 = 𝜔𝑀𝑖 (𝒑), 𝒑 , 𝑝𝜇1 = 𝜔𝐵𝑖 (𝒑), −𝒑 , ) ) ( ( 𝑞2𝜇 = 𝜔𝑀𝑗 (𝒑′ ), 𝒑′ , 𝑝𝜇2 = 𝜔𝐵𝑗 (𝒑′ ), −𝒑′ , (2) with the relative momenta 𝒑, 𝒑′ , and the on-shell energy 𝜔𝑋 (𝒑) ≡ √ 𝑚2𝑋 + 𝒑2 . Furthermore, 𝐸 is the total energy of the meson-baryon system, and 𝑃 = 𝑞1 + 𝑝1 = 𝑞2 + 𝑝2 , 𝐾 = 𝑝1 − 𝑞2 = 𝑝2 − 𝑞1 . Note that the crossed Born term introduces poles when iterated in the scattering equation. To avoid this technical complication, as done in Refs. [50,51], we take the total energy 𝐸 in the denominator as the lowest threshold of the relevant channels. Such approximation does not affect the final results because the contribution of the crossed Born term is rather small to the 𝑆 -wave meson-baryon scattering. After performing the 𝑆 -wave projection of the LO interaction, one can plug it into the coupled-channel scattering equation in the partial wave basis, 2. Formalism ( ( ) ) ∑ 𝑇 𝑗𝑖 𝐸; 𝑝′ , 𝑝 = 𝑉 𝑗𝑖 𝐸; 𝑝′ , 𝑝 + 𝑑𝑘 𝑘2 𝑗𝑛 𝑉 (𝐸; 𝑝′ , 𝑘) ∫ (2𝜋)3 𝑛 In this section, we first lay out the formalism to study the 𝑆 = −1 meson-baryon scattering in the SU(3) unitarized chiral EFT based on a renormalizable framework. The details can be found in Ref. [34]. After that, we investigate the light-quark mass dependence of our model variables by the combination of chiral formulae and lattice QCD data based on the CLS configurations in order to study the evolution of Λ(1405) poles. It is worth noting that the three-body effects from 𝜋𝜋Λ states could be involved in the Λ(1405) region, particularly as we move closer to the physical point. This complicated estimation is beyond the scope of this Letter, and thus we do not include it, as is usually done in the literature. Interested readers can refer to e.g. Refs. [45,46] for more details about the three-body scatterings. (3) × 𝐺𝑛 (𝐸) 𝑇 𝑛𝑖 (𝐸; 𝑘, 𝑝), where 𝑖, 𝑗, 𝑛 denote the initial, final and intermediate particle channels of meson and baryon states. The two-body Green function, which is obtained according to the diagrammatic rules of TOPT [50], has the following form 𝐺𝑛 (𝐸) = 𝑚𝐵𝑛 1 . 2 𝜔𝑀𝑛 (𝒌) 𝜔𝐵𝑛 (𝒌) 𝐸 − 𝜔𝑀𝑛 (𝒌) − 𝜔𝐵𝑛 (𝒌) + 𝑖𝜖 (4) In order to obtain the renormalized scattering 𝑇 -matrix, it is convenient to separate the leading order potential into the one-baryon reducible part and irreducible part, 2.1. Meson-baryon scattering in TOPT 𝑉LO = 𝑉𝑅 + 𝑉𝐼 , (5) where 𝑉𝑅 is the Born term, which can separate into two diagrams by cutting only the single-baryon line, and 𝑉𝐼 includes the remaining terms, At LO, the meson-baryon scattering potential contains the vectormeson exchange contribution, the Born and crossed-Born terms. Note 2 Physics Letters B 855 (2024) 138802 X.-L. Ren which cannot do such separation. Then, one can rewrite the scattering equation (Eq. (3)), 𝑇 = 𝑉 + 𝑉 𝐺𝑇 , as the three coupled equations: 𝑇 = 𝑇𝐼 + (1 + 𝑇𝐼 𝐺) 𝑇𝑅 (1 + 𝐺 𝑇𝐼 ), 𝑇𝐼 = 𝑉𝐼 + 𝑉𝐼 𝐺 𝑇𝐼 , (6) 𝑇𝑅 = 𝑉𝑅 + 𝑉𝑅 𝐺 (1 + 𝑇𝐼 𝐺) 𝑇𝑅 . It is found that the irreducible part 𝑇𝐼 is finite in the removed regulator limit. To demonstrate this fact, we take the vector-meson-exchange potential as an example. First, we investigate the UV behavior of the once-iterated contribution 𝑉 𝐺𝑉 at one-loop level: 𝐼𝑉 𝐺𝑉 = 𝑑3𝑘 𝑉 (𝑝′ , 𝑘) 𝐺(𝑘) 𝑉VME (𝑘, 𝑝). ∫ (2𝜋)3 VME Fig. 1. Left panel: Linear dependence of 𝑀𝐾2 in terms of 𝑀𝜋2 for the quark mass trajectory 𝑚 = 𝑚symm . Right panel: The octet-baryon masses as the functions of the pion mass with the mass trajectory 𝑚 = 𝑚symm . The dots are the lattice results, the stars stand for the experimental masses, corrected for the isospinbreaking effects. (7) When the integral momentum 𝑘 goes to infinity, the one-loop integral becomes 𝐼𝑉 𝐺𝑉 → 𝑑3𝑘 1 1 1 , ∫ (2𝜋)3 𝑘 𝑘3 𝑘 chiral analysis of the meson and baryon masses from the RQCD Collaboration [42] based on the CLS configuration, which is also utilized by the BaSc Collaboration. First, we use a linear function of 𝑀𝜋 and 𝑀𝐾 (8) which converges. This benefit arises from the VME potential, which has a milder UV behavior compared to the often-used WT term. One can easily verify that the infinite iteration of 𝑉VME in the ladder diagram also converges. Besides, the crossed-Born term presents a similar UV behavior to the VME counterpart. Because the half-off-shell potential 𝑉CB (𝑘, 𝑝) approaches 𝑉CB (𝑘, 𝑝) → 𝝈 ⋅ 𝒑𝝈 ⋅ 𝒌 , 𝑘2 𝑀𝐾2 = 𝑎 + 𝑏𝑀𝜋2 , (15) as done in Ref. [53], to parameterize the quark mass dependence of the kaon mass for the quark mass trajectory of 𝑚 = 𝑚symm , keeping the trace of the bare quark-mass matrix fixed 𝑇 𝑟(𝑀) = const., given by the CLS ensembles. The lattice pion and kaon masses in Ref. [42] have been corrected by including the finite-volume effects. The values of the pion and kaon masses at the physical point are (9) when the integral momentum 𝑘 goes to infinity. Regarding the reducible part, because the iteration of the Born term is quadratically divergent, thus the only divergent term in the total 𝑇 -matrix originates from the reducible part 𝑇𝑅 . To achieve renormalization, we first rewrite the Born potential in a separable form, 𝑀𝜋,phys = 134.8(3) MeV, 𝑀𝐾,phys = 494.2(3) MeV, (16) where 𝜉 𝑇 (𝑝) is defined as 𝜉 𝑇 (𝑝) = (1, 𝑝), and 𝐶(𝐸) denotes a 2 ×2 matrix. representing the electrically neutral isospin-averaged results [54]. The best fitting result is shown in the left panel of Fig. 1 with 𝑎 = 0.25189 GeV2 and 𝑏 = −0.42143. Then, the quark mass dependence of 𝑀𝜂 is determined via the Gell-Mann-Okubo relation 𝑀𝜂2 = 𝑇𝑅 (𝑝′ , 𝑝; 𝐸) = 𝜉 𝑇 (𝑝′ ) 𝜒(𝐸) 𝜉(𝑝), As to the lowest-lying octet baryon masses, we employ the mass expressions at next-to-leading order in covariant baryon ChPT 𝑉𝑅 (𝑝′ , 𝑝; 𝐸) = 𝜉 𝑇 (𝑝′ ) 𝐶(𝐸) 𝜉(𝑝), (10) (4𝑀𝐾2 − 𝑀𝜋2 )∕3. Then, the reducible 𝑇𝑅 matrix can be expressed in a separable form (11) where 𝜒(𝐸), as given below, is divergent, [ ]−1 𝜒(𝐸) = 𝐶(𝐸)−1 − 𝜉 𝐺(𝐸) 𝜉 𝑇 − 𝜉 𝐺(𝐸) 𝑇𝐼𝑆 𝐺(𝐸) 𝜉 𝑇 . 𝑚𝐵 = 𝑚0 + (12) + Then, one can apply the subtractive renormalization [33] by replacing the meson-baryon Green function 𝐺(𝐸) with the subtracted one 𝑆 𝐺 (𝐸) = 𝐺(𝐸) − 𝐺(𝑚𝐵 ), 𝑃 =𝜋, 𝐾 𝜉𝑃(2)𝐵 𝑀𝑃2 ∑ 1 𝜉 (3) 𝐻𝐵 (4𝜋𝐹0 )2 𝑃 =𝜋, 𝐾,𝜂 𝑃 𝐵 ( 𝑀𝑃 𝑚0 ) (17) , (2) where 𝑚0 is the baryon mass in the chiral limit. The LO coefficients 𝜉𝑃 𝐵 , as the function of low-energy constants (LECs) 𝑏0, 𝐷, 𝐹 , can be found (13) (3) in Table 1 of Ref. [53]. The NLO coefficients 𝜉𝑃 𝐵 , the combination of the baryon axial coupling constants 𝐷 and 𝐹 , are given in Table 2 of Ref. [53] with the one-loop function 𝐻𝐵 (𝑥) to systematically remove the divergences. This corresponds to taking into account the counter-terms generated by the renormalization of the baryon masses and the meson-baryon coupling constants. Note that such subtraction, which is an analogy to the extended-on-mass-shell (EOMS) scheme [52], corresponds to the expansion around the threshold and can systematically remove the chiral power-counting breaking terms in the iterated amplitudes. Finally, we obtain a renormalized 𝑇 -matrix ( ( ) ) 𝑇 = 𝑇𝐼𝑆 + 𝜉 𝑇 + 𝑇𝐼𝑆 𝐺𝑆 𝜉 𝑇 𝜒 𝑆 (𝐸) 𝜉 + 𝜉 𝐺𝑆 𝑇𝐼𝑆 , ∑ [ 𝐻𝐵 (𝑥) = −2𝑥 3 𝑥 log(𝑥) + 2 √ ] ( ) 𝑥 𝑥2 arccos , 1− 4 2 (18) which is obtained in the EOMS scheme [52]. The 𝐹0 is the pseudoscalar meson decay constant in the chiral limit. To obtain the light-quark mass dependence of the octet baryon mass, we perform the fit of the RQCD results [42], where the finite-volume corrections and the discretization effects are taken into account. Furthermore, we also include the experimental value of octet baryon masses, where the isospin-breaking effects from QED and QCD are removed [42]. We use the SU(6) relation 𝐹 = 2∕3𝐷 and 𝐷 + 𝐹 = 𝑔𝐴 = 1.267 to fix 𝐷 = 0.760 and 𝐹 = 0.507. For the meson decay constant, we use the SU(3) averaged value 𝐹0 = 1.17𝐹𝜋 with 𝐹𝜋 = 92.4 MeV to achieve a relatively better description of lattice data. Then, we have four free parameters, and the best-fit results lead to 𝑚0 = 0.813 GeV, 𝑏0 = −0.669 GeV−1 , 𝑏𝐷 = 0.0254 GeV−1 , and (14) where the superscript 𝑆 demonstrates the replacement of 𝐺 as 𝐺𝑆 in the expressions of 𝑇𝐼 and 𝜒(𝐸). 2.2. Quark mass dependence of our model variables First, we investigate the quark mass dependence of the pseudoscalar meson masses 𝑀𝑃 and the octet baryon masses 𝑚𝐵 , which appear in both of the LO chiral potential and the two-body Green function. Confronting the recent lattice results of Λ(1405) [30,31], we conduct the 3 Physics Letters B 855 (2024) 138802 X.-L. Ren Table 1 ̄ coupled channel calculation Pole positions of two Λ(1405) poles and the corresponding couplings 𝑔𝑖 from the 𝜋Σ-𝐾𝑁 at 𝑀𝜋 = 203.7 MeV and 𝑀𝐾 = 486.4 MeV. Pole positions from the full four-coupled channel calculation are also given. For comparison, the BaSc results are listed with one combined uncertainty. BaSc [30,31] This work Λ(1405) 𝑧𝑅 [MeV] 𝑧𝑅 a [MeV] 𝑧𝑅 [MeV] 𝑔𝜋Σ 𝑔𝐾𝑁 ̄ |𝑔𝜋Σ |∕|𝑔𝐾𝑁 ̄ | Lower pole Higher pole 1392(18) 1455(21) − 𝑖11.5(6.0) 1389.05 1464.55 − 𝑖9.44 1387.14 1469.86 − 𝑖4.71 0.021 + 𝑖1.87 0.038 + 𝑖0.98 0.017 + 𝑖1.55 1.51 − 𝑖1.22 1.21 0.50 a ̄ , 𝜂Λ, and 𝐾Ξ. The full calculation with four coupled channels: 𝜋Σ, 𝐾𝑁 MeV2 and 𝐿𝑟8 = 1.143 × 103 MeV2 . Note that 𝐿𝑟8 is slightly changed, but 𝐿𝑟5 decreases around 1∕3 times in comparison with the value given in Fit I of Ref. [56]. This observation could call for a high-statistics simulation of 𝐹𝜋 at small pion masses. 3. Results and discussion We first present our prediction of the Λ(1405) poles with unphysical quark masses 𝑀𝜋 ≈ 200 MeV and 𝑀𝐾 ≈ 487 MeV. For comparison with ̄ the lattice result of BaSc Collaboration [30,31], we focus on the 𝜋Σ-𝐾𝑁 coupled channel and use the same hadron masses: 𝑀𝜋 = 203.7 MeV, 𝑀𝐾 = 486.4 MeV, 𝑚𝑁 = 979.8 MeV, and 𝑚Σ = 1193.9 MeV to calculate the scattering amplitudes. The vector meson masses appearing in the LO potential are estimated through the chiral expansion Eq. (19) with 𝑀𝜋 = 203.7 MeV. For the decay constant, we take 𝐹0 = 𝐹𝜋 = 93.2 MeV, as used by the BaSc lattice simulation, instead of the chiral limit value. The difference would be of the higher order. The axial vector couplings are fixed as 𝐷 = 0.760 and 𝐹 = 0.507 with 𝐷 + 𝐹 = 𝑔𝐴 = 1.267. To obtain the meson-baryon scattering amplitudes, we first solve the integral equation by including the full off-shell dependence of the kernel potential. Such treatment is always employed in the study of the nucleon-nucleon interaction within the chiral EFT, e.g. Refs. [57,58]. Furthermore, the pole positions of resonances can change in solving the meson-baryon integral loop with the full off-shell dependence [59–61] in comparison with the approximation of the on-shell factorization. Secondly, as mentioned above, we employ the subtractive renormalization to obtain the finite 𝑇 -matrix by taking the cutoff limit Λ → ∞. In practice, a sharp cutoff regulator 𝐹Λ (𝑝′ , 𝑝) = 𝜃(Λ − 𝑝′ ) 𝜃(Λ − 𝑝) is employed with Λ = 10 GeV, which is sufficiently large to get rid of the finiteartifacts (as shown in the following). Therefore, our framework does not have the usual obstacle of the cutoff dependence of the 𝑇 -matrix present in the traditional chiral unitary approaches, which results in parameter-free predictions. ̄ We are in the position to search poles appearing in the 𝜋Σ-𝐾𝑁 coupled-channel 𝑇 -matrix. One can perform an analytic continuation of the 𝑇 -matrix into the complex 𝑠-plane. The 4 Riemann sheets, which are labeled by the sign√of the imaginary part of the c.m. momentum 𝑝𝑖 = 𝜆1∕2 (𝑠, 𝑀𝑃2 , 𝑚2𝐵 )∕(2 𝑠) for each channel, are introduced. The most relevant Riemann sheets in the current work are the physical sheet (+, +) and the second Riemann sheet (−, +) with only the 𝜋Σ channel open for decay. Furthermore, in order to investigate the structures of the poles, one can parameterize the on-shell scattering 𝑇 -matrix around the pole position 𝑧𝑅 , Fig. 2. The vector meson masses (left panel) and the pion decay constant (right panel) as the functions of the pion mass with the quark mass trajectory 𝑚 = 𝑚symm . Notations are the same as Fig. 1. 𝑏𝐹 = −0.399 GeV−1 . The light-quark mass dependence of the baryon masses is shown in the right panel of Fig. 1. Besides, we also require the vector meson mass 𝑀𝑉 and the decay constant 𝐹0 in the chiral limit to obtain the LO potential (Eq. (1)). In practice, we use the experimental masses of vector mesons 𝑀𝜌, 𝜔, 𝐾 ∗ , 𝜙 and the pion decay constant 𝐹𝜋 or the SU(3) averaged decay constant in Ref. [34]. The introduced difference between the chiral limit values and the physical values of the vector meson masses and the meson decay constant is of higher order. As to the light-quark mass dependence of 𝑀𝑉 , only rho meson mass 𝑀𝜌 is simulated with the quark mass trajectory 𝑚 = 𝑚symm of the CLS configuration by Mainz lattice group [43]. To determine the quark mass dependence of the vector meson masses, we use the results of LO ChPT in the SU(3) sector [55], ( ) 𝑀𝜌2 = 𝑀𝜔2 = 𝑀02 + 2𝜆𝑚 𝑀𝜋2 + 𝜆0 2𝑀𝐾2 + 𝑀𝜋2 , ( ) 𝑀𝐾2 ∗ = 𝑀02 + 2𝜆𝑚 𝑀𝐾2 + 𝜆0 2𝑀𝐾2 + 𝑀𝜋2 , ( ) 𝑀𝜙2 = 𝑀02 + 4𝜆𝑚 𝑀𝐾2 − 2𝜆𝑚 𝑀𝜋2 + 𝜆0 2𝑀𝐾2 + 𝑀𝜋2 , (19) where 𝑀𝜌 = 𝑀𝜔 . The three LECs 𝑀0 , 𝜆0 , and 𝜆𝑚 are determined by fitting the physical values of vector meson masses and the lattice data of 𝑀𝜌 , resulting in 𝑀0 = 0.538 GeV, 𝜆𝑚 = 0.468, 𝜆0 = 0.593. The reasonable fitting result is presented in the left panel of Fig. 2. As to the quark mass dependence of the pion decay constant 𝐹𝜋 , the lattice results of the CLS configuration are reported in Ref. [44] with the quark mass trajectory 𝑚 = 𝑚symm . From Fig. 2, it seems that the lattice data for the small pion masses are slightly lower than the empirical value. To describe the lattice results and obtain the pion mass dependence of 𝐹𝜋 , we follow the fitting procedure outlined in Ref. [38]. For further details and formalism, one can refer to the appendix of Ref. [38]. The lowest order 𝑀0𝜋 , 𝑀0𝐾 , and the chiral limit value 𝐹0 are adjusted by reproducing the experimental value and each set of lattice results of 𝑀𝜋 , 𝑀𝐾 and 𝐹𝜋 given in Ref. [44]. We take two strategies to perform the fit: first, the LECs 𝐿𝑟𝑖 , 𝑖 = 4, 5, 6, 8 are fixed as Fit I of Ref. [56], where the experimental and lattice meson-baryon scattering data are fitted. The pion mass dependence of 𝐹𝜋 is presented as the dashed line in Fig. 2-(b). Although the lattice data with small pion masses are well covered, the experimental 𝐹𝜋 is underestimated and the lattice data with large pion masses are overestimated. To achieve a relatively compatible description, we release 𝐿𝑟5 and 𝐿𝑟8 in the fit, which leads to the solid line 𝑇𝑖𝑗 ≃ 4𝜋 𝑔 𝑖 𝑔𝑗 𝑧 − 𝑧𝑅 , (20) where the (complex) 𝑔𝑖 (𝑔𝑗 ) couplings are introduced to denote the coupling strength of the resonance pole to the initial (final) transition channel. Those couplings can be extracted from the residues of the 𝑇 matrix. We find two poles on the (−, +) Riemann sheet, one is resonance ̄ threshold (denoted as “higher pole”), and another is the around 𝐾𝑁 virtual bound state just below 𝜋Σ threshold (denoted as “lower pole”). Their pole positions and the coupling strengths are tabulated in Table 1. One can see that our prediction of pole positions is consistent with the in Fig. 2-(b). In this case, the values of 𝐿𝑟5,8 come to 𝐿𝑟5 = 0.484 × 103 4 Physics Letters B 855 (2024) 138802 X.-L. Ren the right panel of Fig. 4. At the physical point 𝑀𝜋 ≈ 135 MeV, the lower pole is found on the (−, +, +, +) sheet and is recognized as a resonance. Its position 𝑧𝑅 = 1320.5 − 𝑖80.4 MeV is above the 𝜋Σ threshold with a relatively large imaginary part. As 𝑀𝜋 increases, the mass of the lower pole slightly rises (but remains below the 𝜋Σ threshold when 𝑀𝜋 > 146 MeV), while the width decreases rapidly. Correspondingly, the effective coupling of the lower pole to the 𝜋Σ channel also increases. This indicates that the interaction strength of 𝜋Σ is enhanced along with the increase of 𝑀𝜋 . When the interaction is strong enough, the resonance nature becomes a virtual bound state with 𝑀𝜋 ≥ 197 MeV. Because the 𝜋Σ interaction strength is continually enhanced, when 𝑀𝜋 ≥ 269 MeV, the lower pole becomes a bound state, which is located on the physical sheet. And the binding energy increases as the pion mass increases. From the significantly different pictures of both poles’ evaluations, ̄ interaction gradually strengthens with the one could infer that the 𝐾𝑁 increase of pion mass. Meanwhile, the strength of the 𝜋Σ interaction changes rapidly and is enhanced sufficiently to render the lower pole of Λ(1405) as a bound state. We would like to remark that the similar analysis of the evolution of the Λ(1405) pole positions at different quark masses has been reported ̄ in the 𝜋Σ-𝐾𝑁 coupled-channel study [40] by using the WeinbergTomozawa term within the traditional CUA, where the light-quark mass dependence of the decay constant and the adjustable subtraction constants are neglected. In contrast, we incorporate the pion mass dependence of all the model variables based on the CLS configuration. Avoid introducing the finite cutoff or the subtractive constants, we obtain the renormalized scattering amplitudes. In this sense that, our LO results of the quark mass dependence of Λ(1405), illustrated in Fig. 4, is a model-independent prediction. The tendency of pole evolution could be checked by the forthcoming studies from the BaSc Collaboration. Furthermore, it is worth mentioning that in Refs. [36,39,41], the evolution of two Λ(1405) poles has been studied in the SU(3) trajectory. That’s corresponding to a trajectory from the SU(3) flavor limit to the 1 physical point by keeping the average quark mass 𝑚̄ = (2𝑚𝑢∕𝑑 + 𝑚𝑠 ) 3 as a constant and varying a scaling parameter 𝑥 ∈ [0, 1], which is introduced in the pseudoscalar meson masses, octet baryon masses, the meson decay constants, and the subtraction constants. At LO studies, the higher pole changes from a bound state to a resonance varying 𝑥 from 0 to 1; correspondingly, the lower pole undergoes the bound state to a virtual state and finally comes to a resonance. However, in the recent NLO study [41], the evolutions of both Λ(1405) poles along the SU(3) trajectory are changed a lot by including the NLO corrections. It gives us a hint that one needs to go a step further and include the NLO corrections in our framework, at least, to see the changes of the evolution results shown in Fig. 4. Such a study is beyond the scope of the present manuscript and will be reported in future publications. Fig. 3. Evolution of lower pole position as a function of the momentum cutoff Λ. lattice results of BaSc Collaboration [30,31], except for the width of the higher pole, which approaches the lower bound of lattice value. It is interesting to note that the pole positions (𝑧𝑅 = 1389.05 MeV and 1464.55 − 𝑖9.44 MeV) are closer to the lattice results if we perform a full ̄ , 𝜂Λ, and 𝐾Ξ. Furthercalculation with four coupled channels 𝜋Σ, 𝐾𝑁 more, the values of 𝑔𝜋Σ,𝐾𝑁 and their ratios indicate that the lower pole ̄ couples predominantly to the 𝜋Σ channel, while the higher pole couples ̄ channel. The ratios of couplings are also consistent strongly to the 𝐾𝑁 with the BaSc results [30,31]. Additionally, in Fig. 3, we present the evolution of the lower pole position1 by varying the momentum cutoff Λ from 700 MeV to 10 GeV. One can see that the finite-cutoff (Λ = 700 MeV) artifact changes the lower pole position up to 27 MeV. When Λ is up to 3 GeV, the cutoff-independent result is achieved, which guaran̄ scattering amplitudes in our tees the renormalizability of the 𝜋Σ-𝐾𝑁 scheme. Next, we investigate the evolution of Λ(1405) poles by varying the pion mass from 𝑀𝜋 ≈ 135 MeV to 400 MeV along with the quark mass trajectory 𝑚 = 𝑚symm of the CLS configuration. ̄ coupled-channel calculation, we Instead of performing the 𝜋Σ-𝐾𝑁 ̄ , 𝜂Λ, and 𝐾Ξ to obtain prefer to span all four coupled channels 𝜋Σ, 𝐾𝑁 the meson-baryon scattering amplitudes in the 𝑆 = −1 sector. Numerically, the LO potential (Eq. (1)) varies along with the pion mass since the masses of pseudoscalar mesons, octet baryons, octet vector mesons, and the pion decay constant,2 are determined by applying the chiral expansion formulas with the CLS quark mass trajectory 𝑚 = 𝑚symm . Correspondingly, the pion mass dependence also enters the meson-baryon Green function. Thus, by solving the integral equation (Eq. (3)) and utilizing the subtractive renormalization, one can obtain the renormalized 𝑇 -matrix. To search poles of 𝑇 -matrix on the complex 𝑠-plane, one needs ̄ to introduce 16 Riemann sheets for the 𝜋Σ, 𝐾𝑁, 𝜂Λ, and 𝐾Ξ couple channels. Similar to the two coupled-channel case, the most relevant Riemann sheets are the physical sheet (+, +, +, +) and the second Riemann sheet (−, +, +, +). By changing the pion mass from the physical point 𝑀𝜋 = 135 MeV ̄ threshto 400 MeV, we always obtain two poles around the 𝜋Σ and 𝐾𝑁 olds in the coupled-channel 𝑇 -matrix. The corresponding pole positions of Λ(1405) are presented in Fig. 4 as the function of pion mass. The higher pole of Λ(1405) is always a resonance, located on the (−, +, +, +) sheet. Its mass (width) increases (decreases) when the pion mass increases. For the small pion mass region, the higher pole is slightly below ̄ threshold, while for 𝑀𝜋 > 225 MeV, it begins above. This inthe 𝐾𝑁 ̄ channel. Regarding dicates the higher pole couples strongly to the 𝐾𝑁 the lower pole of Λ(1405), it is intriguing to observe that the evolution of its position becomes more complicated as 𝑀𝜋 increase, as shown in 4. Conclusion We applied a renormalizable framework of meson-baryon scattering within covariant chiral effective field theory to investigate the Λ(1405) resonance in the unphysical quark mass region. Confronting the recent lattice calculation of BaSc Collaboration, our leading order analysis focuses on the quark mass trajectory 𝑚 = 𝑚symm of the CLS configuration. First, we compared our parameter-free prediction of Λ(1405) double poles with the lattice data of BaSc at 𝑀𝜋 ≈ 200 MeV and 𝑀𝐾 ≈ 487 MeV. The consistency result indicates the predictive capability of our renormalizable approach. Therefore, we investigated the quark mass dependence of the Λ(1405) state along with the CLS quark mass trajectory 𝑚 = 𝑚symm . The ̄ threshold higher pole of Λ(1405) is always a resonance around the 𝐾𝑁 by changing 𝑀𝜋 from 135 MeV to 400 MeV, while the evolution of the lower pole of Λ(1405) is complicated: transitioning from a resonance to a virtual state, and ultimately to a bound state of 𝜋Σ system. Such behavior could be verified by the forthcoming lattice calculation of the BaSc Collaboration. 1 As to the higher pole, its mass and width have a similar Λ dependence. Note that our description of 𝐹𝜋 is higher than the lattice data when 𝑀𝜋 ≤ 200 MeV, as shown in Fig. 2. 2 5 Physics Letters B 855 (2024) 138802 X.-L. Ren Fig. 4. Λ(1405) poles as the function of pion mass with the CLS quark mass trajectory 𝑚 = 𝑚symm . The upper (lower) panel is the real (imaginary) part of the pole position. The black-, red-, and blue-solid curves denote the pole as the resonance, virtual state, and bound state, respectively. The dashed lines correspond to the ̄ (left panel) and 𝜋Σ (right panel) thresholds. 𝐾𝑁 In order to polish the current results, we plan to extend the study to next-to-leading order (at least) in our renormalizable framework. By incorporating the constraints from the experimental data (see the recent review [15]), such as a large amount of cross sections for 𝐾 − 𝑝 scattering into different final channels, the relevant branching ratios, and the precise energy shift and width of the 1𝑠 state in kaonic hydrogen by the SIDDHARTA Collaboration [62] etc., we can improve the description of ̄ scattering to better determine the pole positions of Λ(1405). At the 𝐾𝑁 this level, one could provide a moderately reliable prediction about the light-quark dependence of Λ(1405) poles. [7] L. Tolos, L. Fabbietti, Strangeness in nuclei and neutron stars, Prog. Part. Nucl. Phys. 112 (2020) 103770, https://doi.org/10.1016/j.ppnp.2020.103770, arXiv: 2002.09223. [8] R.H. Dalitz, S.F. Tuan, A possible resonant state in pion-hyperon scattering, Phys. Rev. Lett. 2 (1959) 425–428, https://doi.org/10.1103/PhysRevLett.2.425. [9] R.H. Dalitz, S.F. Tuan, The phenomenological description of -K -nucleon reaction processes, Ann. Phys. 10 (1960) 307–351, https://doi.org/10.1016/0003-4916(60) 90001-4. [10] M.H. Alston, L.W. Alvarez, P. Eberhard, M.L. Good, W. Graziano, H.K. Ticho, S.G. Wojcicki, Study of resonances of the Sigma-pi system, Phys. Rev. Lett. 6 (1961) 698–702, https://doi.org/10.1103/PhysRevLett.6.698. [11] N. Isgur, G. Karl, P wave baryons in the quark model, Phys. Rev. D 18 (1978) 4187, https://doi.org/10.1103/PhysRevD.18.4187. [12] S. Weinberg, Phenomenological Lagrangians, Physica A 96 (1–2) (1979) 327–340, https://doi.org/10.1016/0378-4371(79)90223-1. [13] J.A. Oller, E. Oset, A. Ramos, Chiral unitary approach to meson meson and meson - baryon interactions and nuclear applications, Prog. Part. Nucl. Phys. 45 (2000) 157–242, https://doi.org/10.1016/S0146-6410(00)00104-6, arXiv:hep-ph/ 0002193. [14] J.A. Oller, U.G. Meissner, Chiral dynamics in the presence of bound states: kaon nucleon interactions revisited, Phys. Lett. B 500 (2001) 263–272, https://doi.org/ 10.1016/S0370-2693(01)00078-8, arXiv:hep-ph/0011146. [15] M. Mai, Review of the Λ(1405) A curious case of a strangeness resonance, Eur. Phys. J. Spec. Top. 230 (6) (2021) 1593–1607, https://doi.org/10.1140/epjs/s11734-02100144-7, arXiv:2010.00056. [16] T. Hyodo, M. Niiyama, QCD and the strange baryon spectrum, Prog. Part. Nucl. Phys. 120 (2021) 103868, https://doi.org/10.1016/j.ppnp.2021.103868, arXiv: 2010.07592. [17] U.-G. Meißner, Two-pole structures in QCD: facts, not fantasy!, Symmetry 12 (6) (2020) 981, https://doi.org/10.3390/sym12060981, arXiv:2005.06909. [18] G. Scheluchin, et al., Photoproduction of K+Λ(1405)→K+𝜋 0Σ0 extending to forward angles and low momentum transfer, Phys. Lett. B 833 (2022) 137375, https:// doi.org/10.1016/j.physletb.2022.137375, arXiv:2108.12235. [19] K. Piscicchia, et al., First simultaneous K−p→Σ0𝜋 0, Λ𝜋 0 cross section measurements at 98 MeV/c, Phys. Rev. C 108 (5) (2023) 055201, https://doi.org/10.1103/ PhysRevC.108.055201, arXiv:2210.10342. [20] S. Aikawa, et al., Pole position of Λ(1405) measured in d(K−,n)𝜋Σ reactions, Phys. Lett. B 837 (2023) 137637, https://doi.org/10.1016/j.physletb.2022.137637, arXiv: 2209.08254. [21] S. Acharya, et al., Constraining the KN coupled channel dynamics using femtoscopic correlations at the LHC, Eur. Phys. J. C 83 (4) (2023) 340, https://doi.org/10.1140/ epjc/s10052-023-11476-0, arXiv:2205.15176. [22] J.-X. Lu, L.-S. Geng, M. Doering, M. Mai, Cross-channel constraints on resonant antikaon-nucleon scattering, Phys. Rev. Lett. 130 (7) (2023) 071902, https://doi. org/10.1103/PhysRevLett.130.071902, arXiv:2209.02471. [23] D. Sadasivan, M. Mai, M. Döring, U.-G. Meißner, F. Amorim, J.P. Klucik, J.-X. Lu, L.-S. Geng, New insights into the pole parameters of the Λ(1380), the Λ(1405) and the Σ(1385), Front. Phys. 11 (2023) 1139236, https://doi.org/10.3389/fphy.2023. 1139236, arXiv:2212.10415. [24] A. Cieply, P.C. Bruns, Constraining the chirally motivated 𝜋Σ-𝐾 N models with the 𝜋Σ photoproduction mass spectra, Nucl. Phys. A 1043 (2024) 122819, https://doi. org/10.1016/j.nuclphysa.2024.122819, arXiv:2305.06205. [25] Y. Nemoto, N. Nakajima, H. Matsufuru, H. Suganuma, Negative parity baryons in quenched anisotropic lattice QCD, Phys. Rev. D 68 (2003) 094505, https://doi.org/ 10.1103/PhysRevD.68.094505, arXiv:hep-lat/0302013. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Data availability No data was used for the research described in the article. Acknowledgements We appreciate the valuable discussions with Daniel Mohler during the MENU 2023 conference. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), in part through the Research Unit (Photonphoton interactions in the Standard Model and beyond, Projektnummer 458854507—FOR 5327), and in part through the Cluster of Excellence (Precision Physics, Fundamental Interactions, and Structure of Matter) (PRISMA+ EXC 2118/1) within the German Excellence Strategy (Project ID 39083149). References [1] R.L. Workman, et al., Review of particle physics, PTEP 2022 (2022) 083C01, https:// doi.org/10.1093/ptep/ptac097. ̄ ̄ [2] N. Barnea, A. Gal, E.Z. Liverts, Realistic calculations of 𝐾𝑁𝑁 , 𝐾𝑁𝑁𝑁 , and ̄ quasibound states, Phys. Lett. B 712 (2012) 132–137, https://doi.org/10. 𝐾̄ 𝐾𝑁𝑁 1016/j.physletb.2012.04.055, arXiv:1203.5234. [3] C. Curceanu, et al., The modern era of light kaonic atom experiments, Rev. Mod. Phys. 91 (2) (2019) 025006, https://doi.org/10.1103/RevModPhys.91.025006. [4] D.B. Kaplan, A.E. Nelson, Strange goings on in dense nucleonic matter, Phys. Lett. B 175 (1986) 57–63, https://doi.org/10.1016/0370-2693(86)90331-X. [5] G.-Q. Li, C.H. Lee, G.E. Brown, Kaons in dense matter, kaon production in heavy ion collisions, and kaon condensation in neutron stars, Nucl. Phys. A 625 (1997) 372–434, https://doi.org/10.1016/S0375-9474(97)00489-2, arXiv:nucl-th/ 9706057. [6] A. Gal, E.V. Hungerford, D.J. Millener, Strangeness in nuclear physics, Rev. Mod. Phys. 88 (3) (2016) 035004, https://doi.org/10.1103/RevModPhys.88.035004, arXiv:1605.00557. 6 Physics Letters B 855 (2024) 138802 X.-L. Ren [26] B.J. Menadue, W. Kamleh, D.B. Leinweber, M.S. Mahbub, Isolating the Λ(1405) in lattice QCD, Phys. Rev. Lett. 108 (2012) 112001, https://doi.org/10.1103/ PhysRevLett.108.112001, arXiv:1109.6716. [27] G.P. Engel, C.B. Lang, D. Mohler, A. Schäfer, QCD with two light dynamical chirally improved quarks: baryons, Phys. Rev. D 87 (7) (2013) 074504, https://doi.org/10. 1103/PhysRevD.87.074504, arXiv:1301.4318. [28] J.M.M. Hall, W. Kamleh, D.B. Leinweber, B.J. Menadue, B.J. Owen, A.W. Thomas, R.D. Young, Lattice QCD evidence that the Λ(1405) resonance is an antikaonnucleon molecule, Phys. Rev. Lett. 114 (13) (2015) 132002, https://doi.org/10. 1103/PhysRevLett.114.132002, arXiv:1411.3402. [29] S. Meinel, G. Rendon, Charm-baryon semileptonic decays and the strange Λ* resonances: new insights from lattice QCD, Phys. Rev. D 105 (5) (2022) L051505, https:// doi.org/10.1103/PhysRevD.105.L051505, arXiv:2107.13084. [30] J. Bulava, et al., Two-pole nature of the Λ(1405) resonance from lattice QCD, Phys. Rev. Lett. 132 (5) (2024) 051901, https://doi.org/10.1103/PhysRevLett.132. 051901, arXiv:2307.10413. ̄ N scattering and the Λ(1405) reso[31] J. Bulava, et al., Lattice QCD study of 𝜋Σ-K nance, Phys. Rev. D 109 (1) (2024) 014511, https://doi.org/10.1103/PhysRevD. 109.014511, arXiv:2307.13471. [32] X.L. Ren, E. Epelbaum, J. Gegelia, U.-G. Meißner, Meson–baryon scattering in resummed baryon chiral perturbation theory using time-ordered perturbation theory, Eur. Phys. J. C 80 (5) (2020) 406, https://doi.org/10.1140/epjc/s10052-020-7991x, arXiv:2003.06272. [33] E. Epelbaum, A.M. Gasparyan, J. Gegelia, U.-G. Meißner, X.L. Ren, How to renormalize integral equations with singular potentials in effective field theory, Eur. Phys. J. A 56 (5) (2020) 152, https://doi.org/10.1140/epja/s10050-020-00162-4, arXiv:2001.07040. [34] X.-L. Ren, E. Epelbaum, J. Gegelia, U.-G. Meißner, The Λ(1405) in resummed chiral effective field theory, Eur. Phys. J. C 81 (7) (2021) 582, https://doi.org/10.1140/ epjc/s10052-021-09386-0, arXiv:2102.00914. ̄ amplitude from 𝜋Σ𝐾 + [35] M. Mai, U.-G. Meißner, Constraints on the chiral unitary 𝐾𝑁 photoproduction data, Eur. Phys. J. A 51 (3) (2015) 30, https://doi.org/10.1140/ epja/i2015-15030-3, arXiv:1411.7884. [36] D. Jido, J.A. Oller, E. Oset, A. Ramos, U.G. Meissner, Chiral dynamics of the two Lambda(1405) states, Nucl. Phys. A 725 (2003) 181–200, https://doi.org/10.1016/ S0375-9474(03)01598-7, arXiv:nucl-th/0303062. [37] C. Garcia-Recio, M.F.M. Lutz, J. Nieves, Quark mass dependence of s wave baryon resonances, Phys. Lett. B 582 (2004) 49–54, https://doi.org/10.1016/j.physletb. 2003.11.073, arXiv:nucl-th/0305100. [38] R. Molina, M. Döring, Pole structure of the Λ(1405) in a recent QCD simulation, Phys. Rev. D 94 (5) (2016) 056010, Addendum: Phys. Rev. D 94 (2016) 079901, https://doi.org/10.1103/PhysRevD.94.079901, arXiv:1512.05831. ̄ N coupled [39] P.C. Bruns, A. Cieplý, SU(3) flavor symmetry considerations for the K channels system, Nucl. Phys. A 1019 (2022) 122378, https://doi.org/10.1016/j. nuclphysa.2021.122378, arXiv:2109.03109. [40] J.-M. Xie, J.-X. Lu, L.-S. Geng, B.-S. Zou, Two-pole structures as a universal phenomenon dictated by coupled-channel chiral dynamics, Phys. Rev. D 108 (11) (2023) L111502, https://doi.org/10.1103/PhysRevD.108.L111502, arXiv:2307.11631. [41] F.-K. Guo, Y. Kamiya, M. Mai, U.-G. Meißner, New insights into the nature of the Λ(1380) and Λ(1405) resonances away from the SU(3) limit, Phys. Lett. B 846 (2023) 138264, https://doi.org/10.1016/j.physletb.2023.138264, arXiv:2308. 07658. [42] G.S. Bali, S. Collins, P. Georg, D. Jenkins, P. Korcyl, A. Schäfer, E.E. Scholz, J. Simeth, W. Söldner, S. Weishäupl, Scale setting and the light baryon spectrum in N𝑓 = 2 + 1 QCD with Wilson fermions, J. High Energy Phys. 05 (2023) 035, https:// doi.org/10.1007/JHEP05(2023)035, arXiv:2211.03744. [43] M. Cè, A. Gérardin, G. von Hippel, H.B. Meyer, K. Miura, K. Ottnad, A. Risch, T. San José, J. Wilhelm, H. Wittig, The hadronic running of the electromagnetic coupling and the electroweak mixing angle from lattice QCD, J. High Energy Phys. 08 (2022) 220, https://doi.org/10.1007/JHEP08(2022)220, arXiv:2203.08676. [44] M. Cè, et al., Window observable for the hadronic vacuum polarization contribution to the muon g-2 from lattice QCD, Phys. Rev. D 106 (11) (2022) 114502, https:// doi.org/10.1103/PhysRevD.106.114502, arXiv:2206.06582. [45] D. Sadasivan, A. Alexandru, H. Akdag, F. Amorim, R. Brett, C. Culver, M. Döring, F.X. Lee, M. Mai, Pole position of the a1(1260) resonance in a three-body unitary framework, Phys. Rev. D 105 (5) (2022) 054020, https://doi.org/10.1103/PhysRevD.105. 054020, arXiv:2112.03355. [46] Z.T. Draper, M.T. Hansen, F. Romero-López, S.R. Sharpe, Three relativistic neutrons in a finite volume, J. High Energy Phys. 07 (2023) 226, https://doi.org/10.1007/ JHEP07(2023)226, arXiv:2303.10219. [47] K. Kawarabayashi, M. Suzuki, Partially conserved axial vector current and the decays of vector mesons, Phys. Rev. Lett. 16 (1966) 255, https://doi.org/10.1103/ PhysRevLett.16.255. [48] Riazuddin, Fayyazuddin, Algebra of current components and decay widths of rho and K* mesons, Phys. Rev. 147 (1966) 1071–1073, https://doi.org/10.1103/PhysRev. 147.1071. [49] D. Djukanovic, M.R. Schindler, J. Gegelia, G. Japaridze, S. Scherer, Universality of the rho-meson coupling in effective field theory, Phys. Rev. Lett. 93 (2004) 122002, https://doi.org/10.1103/PhysRevLett.93.122002, arXiv:hep-ph/0407239. [50] V. Baru, E. Epelbaum, J. Gegelia, X.L. Ren, Towards baryon-baryon scattering in manifestly Lorentz-invariant formulation of SU(3) baryon chiral perturbation theory, Phys. Lett. B 798 (2019) 134987, https://doi.org/10.1016/j.physletb.2019.134987, arXiv:1905.02116. [51] X.L. Ren, E. Epelbaum, J. Gegelia, Λ-nucleon scattering in baryon chiral perturbation theory, Phys. Rev. C 101 (3) (2020) 034001, https://doi.org/10.1103/PhysRevC. 101.034001, arXiv:1911.05616. [52] T. Fuchs, J. Gegelia, G. Japaridze, S. Scherer, Renormalization of relativistic baryon chiral perturbation theory and power counting, Phys. Rev. D 68 (2003) 056005, https://doi.org/10.1103/PhysRevD.68.056005, arXiv:hep-ph/0302117. [53] X.L. Ren, L.S. Geng, J. Martin Camalich, J. Meng, H. Toki, Octet baryon masses in next-to-next-to-next-to-leading order covariant baryon chiral perturbation theory, J. High Energy Phys. 12 (2012) 073, https://doi.org/10.1007/JHEP12(2012)073, arXiv:1209.3641. [54] S. Aoki, et al., Review of lattice results concerning low-energy particle physics, Eur. Phys. J. C 77 (2) (2017) 112, https://doi.org/10.1140/epjc/s10052-016-4509-7, arXiv:1607.00299. [55] Y. Zhou, X.-L. Ren, H.-X. Chen, L.-S. Geng, Pseudoscalar meson and vector meson interactions and dynamically generated axial-vector mesons, Phys. Rev. D 90 (1) (2014) 014020, https://doi.org/10.1103/PhysRevD.90.014020, arXiv:1404.6847. [56] J. Nebreda, J.R. Pelaez, Strange and non-strange quark mass dependence of elastic light resonances from SU(3) unitarized chiral perturbation theory to one loop, Phys. Rev. D 81 (2010) 054035, https://doi.org/10.1103/PhysRevD.81.054035, arXiv:1001.5237. [57] E. Epelbaum, H.-W. Hammer, U.-G. Meissner, Modern theory of nuclear forces, Rev. Mod. Phys. 81 (2009) 1773–1825, https://doi.org/10.1103/RevModPhys.81.1773, arXiv:0811.1338. [58] R. Machleidt, D.R. Entem, Chiral effective field theory and nuclear forces, Phys. Rep. 503 (2011) 1–75, https://doi.org/10.1016/j.physrep.2011.02.001, arXiv:1105. 2919. [59] P.C. Bruns, M. Mai, U.G. Meissner, Chiral dynamics of the S11(1535) and S11(1650) resonances revisited, Phys. Lett. B 697 (2011) 254–259, https://doi.org/10.1016/j. physletb.2011.02.008, arXiv:1012.2233. [60] M. Mai, U.-G. Meissner, New insights into antikaon-nucleon scattering and the structure of the Lambda(1405), Nucl. Phys. A 900 (2013) 51–64, https://doi.org/10. 1016/j.nuclphysa.2013.01.032, arXiv:1202.2030. [61] O. Morimatsu, K. Yamada, Renormalization of the unitarized Weinberg-Tomozawa ̄ N–𝜋Σ coupled channels, Phys. interaction without on-shell factorization and I=0 K Rev. C 100 (2) (2019) 025201, https://doi.org/10.1103/PhysRevC.100.025201, arXiv:1903.12380. [62] M. Bazzi, et al., A new measurement of kaonic hydrogen X-rays, Phys. Lett. B 704 (2011) 113–117, https://doi.org/10.1016/j.physletb.2011.09.011, arXiv:1105. 3090. 7