The $\Lambda(1405)$ in resummed chiral effective field theory

We study the unitarized meson-baryon scattering amplitude at leading order in the strangeness $S=-1$ sector using time-ordered perturbation theory for a manifestly Lorentz-invariant formulation of chiral effective field theory. By solving the coupled-channel integral equations with the full off-shell dependence of the effective potential and applying subtractive renormalization, we analyze the renormalized scattering amplitudes and obtain the two-pole structure of the $\Lambda(1405)$ resonance. We also point out the necessity of including higher-order terms.


I. INTRODUCTION
With the breakthrough in the observation of gravitational waves by the Advanced LIGO and Virgo Collaborations [1], the study of neutron stars enters a new era of multi-message observations.
As the density increases, the strange degrees of freedom are expected to become active in the interior of dense objects like e.g. neutron stars. In addition to the appearance of hyperons, antikaon condensation can soften the equation of state and modify the bulk properties of neutron stars [2,3].
Besides, bound systems of antikaonic and multi-antikaonic nuclei have been studied e.g. in Ref. [4]. Those phenomena attract attention to studying the dynamics of the strongKN interaction.
Such investigations are also essential to deepen our understanding of the SU(3) dynamics in nonperturbative QCD.
Among the above mentioned theoretical methods, the chiral unitary approach that relies on chiral perturbation theory [50,51] retains a special place as it incorporates important constraints from the chiral symmetry on the dynamical generation of Λ(1405). The most-interesting phenomenon of the two-pole structure of Λ(1405), i.e. that two poles are found on the same Riemann sheet, was first reported in Ref. [27]. The origin of this two-pole structure is attributed to the two attractive channels (πΣ andKN ) in the SU(3) basis. Details of the two-pole structure can be found in the dedicated review article [52]. Various studies have revealed that the higher pole (i.e. the one with the larger real part) is slightly below the threshold ofKN with the narrow width of the order of 10 MeV. However, for the lower pole of Λ(1405), there is about 50 MeV uncertainty in the real and imaginary parts obtained in different works [35][36][37][38][39], because the current experimental data are not very sensitive to the lower pole. This raised a debate on the one-pole versus two-pole structure of Λ(1405) state, see, e.g., Refs. [53][54][55][56][57]. Note that the mentioned studies finding only a single pole have some deficiencies as detailed e.g. in Ref. [52]. Notice further that in chiral unitary models, the scattering amplitudes depend on the momentum cutoff parameter (Λ) [26] or subtraction constant(s) [27,33,43] introduced to deal with the ultraviolet divergences in the unitarization procedure. This results in some model dependence of the pole position(s) of the Λ(1405).
Recently we have proposed a renormalizable approach to study meson-baryon scattering by utilizing time-ordered perturbation theory for a manifestly Lorentz-invariant formulation of chiral perturbation theory [58]. Effective potentials are defined as the sums of all possible two-particle irreducible time-ordered diagrams, and the integral equations for the meson-baryon scattering amplitudes are derived in time-ordered perturbation theory. Renormalized amplitudes are obtained by applying the subtractive renormalization to the solutions of the integral equations at leading order (LO), while higher-order corrections are included perturbatively in a similar fashion to Refs. [59][60][61]. As shown in Ref. [58], our approach can be successfully applied to the pion-nucleon system.
In the current work, we apply this framework to study meson-baryon systems with strangeness S = −1 at LO and investigate the nature of the S-wave Λ(1405) resonance. This study should be considered as a first step. In the future, we plan to extend it to include higher-order corrections and further experimental data in order to sharpen our conclusions.
The manuscript is organized as follows. In Sect. II we lay out the formalism to study mesonbaryon scattering in SU(3) unitarized chiral effective field theory based on a renormalizable approach. Our results are presented and discussed in Sect. III. We end with a summary and outlook in Sect. IV.

II. THEORETICAL FRAMEWORK
In this section we briefly outline the theoretical framework of our work, which is the three-flavor extension of the SU(2) formalism developed in Ref. [58].

A. Meson-baryon scattering amplitude
The on-shell amplitude of the elastic meson-baryon scattering process M 1 (q 1 ) + B 1 (p 1 ) → M 2 (q 2 ) + B 2 (p 2 ) can be parameterized as The Dirac spinor u(p, s) of a baryon is normalized according to where χ s is a two-component spinor with spin s, and ω B (p) = √ p 2 + m 2 is the energy. Following Ref. [58], we decompose the Dirac spinor as and consider u ho as a higher-order contribution. Using the leading approximation for the Dirac spinor we obtain the reduced amplitude which reads One essential feature of our framework is that it incorporates the fields corresponding to lowest-lying vector mesons as dynamical degrees of freedom of the effective Lagrangian. In this formulation, the Weinberg-Tomozawa term in the effective meson-baryon Lagrangian is saturated by the vector-meson exchange, which has a better ultraviolet behavior. The LO chiral Lagrangian used in our calculations has the form where . . . denotes the trace in the flavor space, and with the pion decay constant F 0 in the three-flavor chiral limit, and D and F -the axial-vector couplings. We use the coupling constant of the vector-field self-interaction g determined via the KSFR relation, M 2 V = 2g 2 F 2 0 . M denotes the quark-mass matrix and B 0 is related to the scalar quark condensate, m and M V stand for the octet baryon and the vector meson masses in the chiral limit, respectively. The SU(3) matrices collecting the pseudoscalar mesons, the octet baryons and the vector mesons are given, respectively, by At leading order, the meson-baryon scattering amplitude for the process is given by the time-ordered diagrams shown in Fig. 1. Besides the Born and crossed-Born diagrams, there are also the vector meson exchange contributions that replace the Weinberg-Tomozawa contact term. Notice that the contributions stemming from the second term in the vector meson propagator ∝ g µν − q µ q ν /M 2 V are suppressed compared to the ones of the first term for considered low-energy processes and need to be taken into account together with other higher-order corrections. We further emphasize that possible power-counting-breaking contributions stemming from the decays V → P P [62] in loop diagrams can be absorbed in vertex corrections [63].
In this LO calculation we employ the scattering amplitudes in the isospin limit using the averaged masses for the mesons and baryons. The LO potential in the isospin formalism is given by where the expressions are summed over all vector mesons V in the vector-meson-exchange contributions, and over the internal baryons B in the Born and crossed-Born terms. Further, E = √ s is the total energy of the meson-baryon system, and P = q 1 + p 1 = q 2 + p 2 , K = p 1 − q 2 = p 2 − q 1 .
The on-shell energy of a particle is given by ω X (p) ≡ m 2 X + p 2 . In the S = −1 sector, there are four coupled channels with isospin I = 0, namely πΣ,KN , ηΛ and KΞ. The various coefficients Tables I,  II and III, where the indices i, j represent the particle channels. Here, we use the phase convention We rewrite the LO potential as the central and spin-orbital parts, It is convenient to calculate the amplitude in the center-of-mass (CMS) frame with The partial wave projection of the potential in the isospin basis is given by where z = cos θ, with θ the angle between p and p , p ≡ |p| and P L (z) denotes the Legendre polynomial.

C. Partial wave integral equations and subtractive renormalization
In time-ordered perturbation theory one obtains the coupled-channel integral equation for the T -matrix, which is visualized in Fig. 2, where M i B i , M j B j and M B denote the initial, final and intermediate particle channels, and the two-body Green functions read Projecting onto specific partial waves in the |LJ basis, the integral equation is written as where p, p , k are defined as the magnitudes of the momenta, p = |p|, p = |p |, k = |k|. Since the LO potential can be divided into the one-baryon-reducible and irreducible parts, with V R = V (c) and V I = V (a+b) + V (d) , we can apply a subtractive renormalization to obtain the finite on-shell T -matrix. Using a symbolic notation, the meson-baryon integral equation can be rewritten as a system of coupled equations as In order to obtain the renormalized finite T-matrix, we replace the meson-baryon propagator . This corresponds to including the contributions of an infinite number of meson-baryon counter-terms (details can be found in Refs. [58,59]). As discussed in Ref. [58], the subtractive renormalization does not affect the dynamics of bound states or resonances.  [5]. Note that at this order, there is no free parameter (see also the discussion below). Thus, at this order we have pure predictions.
To obtain the I = 0KN scattering T -matrix with coupled-channel effects taken into account  Ref. [38], Fit II 1388 +9 Ref. [66], sol-2 Ref. [66], sol-4 It is further interesting to investigate the structure of the above two poles. Approaching the pole position z R , the on-shell scattering T -matrix can be approximated by where g i (g j ) represents the contribution to the coupling strength of the initial (final) transition  channel. In general, the couplings g i , g j , which can be extracted from the residues of the T -matrix, are complex-valued numbers. The couplings obtained for the Λ(1405) resonance are tabulated in Table V. One can see that the lower pole couples predominantly to the πΣ channel, while the higher pole couples strongly to theKN channel. This could explain the large imaginary part of the lower pole, which is the consequence of the strong πΣ coupling. Thus, the different coupling nature to the meson-baryon channels form the two-pole structure of Λ(1405).
Furthermore, we present the shape of the Λ(1405) spectrum in Fig. 3, where the calculated invariant mass distribution is compared with the experimental data of π − Σ + channel from Ref. [67].
The event distribution is calculated by taking into account both the πΣ → πΣ andKN → πΣ channels as described in Ref. [27].
Finally, we note that we have also calculated theKN scattering lengths for I = 0 and I = 1.
The isospin-zero scattering length turns out to be a 0 = −2.50+i1.37 fm, which is somewhat outside of the region allowed by combining the scattering data with the SIDDHARTA kaonic hydrogen result, see e.g. Ref. [68]. We expect this issue to be resolved upon including NLO corrections. Our LO result for the isovector scattering length a 1 = 0.33 + i0.72 fm is, on the other hand, within the allowed region mapped out from scattering and kaonic hydrogen data.

IV. SUMMARY AND OUTLOOK
In this paper we have studied meson-baryon scattering in the strangeness S = −1 sector to investigate the structure of the Λ(1405) resonance using time-ordered perturbation theory applied to the Lorentz-invariant effective chiral Lagrangian with the explicit inclusion of low-lying vector mesons. In the considered framework, the effective potential of the meson-baryon scattering is defined as a sum of two-particle irreducible time-ordered diagrams. The renormalized S-wave amplitudes with I = 0 are obtained by taking into account the full off-shell dependence of the potential in the coupled-channel integral equations and applying subtractive renormalization.
In our leading-order study with no free parameters, we obtain the two-pole structure of the Λ(1405) state. The higher pole at E R = 1431 − i 8 MeV is mainly coupled to theKN channel, while the lower one located at E R = 1338 − i 79 MeV couples mainly to the πΣ channel. It is worth noticing that our results are independent on the momentum cutoff. The obtained πΣ invariant mass distribution agrees well with the experimental data while theKN isoscalar scattering length is found to have a somewhat too large (in magnitude) real part.
The existing data and the upcoming experiments focused on investigating theKN dynamics, such as the lowest-energy beam of strange hadron production in JLab experiments [69], the kaonic hydrogen SIDDHARTA experiment [15], the photoproduction data from JLab [70] and the experiments with kaonic nuclear bound states [71,72] provide a strong motivation to extend our renormalizable framework to next-to-leading order. Work along these lines is in progress.