K⁺Λ(1405) photoproduction and line shape

The Λ(1405) has remained an enigmatic state since its discovery 60 years ago. Located directly at the KN threshold, before the advent of the quark model it initially was predicted by Dalitz and Tuan as a KN molecule [1], later taken as a genuine 3-quark state, but its nature always remained disputed due to its unusally low mass in relation to the non-strange negative parity partner S₁₁(1535) and its strange angular momentum partner Λ(1520). The 2010 edition of the PDG then stated: “... settle the decades-long discussion about the nature of the Λ(1405)– true 3-quark state or mere KN threshold effect? – unambiguously in favor of the first interpretation.“ Only six years later then the backward roll: “The Λ(1405) resonance ... is the archetype of what is called a dynamically generated resonance, as pioneered by Dalitz and Tuan“ (PDG 2016).

Chiral unitary approaches describe the Λ(1405) as a superposition of two isospin-0 poles [2–5]. Due to different couplings of the two poles to the three charge combinations in the decay Λ(1405) → (Σπ)⁰, different Σ⁺π⁻, Σ⁻π⁺ and Σ⁰π⁰ invariant mass distributions (commonly referred to as line shapes) have been predicted and experimentally confirmed in K⁺Λ(1405) photoproduction [6]. The detailed analysis of Ref. [5] extracted poles at approximately 1330 and 1430 MeV with widths of 50–90 MeV for the lower pole and 10–12 MeV for the higher one. It had been shown previously that, in principle, the two-pole structure should be visible from the Σ⁰π⁰ decay channel alone [7].

In addition to the line shape, also the mechanism of how the Λ(1405) is created in photoproduction is of high interest. Commonly, it was assumed to mainly proceed through kaon t-channel exchange (c.f. Fig. 1 left). Such a processes should exhibit forward peaking of the cross section and dominance of small t, in particular if the Λ(1405) was a spacially extended molecular type state as suggested by meson-baryon dynamics. In contrast, if Λ(1405) photoproduction was proceeding through a triangle singularity as studied in Ref. [8] (Fig. 1 right), then the K+ angular distribution is expected flat with a drop at very forward directions. The triangle singularity is driven by a broad resonance [8] labelled N*(2030) in Fig. 1. This object is assumed to have a width of about 300 MeV due to mixing of vector-baryon and pseudoscalar-baryon states associated with N*(2080)(3/2)⁻ and N*(2090)(1/2)⁻.

Fig. 1: Mechanisms for γp → K⁺Λ(1405) photoproduction (taken from Ref. [8]).
Left: Kaon t-channel contribution.
Right: Triangle singularity involving the formation of N*(2030) and its decay to K*Σ.

To feed the triangle, according to the Coleman-Norton theorem [9] the N*(2030) must strongly couple to K*Σ, and it can be identified with the vectormeson-baryon type state involved in the cusp/bump structures observed in K⁰_sΣ^+/0 photoproduction [10]. With the BGOOD experiment we measured the reaction γp →K⁺Λ(1405) and reconstructed the subsequent decay Λ(1405) →π⁰Σ⁰. In contrast to the charged decay modes, this one is purely isospin I = 0, and contributions of the physically overlapping Σ(1385) are thus prohibited due to isospin conservation. After the decays π⁰ →γγ (B.R. ≅100 %), Σ⁰ →γΛ (100 %), and subsequently Λ →π⁻p (66 %), the total reaction to be observed is γp→K⁺(γγ)γπ−p. The detector is ideally suited to detect this complex final state.

In the analysis all event topologies comprising 3 neutral and 3 charged particles were used. A kinematic fit was applied to all possible combinations of particle assignments with the constraints of four-momentum conservation, and π⁰ and Λ masses. Events with a confidence level below 0.2 were discarded to improve the signal to background ratio. If the K⁺ was in the range of the forward spectrometer, then it was identified ther from momentum and β. This suppressed combinatorial background for forward kaon angles cos θ^K_cm >0.86. Generally, combinatorial background was determined and subtracted. Multi-dimensional background distributions were simulated and fitted to the observed distributions using CERN’s RooFit package [11].

Fig. 2: Line shape of π⁰Σ⁰ in the energy region of the Λ(1405). Only statistical errors shown.
Left: K⁺ in full angular range cos θ_cm = −1 to 1 for E_γ = 1550...2300 MeV. BGOOD data (black dots) in comparison to earlier results of ANKE [12] (blue dots, arbitrarily scaled) and CLAS [6] (red squares).
Right: K⁺in forward directions cos θ_cm > 0.86 for the two energy bins labelled inset. Vertical dotted linesindicate the position of the two poles predicted in Ref. [3]. The bumps above 1450 MeV are associated with the Λ(1520).

After subtraction of the background the line shape depicted in Fig. 2 was obtained. The Λ(1520), which has the same π⁰Σ⁰ decay as the Λ(1405), is not suppressed and later used as a reference. Within the statistical errors the BGOOD data is in general agreement with ANKE [12] and CLAS [6] measurements.
It is intersting to observe that together with the ANKE results our new data seem to suggest a two peak structure in the π⁰Σ⁰ line shape, with the first peak around 1395 MeV and the second at 1425 MeV. This is close to the two poles of the Λ(1405) predicted in [3]. The mass resolution obtained in the present BGOOD analysis of σ ≅13 MeV would allow to resolve the two poles if the higher one was narrower than in the original predictions [3]. The observation would be consistent with the above mentioned analysis based on the CLAS data of all charge channels of Ref. [5].

Fig. 2 (right) shows the line shape when the K⁺ is detected in the forward spectrometer, corresponding to cos θ^K_cm > 0.86, a kinematic region inaccessible to the CLAS experiment. The vertical dotted lines indicate the predicted position of the lower pole at 1390 MeV (red) and the higher one at 1425 MeV (blue). It seems there is indeed excess strength at those pole positions. In addition, excitation of the lower pole appears stronger relative to the higher one in the low-energy bin and vice-versa in the high-energy bin. This would be a new and very interesting finding if a firm conclusion was possible, however this is currently prohibited by the limited statistics. Based on these results it is one of the goals of the present proposal to significantly increase statistics for the K⁺Λ(1405) channel.

In addition to the line shape, differential and total cross sections were determined for K⁺Λ(1405) photoproduction. As mentioned above, K⁺Λ(1520) was analysed in parallel as a reference channel, and the cross sections for Λ(1520) are in satisfactory agreement with CLAS [13].
Through the process of γp → K⁺Λ(1405) photoproduction it is possible from a perspective different to the line shape to get a hand on the hyperon’s structure. This is because the two mechanisms depicted in Fig. 1 are expected to contribute in a different manner to both energy and K⁺ angular distributions.

Fig. 3: Differential cross section of γp →K⁺Λ(1405). Black squares are BGOOD, red circles are CLAS [13].

3_L1405_TotalCS_TJ-eps-converted-to.pdf-1.png

Fig. 4: Total cross sections for the photoproduction of (a) K⁺Λ(1405) and (b) K⁺Λ(1520). BGOOD data are the black squares with the systematic uncertainties shown as grey bars on the abscissa. CLAS data are the red circles [13]. The model of Wang et al. [8] are the superimposed lines in (a). The dashed green line shows the triangle singularity mechanism contribution, dashed blue line is the t-channel production via K* exchange, and the dashed line is t-channel production via K exchange. All the contributions were fitted to the CLAS data and the total is shown as the purple line.

The t-channel graph (Fig. 1 left) is expected to contribute over a wide kinematic range with the typical exponential fall-off with Mandelstam t. In addition, large t (at least large 3-momentum transfers) could be suppressed, if the Λ(1405) was an extended molecular-type structure. In contrast, the triangle singularity feeds Λ(1405) production through the precursor N* state(s), virtually independent of t determined from beam photon and K⁺. It selects a narrow kinematic range, because any triangle singularity occurs only if the relevant Feynman diagram can be interpreted as an energy and momentum conserving process in space-time, with all internal particles real, on the mass shell, and moving forward in time (this is the Coleman-Norton theorem [9]).

Fig. 3 shows the differential cross sections for the reaction γp →K⁺Λ(1405) in comparison to CLAS [13]. There is general agreement between the data sets. BGOOD extents the angular range of the K⁺ to very forward angles. We confirm a flat angular distribution for E_γ = 1550 ...2000 MeV, corresponding to cm energies of E_cm = 1.95 ...2.15 GeV. This would be consistent with a triangle mechanism, including the drop visible at very forward angle. At higher energies forward peaking is observed consistent with t-exchange. This suggests the triangle singularity drives the lower pole rather than the higher one, and vice-versa for the t-channel mechanism.
The complete kinematic coverage of the BGOOD experiment allows to determine the integrated total photoproduction cross sections with virtually no extrapolations. This is shown in Fig. 4 for Λ(1405) and Λ(1520). In particular the Λ(1405) total cross section could be determined with unprecedented energy resolution. It provides clear evidence for the significant role of the triangle mechanism in Λ(1405) photoproduction. This vice versa makes the N*(2030) likely to exist as a dynamically generated K*Σ molecular type state.

Fig. 5: Mandelstam t-dependence of photoproduction of Λ(1405) (left) and Λ(1520) from the present BGOOD measurement. Vertical dotted line in left graph marks the K⁺ mass. Photon energy bins depicted by colors as labelled inset.

The t-dependence of the process exhibits interesting features. It is shown in Fig. 5 for photoproduction of Λ(1405) (left) and Λ(1520) (right). As expected, there is an enhancement towards small t down to a minimum which corresponds to the K⁺ mass (vertical line in left diagram). It appears narrower with the steepest high-energy tail in the highest energy interval, and slightly flatter towards higher t at lower energies, again consistent with a stronger triangle contribution at this region. The right panel of Fig. 5 shows in comparison the situation for Λ(1520). It appears similar on the whole and, in particular, the width similar to the lower pole region of the Λ(1405). In the light of the previous discussion this might suggest a possible trianglemechanism in Λ(1520) photoproduction as well. Indeed, an earlier LEPS result seems to indicate a strong falloff of the cross section at forward angles [14]. Albeit interpreted in a different manner in Ref. [14], this would also fit to a triangle scenario where in Fig. 1 (right) the Σ(q) is the Σ(1385) and instead of Λ(1405) the final state contains the Λ(1520). It will be another goal of the present proposal to shed further light on this.

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