Constraints on the threshold K- nuclear potential from FINUDA (stopped K-, pi-) hypernuclear spectra

1s(Lambda) hypernuclear formation rates in stopped K- reactions on several p-shell targets are derived from hypernuclear formation spectra measured recently by the FINUDA Collaboration and are compared with calculated 1s(Lambda) formation rates based on a KbarN coupled channels chiral model. The calculated rates are about 15% of the derived rates, depending weakly on the depth of the threshold K- nuclear potential. The A dependence of the calculated rates is in fair agreement with that of the derived rates, showing a slight preference for a deep density dependent K- nuclear potential of depth (150-200) MeV at nuclear matter density over a shallow potential of depth about 50 MeV. These new features originate from a substantial energy and density dependence found for the in-medium subthreshold (K- n -->pi- Lambda) branching ratio that serves as input to the K- capture at rest calculations.


Introduction
How strong is the K − nuclear interaction? Various scenarios proposed for kaon condensation in dense neutron-star matter [1], and more recently for quasibound K − nuclear clusters [2] and for self-bound strange hadronic matter [3] depend on the answer to this question which has not been resolved todate. A modern theoretical framework for the underlying low-energyKN interaction is provided by the leading-order Tomozawa-Weinberg vector term of the chiral effective Lagrangian which, in Born approximation, yields a moderately attractive K − nuclear potential V K − : where ρ is the nuclear density, ρ 0 = 0.17 fm −3 , and f π ∼ 93 MeV is the pion decay constant. This attraction is doubled, roughly, within chirally based coupled-channelKN-πΣ-πΛ calculations which provide also for a strong absorptivity [4]. Shallower potentials, Re V K − (ρ 0 ) ∼ −(40-60) MeV at threshold, are obtained by requiring that the in-medium K − N t(ρ) matrix is derived self-consistently with the potential V K − = t(ρ)ρ it generates [5,6]. In contrast, comprehensive global fits to K − -atom strong-interaction shifts and widths yield very deep density dependent K − nuclear potentials at threshold, in the range Re V K − (ρ 0 ) ∼ −(150-200) MeV [7]. In this Letter we discuss recent FINUDA measurements that might bear on this issue by providing constraints on how deep Re V K − is at threshold. In the preceding Letter [8], the FINUDA Collaboration at DAΦNE, Frascati, reported on Λ-hypernuclear excitation spectra taken in the K − stop + A Z → π − + A Λ Z reaction on several p-shell nuclear targets. Formation rates were given per stopped K − for bound states and for low lying continuum states. In 16 Λ O the bound state formation rates agree nicely with a previous KEK measurement [9]. The recent FINUDA data allow for the first time to consider the A dependence of the formation rates in detail within the nuclear p shell where nuclear structure effects may be reliably separated out. It is our purpose in this companion Letter to apply one's knowledge of the nuclear structure aspect of the problem in order to extract the dynamical contents of the measured formation rates, particularly that part which concerns the K − nuclear dynamics at threshold. In doing so we transform the partial formation rates reported for well defined and spectroscopically sound final Λ hypernuclear states into 1s Λ hypernuclear formation rates that allow direct comparison with DWIA calculations.
The expression for the formation rate of hypernuclear final state f in capture at rest on target g.s. i, apart from kinematical factors, is a product of two dynamical factors [6,10,11,12]: (i) the branching ratio for K − n → π − Λ in K − absorption at rest in the nuclear medium, here denoted BR; and (ii) the absolute value squared of a DWIA amplitude given by divided for a proper normalization by the integral ρ of the K − atomic density overlap with the nuclear density ρ(r) Here ρ f i stands for the nuclear to hypernuclear transition form factor, χ (−) q f is an outgoing pion distorted wave generated by a pion optical potential fitted to scattering data, and Ψ nLM is a K − atomic wavefunction obtained by solving the Klein-Gordon equation with a K − nuclear strong interaction potential V K − added to the appropriate Coulomb potential. The integration on the r.h.s. of Eq. (2) is confined by the bound-state form factor ρ f i to within the nucleus, where Ψ nLM is primarily determined by the strong-interaction V K − , although Ψ nLM is an atomic wavefunction that peaks far outside the nucleus. The sensitivity of the DWIA amplitude Eq. (2) to V K − arises from the interference of Ψ nLM with the pion oscillatory distorted wave χ (−) q f . In particular, once V K − is sufficiently deep to provide a strong-interaction bound state for a given L, the atomic Ψ nLM also becomes oscillatory within the nucleus which magnifies the effects of interference, as verified in past DWIA calculations [6,12].
In this Letter we point out another strong sensitivity to the initial-state K − nuclear dynamics arising from the energy and density dependence of the K − n → π − Λ BR. We show how to incorporate this energy and density dependence into the calculation of a properly averaged value BR which depends on the K − atomic orbit through L and on the mass number A of the target. The resulting calculated 1s Λ formation rates are then compared to those derived from the FINUDA data and conclusions are made on the deep vs. shallow K − nuclear potential issue.

Derivation of 1s Λ capture rates from FINUDA data
The FINUDA spectra show distinct peaks for several 1s Λ and 1p Λ states in the nuclear p shell. In general, the derivation of the 1p Λ formation rate is ambiguous given that the 1p Λ formation strength is often obscured by a rising Λ continuum. In 9 Λ Be and in 13 Λ C it is also mixed with a substantial part of the 1s Λ formation strength owing particularly to high lying T = 1 parent states in the corresponding core nuclei. For this reason, we here deal only with the 1s Λ formation strength, deriving it in each p-shell Λ hypernucleus from unambiguously identified low lying 1s Λ states. According to Ref. [13], the corresponding hypernuclear formation rates are given by a 1s Λ formation rate R(1s Λ ), which is independent of the particular hypernuclear excitation considered, times a structure fraction derived from neutron pick-up spectroscopic factors in the target nucleus. This theoretical framework is also applicable to forward cross sections of in-flight reactions such as (π + , K + ) and (e, e ′ K + ). In Table 1 we present 1s Λ formation rates derived from the FINUDA K − capture at rest hypernuclear spectra [8,14] for a procedure denoted (a). In each spectrum we focus on the strongest low-lying particlestable hypernuclear excitation which is also well described in terms of a Λ hyperon weakly coupled to a nuclear core parent state. These core parent states are listed in the table. The measured formation rates for the corresponding hypernuclear excitations from Refs. [8,14] are then divided by the structure fractions listed in the table to obtain values of R(1s Λ ). For comparison, we display in the last column the 1s Λ component of forwardangle integrated (π + , K + ) cross sections, also derived using the peaks listed in the second and third columns. These (π + , K + ) strengths show little A dependence, in contrast to the K − capture at rest 1s Λ formation rates that decrease by a factor 3.5 in going from 7 Li to 16 O.  [8], and for 12 Λ C from [14]. The errors are statistical and systematic, in this order. The 1s Λ structure fractions are from [15] and, if unlisted there, from [13]. Listed in the last column, for comparison, are 1s Λ forward-angle integrated (π + , K + ) cross sections, also derived by using procedure (a) from KEK-E336 measurements [16]. In the second procedure, denoted (b) and presented in Table 2, we consider all the particle-stable 1s Λ states corresponding to observed peaks for which the shell model offers reliable identification. For three of the five targets listed, this procedure saturates or is close to saturating the 1s Λ formation strength. However, in both 9 Λ Be and 13 Λ C the 1s Λ particle stable hypernuclear states represent less than half of the full 1s Λ strength. In the last column of Table 2 we assembled 1s Λ forward-angle integrated (π + , K + ) cross sections, derived this time by applying procedure (b). Similarly to Table 1, the weak A dependence of these 1s Λ (π + , K + ) cross sections is in stark contrast to the fast decrease of the 1s Λ formation rates, again by a factor 3.5, going from 7 Li to 16 O in K − capture at rest. The strong A dependence of the (K − stop , π − ) rates with respect to the weak A dependence of the (π + , K + ) cross sections reflects the sizable difference between the strongly attractive K − nuclear interaction at threshold and the weakly repulsive K + nuclear interaction. It is encouraging to see that both sets of R(1s Λ ) values in Tables 1  and 2 are consistent within statistical uncertainties with each other, except marginally for 12 C which dates back to a separate FINUDA run [14]. Procedure (a) yields a value for 12 C that compares well with R(1s Λ )[ 12 C] = (1.11 ± 0.14) × 10 −3 per K − stop , the latter value corresponding to E x 7 MeV in the KEK 12 Λ C spectrum. 1 Therefore, in the present study we adopt the R(1s Λ ) values listed in Table 1. 3. Energy and density dependent K − n → π − Λ branching ratios  Figure 1 shows the subthreshold energy dependence of the free-space K − n → π − Λ BR generated by the CS30 version of the chirally motivated coupled channel model of Ref. [18]. 2 This I = 1 BR is about 10% at threshold, decreasing to roughly half of its value as the I = 0 Λ(1405) subthreshold resonance is traversed, and then increases to approximately 40% on approaching the πΣ threshold about 100 MeV below theKN threshold. The figure also shows the in-medium BR below threshold at densities 50% and 100% of nuclear matter density ρ 0 , in two versions of medium modifications. In the l.h.s. plots the only medium effect is Pauli blocking, which acts in interme-diateKN states in the coupled channel equations. This is known to have the effect of pushing the dynamically generated Λ(1405) to energies above threshold [19,20], thus weakening the I = 0 interaction and consequently increasing the I = 1 BR. The energy dependence in the subthreshold region is seen to be monotonic. The r.h.s. plots include in addition to Pauli blocking also meson and baryon self-energy (SE) terms in intermediate state propagators. This pushes back the Λ(1405) [5,21] and in the chirally based model used here [18] results in stronger energy and density dependencies. The same chiral model was used in Ref. [12] to generate a K − n → π − Λ BR which, however, was fixed at its threshold value, thus neglecting any possible energy dependence. Since the in-medium BRs plotted in Fig. 1 exhibit a sizable energy and density dependence, it is essential to consider the implied effects in the evaluation of the 1s Λ formation rates.
The K − n → π − Λ BR depends on the initial K − n invariant energy √ s, with s = (E K + E N ) 2 − ( p K + p N ) 2 in obvious notation. In the two-body c.m. system p K + p N = 0, but in the K − -nucleus c.m. system (approximately nuclear lab system) p K + p N = 0 and averaging over angles yields ( For K − capture at rest, we further neglect the atomic B K with respect to B N and replace the K − kinetic energy p 2 K /(2m K ) in the local density approximation by −Re V K − (ρ) which dominates over the K − Coulomb potential within the range of densities of interest. The neutron kinetic energy p 2 N /(2m N ) is approximated in the Fermi gas model by 23 (ρ/ρ 0 ) 2/3 MeV. Altogether the energy argument of the K − n → π − Λ BR assumes the form 3 which unambiguously prescribes the subthreshold two-body energy as a function of nuclear density at which BR( √ s, ρ) of Fig. 1 is to be evaluated. 4 Note that Eq. (5) leads to implicit density dependence of BR( √ s, ρ) through the invariant energy variable √ s, in addition to the explicit ρ dependence. The input BRs for our 1s Λ hypernuclear formation rates calculation were obtained by averaging this chiral-model BR( √ s, ρ), for a given V K − (ρ), over the K − nuclear density overlap ρ(r) | Ψ nLM (r) | 2 of Eq. (3): For B N we used target neutron separation energies. The nuclear densities used were obtained from modified harmonic oscillator nuclear charge densities by unfolding the finite size of the proton. The structure of Eqs. (5), (6), together with the plots of Fig. 1, imply that deep K − nuclear potentials lead to significantly higher values of BR than the threshold value used in Ref. [12], which indeed is borne out by the present calculations.

Confronting data with calculations
The 1s Λ formation rates for a shallow K − nuclear potential V SH MeV have been recalculated with refined K − atomic wavefunctions and π − distorted waves. 5 A major change here with respect to Refs. [12,24] is the use of energy and density dependent BRs as outlined in Sect. 3. The resulting BRs for the deep K − potential V DD K − display considerable A dependence, with values higher than the threshold value used in Ref. [12], particularly from 12 C on. In contrast, the BRs for the shallow potential V SH K − show little A dependence, with values lower than the threshold value. Consequently, the difference between the DD and SH rates is no longer as large as calculated for a fixed BR threshold value [12]. For example, the calculated rates are (15)(16)(17)(18)% of the experimentally derived rate for 7 Li under procedure (a) using the best-fit pion optical potential π e , Eqs. (19), (20) of Ref. [6], and (23-26)% of it using the pion optical potential π b employed in Ref. [12].
Here we focus on the A dependence of the 1s Λ formation rates. For given K − and π − potentials the calculated rates are scaled up by a normalization factor to achieve agreement for 7 Li with the 1s Λ rate derived from the data under procedure (a) in Table 1. This is shown in Fig. 2 where the uncertainties of the experimentally derived 1s Λ rates consist only of statistical errors that vary from one target to another. The systematic errors, on the other hand, are the same for all targets and drop out when considering A dependence within the present set of FINUDA data. The normalized calculated 1s Λ rates shown in the figure are based on BRs calculated according to Eq. (6) from the BRs plotted on the l.h.s. of Fig. 1 (CS30, no-SE). Results 5 The complex K − nuclear potentials V SH K − and V DD K − were denoted K χ and K DD , respectively, in Ref. [12] where a complete listing of their parametrization is available.  Figure 2: Comparison between 1s Λ formation rates derived from the FINUDA K − capture at rest data [8,14] and DWIA calculations normalized to the 1s Λ formation rate of 7 Λ Li listed in Table 1 for shallow (SH, solid) and deep (DD, dashed) K − nuclear potentials. The calculated 1s Λ formation rates use K − n → π − Λ in-medium BRs without self energies, see Sect. 3, and pion optical potential π e from Ref. [6]. The error bars consist of statistical uncertainties only.
are shown for the pion optical potential π e which was fitted to π − -12 C angular distributions at 162 MeV [6], and for the two K − nuclear potentials V SH K − and V DD K − . We note that the decrease of the experimentally derived 1s Λ rates from 7 Li to 9 Be, followed by increase for 12 C and subsequently decreasing through 13 C down to 16 O, is well reproduced by both calculations shown in Fig. 2. However, the deep V DD K − calculated rates reproduce better the A dependence of the experimentally derived rates than the shallow V SH K − potential does. In reaching this conclusion on V DD K − , the increase of the K − n → π − Λ BR values between 7 Li and 16 O is essential, by moderating the fall off of the rates calculated using A independent BRs. Similar conclusions hold for the A dependence of rates calculated using BRs that are derived according to Eq. (6) from the BRs plotted on the r.h.s. of Fig. 1 (CS30, with SE). On the other hand, if the pion optical potentials π b or π c (applied in Ref. [12]) are used in these calculations, neither V DD K − nor V SH K − do as good a job as the combination V DD K − and the best-fit π e does, and no firm conclusion can be drawn.

Conclusion
In conclusion, we have derived 1s Λ hypernuclear formation rates from peak formation rates associated with the A Z(K − stop , π − ) A Λ Z spectra presented recently by the FINUDA Collaboration on several nuclear targets in the p shell [8,14]. We then compared the A dependence of these derived rates with that provided by calculations for the two extreme V K − scenarios discussed at present, a shallow potential [5,6] and a density dependent deep potential [7]. The calculations use K − n → π − Λ in-medium BRs generated by applying a recent chirally motivated coupled channel model [18]. These BRs exhibit a strong subthreshold energy and density dependence, as shown in Fig. 1, and therefore result in A dependent input values BR that depend sensitively on the initial-state K − nuclear potential V K − . The calculations also demonstrate additional strong sensitivity to V K − through the atomic wavefunctions it generates which enter the DWIA amplitude Eq. (2), as discussed extensively in previous calculations of K − stop hypernuclear formation rates [6,12]. The comparison between the calculated A dependence and that derived from the FINUDA data slightly favors a deep K − nuclear potential over a shallow one. This conclusion outdates the one reached in an earlier version in which the energy and density dependence of the BRs, resulting here in a new source of sensitivity to V K − , was disregarded [24]. In future work, it would be interesting to use other versions of K − N chirally motivated models and to extend the range of nuclear targets used in stopped K − reactions to medium and heavy weight nuclei in order to confirm the present conclusion, and to look for more subtle effects of density dependence.