Precision measurement of the $3d \to 2p$ x-ray energy in kaonic $^4$He

We have measured the Balmer-series x-rays of kaonic $^4$He atoms using novel large-area silicon drift x-ray detectors in order to study the low-energy $\bar{K}$-nucleus strong interaction. The energy of the $3d \to 2p$ transition was determined to be 6467 $\pm$ 3 (stat) $\pm$ 2 (syst) eV. The resulting strong-interaction energy-level shift is in agreement with theoretical calculations, thus eliminating a long-standing discrepancy between theory and experiment.


Introduction
The measurement of the strong-interaction energy-level shift and width of kaonic atom x-rays offers a unique possibility to precisely determine the Knucleus strong interaction in the low energy limit.Therefore many experiments have been performed to collect data on a variety of targets from hydrogen to uranium.It has been known that most of the available kaonic-atom data can be fitted fairly well for Z ≥ 2 by optical-potential models [1] with the exception of kaonic helium and oxygen.
The strong-interaction shift of the 2p level ∆E 2p for kaonic 4 He has been previously measured in three experiments.Note that ∆E 2p is defined as ∆E 2p ≡ −(E (2,p) −E EM (2,p) ), where E (n,l) is the energy of the level with principal quantum number n and the orbital angular momentum l, and E EM (n,l) is the energy calculated using only the electromagnetic interaction (EM).The average of the three previous results gives ∆E 2p = −43 ±8 eV [2,3,4], while most of the theoretical calculations give ∆E 2p ∼ 0 eV [5,6,7] (e.g.∆E 2p = −0.13±0.02eV [5]).They disagree by more than five standard deviations, and this discrepancy is known as the "kaonic helium puzzle".Therefore an accurate re-measurement of the energy shift of the 2p level of kaonic helium has been long awaited.
In the present experiment we have performed a measurement of the Balmerseries x-rays of kaonic 4 He atoms, setting as our experimental objective a precision of ∼ 2 eV, thus shedding new light on the kaonic helium puzzle.

Experiment
The experiment E570 was carried out at the K5 beamline of the KEK 12-GeV proton synchrotron (PS).We accumulated data in two periods -520 hours in October 2005 (cycle 1) and 260 hours in December 2005 (cycle 2).The experimental apparatus was essentially the same as that of the former KEK-PS E549 experiment [8] performed at the same beamline, except for the inclusion of x-ray detectors and energy calibration foils in the helium-target cryostat.A schematic view of the E570 setup around the target is shown in Fig. 1.The detailed description of the experimental setup is given in a separate paper [8].
The kaonic 4 He atoms were produced by means of the stopped-K − reaction using a superfluid 4 He target (cylindrical shape 15 cm long and 20 cm in diameter at a density of 0.145 g/cm 3 ).Incident negatively-charged kaons with momentum ∼650 MeV/c were degraded in carbon degraders, counted with beamline counters, tracked by a high-rate beamline drift chamber and stopped inside the 4 He target.The energy losses before stopping were measured in a set of scintillation counters, T0.X-rays emitted from the kaonic 4 He atoms were detected by eight x-ray detectors which viewed the target from downstream through the 75 µm-thick Mylar window of the target vessel.Secondary charged particles produced in the kaon-absorption process following emission of kaonic x-rays were detected by charged-particle trigger/tracking systems placed on the left, right, top and bottom of the target.
In the present experiment, a significant improvement over the past experiments was achieved by incorporating the following features:

Silicon drift detectors
As x-ray detectors, we employed eight silicon drift detectors (SDDs) produced by KETEK GmbH [9], each having an effective area of 100 mm 2 and a 260 µmthick active layer with an energy resolution of ∼190 eV (FWHM) at 6.4 keV, which corresponds to the kaonic-helium 3d → 2p x-ray energy.The tempera-ture of the SDDs was kept at ∼83 K during the experiment by a connection to the thermal-radiation shield for the helium target cooled by liquid nitrogen.
In the SDD, the electrons produced by an x-ray hit drift radially toward the central anode where they are collected.The small anode size (and hence small capacitance) is essential to realize the good energy resolution despite the large effective area.The energy resolution is about twice as good as that of the Si(Li) x-ray detectors used in the previous three experiments.The time resolution is comparable with that of a Si(Li) detector.
The small anode area also makes it possible to reduce the active layer thickness, while the capacitance is still kept small.The thin active layer of 260 µm (less than 1/10th of the previously used detectors) helps to reduce continuum background caused by the Compton scattering occurring inside the detector.

Cuts applied to reduce background
We required that the reaction vertices reconstructed from an incident kaon track and an outgoing secondary charged particle track should be within the target, which is called the "fiducial volume cut".Moreover, in-flight kaon decay/reaction events were rejected by applying a correlation cut between the z-coordinate of the reaction vertex and the energy loss in T0.As a result, continuum background events were drastically reduced.

In-beam energy calibration
The energy calibration was done by using characteristic x-rays induced by charged particles (i.e.π − , which abundantly existed in the incident beam) on high-purity titanium and nickel foils placed just behind the target cell.The energy of the kaonic-helium 3d → 2p x-ray, ∼6.4 keV, lies between the characteristic x-ray energies, 4.5 keV(Ti) and 7.5 keV(Ni).To obtain highstatistics energy calibration spectra, we accumulated SDD self-triggered events together with the stopped-K − triggered events, which provide high-accuracy in-situ calibration spectra.
To avoid detecting the background characteristic x-rays from other than the titanium and nickel, high-purity aluminum foils were placed on all objects in the view of the SDDs.

Analysis
Figure 2 shows the correlation between the z-coordinate of the reaction vertex and the light output of T0.Each component of the target assembly (a carbon degrader, a target cell and SDDs/foils) is clearly seen.We applied a fiducial volume cut of −7.0 < z < 9.0 cm on the z coordinate as shown in Fig. 2, and of √ x 2 + y 2 < 11.0 cm on the radius from the target center.Slower incident kaons, which give larger light output on T0, stop upstream in the target, while faster kaons (hence smaller pulse height) stop downstream.Events which follow this trend were selected as stopped-K − events when lying within the solid-lined box in Fig. 2.
Stopped-K − -timing events were selected using SDD timing information to reduce the accidental background.Time resolution of the SDD after timewalk correction was ∼ 160 ns (σ) at ∼83 K, which reflected the drift-time distribution of the electrons in the SDD.Data within ± 2 standard deviations from the average SDD hit timing were selected.
Figure 3 (a) shows a typical x-ray spectrum for SDD self-triggered events, which is used for the energy calibration.Characteristic x-ray peaks of titanium and nickel were obtained with high statistics.Typical yields of titanium K α peaks are 5 ×10 2 events per hour for each SDD.Time-dependent gain drift was corrected about every 20 hours.The energy scale was calibrated by K α lines of titanium and nickel with the well-known energies [10] and intensity ratios [11] of K α1 and K α2 .
After applying the event selections described above and calibrating the energy scale, we obtained x-ray energy spectra for stopped-K − triggered events shown in Fig. 3. Kaonic-helium 3d → 2p, 4d → 2p and 5d → 2p transitions are clearly observed, while the Ti and Ni x-ray peaks are greatly suppressed.Figure 3 (b) and (c) respectively show the x-ray spectra taken in the runs in October 2005 (cycle 1) and December 2005 (cycle 2).In cycle 1, only 3 out of 8 SDDs yielded useful data; the faulty detectors were then replaced, and 7 SDDs were functional in cycle 2.In total, ∼ 7 × 10 2 (cycle 1) and ∼ 8 × 10 2 (cycle 2) events of 3d → 2p x-rays were accumulated for each cycle.The total yields are approximately equivalent for both runs since the beam time of the October run was twice that of the December run.In comparison to the most recent measurement of the kaonic 4 He spectrum [4], we achieved ∼2 times better energy resolution, ∼3 times higher statistics, and ∼6 times better signal-tonoise ratio.

Spectral fitting and results
During the course of the present analysis of spectra from SDDs, many fine details of the signal and calibration pulses were found necessary to attain eV accuracy and will be discussed extensively in a paper to follow.Here, we briefly summarize the spectral fitting method with a SDD-response function studied in detail, and show the fit results.
For fitting the spectra of kaonic-helium x-ray peaks, a convolution of a Gaussian (representing the detector response) and a Lorentz function (natural width), "Voigt function", was adopted as the main-peak function, whereas a Gaussian response was employed as a main-peak function for fitting the characteristic x-ray peaks since their natural width is much less than the energy resolution of the SDD.
An energy-dependent experimental energy resolution was employed as is usually the case for silicon detectors: ∆E(FWHM) = 2.35ω W 2 N + F E/ω, where W N denotes the contribution of noise to the resolution (independent of the x-ray energy), E is the x-ray energy, F is the Fano factor (≈ 0.12 for silicon), and ω is the average energy for electron-hole creation in silicon.Here, ω was fixed to be 3.81 eV, and F and W N were introduced as free fit-parameters for the self-triggered-event spectra.
Because of the large incoherent (Compton) total scattering cross section of liquid 4 He (∼ 1 barn/atom at photon energies of ∼10 keV), a low-energy tail structure due to the Compton scattering must be taken into account.The convolution of an exponential function with a Gaussian was adopted as the spectral shape (called the "Compton tail function").All parameters of the Compton tail function relative to the main peak were estimated by fitting the simulated x-ray energy spectra smeared with a Gaussian resolution function.The x-ray spectra were simulated with the GEANT code using the Low Energy Compton Scattering (LECS) package [12] with a realistic setup of E570 and the measured stopped-K − distribution.
We also have waveform data available from flash ADCs (FADCs), which were accumulated as well as ordinary comparator-type pulse-height ADC data, although the FADC data are available only for about half of cycle 1.Using the waveform analysis, it is shown that there is a non-negligible pileup effect due to the high-rate beam condition of E570.The spectral function attributed to the pileup events could be estimated as a Gaussian on the right flank of the main-peak function (called the "pileup Gaussian").The mean and sigma parameters of the pileup Gaussian were estimated using FADC data.
There are many empirical investigations of the response function of silicon detectors using monoenergetic x-rays (e.g.[13,14]), and it is known that there is an exponential-like feature decreasing steeply in intensity towards lower energy on the left flank of the main-peak function (called the "tail function") and a flat shelf-like feature which extends to near zero energy (called the "shelf function") [14], due to electron transport processes and imperfections in the fabrication processes.These effects were also taken into account in the spectral fitting separately from the Compton tailing effect in the liquid 4 He target mentioned above.
The intensity ratios of these components (pileup Gaussian, tail and shelf functions) to the main-peak component were estimated by fitting the highstatistics x-ray spectra for self-triggered events.The estimated intensity ratios were then fixed in the fits of the x-ray spectra for stopped-K − triggered events.
Resulting fit-lines are overlaid on the x-ray spectra shown in Fig. 3 (b) and (c) with each contribution: main Voigtian, Compton tail function, pileup Gaussian, tail function and shelf function.The fit residuals are also shown under each spectrum, with thin lines denoting the ±2σ values of the data, where σ is the standard deviation due to the counting statistics.In the fits, the intensity and mean parameters for each kaonic helium x-ray transition are independent.As a result, the kaonic 4 He x-ray energy of the 3d → 2p transition was determined to be where the first error is statistical and the second is systematic.The quoted systematic error is a linear summation of the contributions from the intensity ambiguities of the Compton tail , pileup, shelf and tail functions for kaonichelium x-rays.The other transition energies (4d → 2p and 5d → 2p) obtained in this fit are listed in table 1 with only statistical errors.In this table, we also tabulate the EM values updated from Refs.[2,3,4] by Koike [15] using the latest kaon mass given by the particle data group (PDG) [16].These values are consistent with another recent calculation [17] and differ slightly from the ones used in previous experiments [2,3,4].The intrinsic width obtained in the fits seems to be very small and it needs more study to disentangle it from instrumental effects.Since the strong-interaction shifts are negligibly small for the levels with the principal quantum number n larger than two, the 2p-level shift ∆E 2p can be derived from the Balmer-series x-ray energies using the equation: where p) correspond to the measured and calculated EM x-ray energies, respectively.To combine all statistics of the observed Balmer-series lines, we calculated ∆E 2p for each line using Eq. 2 and took their statistical averages; the 2p-level shift was then derived as where EM values listed in table 1 [15] were adopted, and the systematic error was estimated in the way mentioned above.Note that the EM energy and thus the energy shift are sensitive to the value of the kaon mass, for which two slightly disagreeing measurements exist [16] leading to a large error in the PDG value.If the kaon mass changes by one standard deviation from the current value of 493.677 (16) MeV/c 2 [16], ∆E 2p changes by about 0.2 eV [15].

Conclusion
In conclusion, we have measured the Balmer-series x-rays of kaonic 4 He atoms using silicon drift detectors which lead to a much improved energy resolution and signal-to-noise ratio compared to the Si(Li) x-ray detectors used in the past experiments.The kaonic 4 He x-ray energy of the 3d → 2p transition was determined to be 6467 ± 3 (stat) ±2 (syst) eV.
The theoretical calculations of the shift are very close to zero (∼ −0.1) eV by an analysis with global fits to existing kaonic-atom x-ray data on various nuclei using an optical potential [1,5], and also by a calculation using an SU(3) chiral unitary model [6].A recent calculation by Friedman gives a value of −0.4 eV as the lowest possible one [7], when the non-linear density dependence is included [18].On the other hand, Akaishi calculated the shift as a function of the real part (U 0 ) of the KN potential depth at a certain coupled potential depth (U coupl = 120 MeV) [19].The calculation was based on the coupledchannel approach between the KN channel and the Σ π decay channel.A large shift (|∆E 2p | ∼ 10 eV) is predicted near the resonance between atomic and nuclear poles, when the potential depth is at around ∼ 200 MeV.The presently observed small shift disfavors the values of (U 0 , U coupl ) = (∼ 200 MeV, 120 MeV) within his framework.
Our careful and precise determination of the 2p-level shift resolved the longstanding kaonic helium puzzle.The present data alone are not sufficient to deduce the K-nucleus potential strength at the center of the nucleus.A unified study with the 2p width to be determined in further analysis and with data to be collected in kaonic 3 He x-ray spectroscopy [20] will indubitably yield invaluable constraints for the theories.

Fig. 1 .
Fig. 1.(a) A schematic side view of the E570 setup around the cylindrical target with the x-ray detection system.(b) A front view of the silicon drift detector (SDD) assembly.Eight SDDs are mounted on holders tilted at a 45 degree angle to the beam center in an annular-shaped pattern.Fan-shaped high-purity titanium and nickel foils are put alternately on a cone-shaped support located on the beam axis.

Fig. 2 .
Fig. 2. A typical density plot between the z-coordinate of the reaction vertex and the light output on T0, used to reject in-flight kaon decay/reaction events.

Fig. 3 .
Fig. 3. (a) A typical x-ray spectrum for self-triggered events which provides high-statistics energy-calibration information.(b)(c) Measured x-ray spectra for stopped-K − events obtained from the runs in October 2005 (cycle 1) and December 2005 (cycle 2) respectively.A fit line is also shown for each spectrum, along with individual functions of the fit.The fit residuals are shown under each spectrum, with thin lines denoting the ±2σ values of the data, where σ is the standard deviation due to the counting statistics.

Fig. 4 .
Fig.4.The 2p-level shift of kaonic4 He, ∆E 2p , obtained from this work and the past three experiments (WG71[2], BT79[3], BR83[4]).Error bars show quadratically added statistical and systematic errors.The average of these past experiments is indicated by the horizontal gray band.