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Kaonic hydrogen atom and kaon-proton scattering length

Kaonic hydrogen is studied with various realistic potentials in an accurate numerical approach based on Sturmian functions. The kaon-proton scattering length extracted from the 1s energy shift of the kaonic hydrogen by applying the Deser-Trueman formula is severely inconsistent with the one derived by directly solving the scattering Schoedinger equation. We pay special attention to the recent measurement of the energy shift and decay width of the 1s kaonic hydrogen state by the DEAR Collaboration. After taking into account the large discrepancy between the extracted and directly-evaluated scattering lengths, we found theoretical predictions of most chiral SU(3) based models for the kaonic hydrogen decay width are consistent with the DEAR data. We warn the SIDDHARTA collaboration that it may not be reasonable to extract kaon-nucleon scattering lengths, by using the Coulomb-interaction corrected Deser-Truemab formula, from the planned measurement of kaonic hydrogen.

compared with the DEAR data of the energy and decay width of kaonic hydrogen, by using the Deser-Trueman formula [11,12] ∆E 1s + i Γ 1s 2 = 2α 3 µ 2 a p (1) or the Coulomb-interaction corrected Deser-Trueman formula [5] ∆E 1s + i Γ 1s 2 = 2α 3 µ 2 a p [1 − 2α µ a p (ln α − 1)] (2) where µ is the reduced mass of the K − p system, ∆E 1s * Electronic address: yupeng@sut.ac.th and Γ 1s are the energy shift and decay width of the 1s kaonic hydrogen due to the strong interaction, and a p stands for the S-wave K − p scattering length. A general result is that the energy shifts and decay widths extracted, by using eq. (1) or (2), from theoretical K − p scattering lengths are much larger than the DEAR data.
In this work we show that eqs. (1) and (2) may not well hold for the KN system and the discrepancy between the theoretical results and the DEAR data is not that large and particularly the theoretical decay widths are consistent with the DEAR data.
We derive the K − p scattering length and kaonic hydrogen energy shift and decay width by solving the same dynamical equation, where V em is the electromagnetic potential, V h 0 and V h 1 are respectively the isospin I=0 and 1 strong interactions of the KN system, R K − p (r) and RK0 n (r) are respectively the K − p andK 0 n parts of the radial wave function of the KN system. Eq. (3) embeds into the Schrödinger equation the relativistic effect and the mass difference between the K − p andK 0 n components. The relativistic modification of the Schrödinger equation to eq.(3) has been discussed in the works [13,14,15,16,17,18].
We study kaonic hydrogen and kaon-nucleon scattering with the phenomenological KN potential taken from the work [19,20] and the various effective potentials which are worked out in the work [21]. The interaction [19,20] is constructed by fitting the freeKN scattering data [22], the KpX data of kaonic hydrogen by the KEK Collaboration [23] and the binding energy and decay width of Λ(1405), which is regarded as an isospin I = 0 bound state ofKN . Since the interaction gives one molecular state Λ(1405), it must be much stronger than the strong pion-pion interaction.
In the work [21] an effective local potential in coordinate space is constructed such as the solution of the Schrödinger or Lippmann-Schwinger equation with such a potential approximates as closely as possible the scattering amplitude derived from the full chiral coupledchannel calculation. Several realistic chiral SU(3) based models [3,4,8,10] have been studied.
The accurate evaluation of energy shifts, decay widths and especially wave functions of exotic atoms has been a challenge to numerical methods [17,24]. An approach is required, which is able to account accurately for both Energy shifts and decay widths are given in eV. the strong short-range interaction and the long-range Coulomb force. The numerical approach based on Sturmian functions [25] has been found effective and accurate. In this work we use the numerical method which has been carefully studied and discussed in [25,26,27] to study kaonic hydrogen.
Shown in Table I are our theoretical results with various realistic KN potentials. The energy shift ∆E 1s and decay width Γ 1s of the 1s kaonic hydrogen state are derived by solving eq. (3) in the above mentioned Sturmian function approach [25,26,27]. The negative energy shifts in Table I Table I are directly evaluated from eq. (3). Listed in the last column of Table I are the extracted K − p scattering lengthã K − p from the energy shifts ∆E 1s and decay width Γ 1s in Column 1 and 2 by applying the Deser-Trueman formula of eq. (1).
It is clear that with the same interaction [20,21] the scattering length derived by directly solving the Schrödinger equation in eq. (3) is rather different from the one extracted from the energy shift and decay width of the 1s kaonic hydrogen by applying the Deser-Trueman formula of eq. (1). One finds from Table I that for the imaginary part the extracted scattering length is smaller by a factor of 20% to 30% than the directly-derived scattering length with the same interaction. Averaging over the results for all the five potentials we get the averaged factor to be 0.26. The result implies that both the lowest Deser-Trueman formula in eq. (1) and the Coulomb interaction corrected one in eq. (2) may not well apply to the KN system. The Coulomb interaction leads to a correction, less than 10%, to the lowest Deser-Trueman formula in eq. (1). Hence one may argue that for the KN system an extraction, by using the formulas in eqs. (1) or (2), may not be accurate, if not say, unreliable.
It has been puzzling that the energy shifts and decay widths, extracted from the scattering lengths derived in a number of chiral SU(3) based models [2,8,9,10], are inconsistent with the DEAR data of kaonic hydrogen. However, we find, after considering the large discrepancy between the directly evaluated and the extracted scattering lengths, that the inconsistence between the theoretical decay width and the DEAR data is not that obvious.
Listed in the second and third columns of Table II are respectively the isospin 0 and 1 KN scattering lengths predicted by various chiral SU(3) based models [2,6,7,8,9,10] where the isospin symmetry limit is applied. In the fourth column we give the corresponding K − p scattering lengths evaluated from the isospin based scattering lengths a 0 and a 1 , by using the formula with where µ 0 is the reduced mass of theK 0 n system and ∆ the mass difference between theK 0 n and K − p systems. The formula in eq. (13) holds with a high accuracy. We found a K − p scattering length evaluated with the formula in eq. (13) has a difference about 1% from the one derived by directly solving the particle based dynamical equation in eq. (3) for the potentials listed in Table I.
The decay widths Γ 1s in the last column of Table II are extracted from the scattering lengths listed in the fourth column with the formula with R = 0.35. We choose R = 0.35 based on the facts that the Coulomb interaction correction to the lowest Deser-Trueman formula is up to 10% and on average over the five realistic KN potentials the extracted K − p scattering length with the lowest Deser-Trueman formula is smaller by a factor of 0.26 than the directly evaluated one. (15), to a decay width Γ 1s = 202 ± 20 eV.
We conclude that for all the realistic local potentials employed in the work, the K − p scattering length extracted from the energy shift and decay width of the 1s kaonic hydrogen by applying the lowest and Coulomb interaction corrected Deser-Trueman formulas is severely inconsistent with the one derived by directly solving the scattering Schödinger equation. After taking into account the large discrepancy between the extracted and directly-evaluated scattering lengths, we found theoretical predictions for the kaonic hydrogen decay width by most chiral SU(3) based models are consistent with the DEAR data. We warn the SIDDHARTA collaboration that it may not be reasonable to extract kaon-nucleon scattering lengths, by using the Coulomb-interaction corrected Deser-Truemab formula, from the planned measurement of kaonic hydrogen.