A New Measurement of Kaonic Hydrogen X rays

The $\bar{K}N$ system at threshold is a sensitive testing ground for low energy QCD, especially for the explicit chiral symmetry breaking. Therefore, we have measured the $K$-series x rays of kaonic hydrogen atoms at the DA$\Phi$NE electron-positron collider of Laboratori Nazionali di Frascati, and have determined the most precise values of the strong-interaction energy-level shift and width of the $1s$ atomic state. As x-ray detectors, we used large-area silicon drift detectors having excellent energy and timing resolution, which were developed especially for the SIDDHARTA experiment. The shift and width were determined to be $\epsilon_{1s} = -283 \pm 36 \pm 6 {(syst)}$ eV and $\Gamma_{1s} = 541 \pm 89 {(stat)} \pm 22 {(syst)}$ eV, respectively. The new values will provide vital constraints on the theoretical description of the low-energy $\bar{K}N$ interaction.

Introduction 2p 1s (only Coulomb) 1s due to strong int. Ref. [44]. For ease of reference, the complete data set listed in [44] will be referred The data set with 180 and 98Mo omitted will be denoted LESS, whilst the measureme two isotope pairs 160-180 and 92Mo-98Mo will be referred to as ISO.

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Kaonic atoms for Z = 1 & 2 The final goal of the DEAR experiment is to extract precise values of the KN S-wave scattering lengths from the data by using some counterpart of Deser-type relations [7]. Neglecting isospin-breaking corrections altogether, in the case of kaonic hydrogen these relations are given by where µ c denotes the reduced mass of the K − p system, and a 0 , a 1 stand for the I = 0, 1 S-wave KN scattering lengths in QCD in the isospin limit (α = 0, m d = m u ). In addition, our definition of the isospin limit implies that the particle masses in the multiplets are taken to be equal to the charged particle masses in the real world (proton, * On leave of absence from HEPI, Tbilisi State University, University st. 9, 380086 Tbilisi, Georgia S-wave scattering length "a K-p " expressed with isospin dependent scattering lengths a 0 (I=0), a 1 (I=1)

Deser-Truman Formula
Here, ∆E s 1 stands for the strong-energy-level shift of the ground state of the kaonic hydrogen (total energy shift minus certain electromagnetic contributions), and Γ 1 denotes the width of the ground state. It should be pointed out that these results are in contradiction with the earlier measurements [3-6]; see also Fig. 3 below.
The final goal of the DEAR experiment is to extract precise values of the KN S-wave scattering lengths from the data by using some counterpart of Deser-type relations [7]. Neglecting isospin-breaking corrections altogether, in the case of kaonic hydrogen these relations are given by where µ c denotes the reduced mass of the K − p system, and a 0 , a 1 stand for the I = 0, 1 S-wave KN scattering lengths in QCD in the isospin limit (α = 0, m d = m u ). In addition, our definition of the isospin limit implies that the particle masses in the multiplets are taken to be equal to the charged particle masses in the real world (proton, * On leave of absence from HEPI, Tbilisi State University, University st. 9, 380086 Tbilisi, Georgia able char matches t lengths fo kaon-nuc present p It tur lowest-or these are hydrogen can be im comparin this qual the exist are done (see [9-12 subject), particula in some o systemat possible l to say, al of the acc difficult t The a relation b ly used for determining 2)) from which the inng lengths is extracted. as a difference between romagnetic shift. In the strong shift only. problem completely: the f the effective couplings e observable quantities. achieved by performing ring amplitudes in the nce of isospin breaking r, as one sees from (9), 4 ),d 1 should be known of (10) suffices). At the can be determined from lastic scattering amplietic and strong isospinesponding procedure is the first step, one re- The quantity T KN should be matched to its non-relativistic counterpart T NR KN , written in terms of the couplingsd i . A direct calculation with the Lagrangian (4), which is carried out in a similar way as in [16], yields (15) The matching condition 2M K + T NR KN = T KN enables one to determine the couplingd 1 at the required accuracy. Substituting this value ofd 1 into the expression for the strong shift, we finally get the formula in terms of the observable quantities, which contains all isospin-breaking terms up-to-and-including O(δ 4 ): Although (16)  matrix and should not be neglected as in previous coupled-channels calculations [3][4][5][6]8]. We account for these corrections by adding the quantum mechanical Coulomb scattering amplitude to the strong elastic K ÿ p amplitude, f str K ÿ p!K ÿ p 1=8 s p T str K ÿ p!K ÿ p . The total elastic cross section is then obtained by performing the integration over the center-of-mass scattering angle. Since this integral is infrared divergent in the presence of the Coulomb ampli- integrals in each channel. This amounts to a summation of a bubble chain to all orders in the s channel, equivalent to solving a Bethe-Salpeter equation with V as driving term. We perform a global 2 fit to a large amount of data, including K ÿ p scattering into coupled S ÿ1 channels, the threshold branching ratios of K ÿ p into and 0 channels, the mass spectrum, and the shift and width of kaonic hydrogen recently measured at DEAR [1]. The resulting values of the subtraction constants a at 1 GeV are a KN 0:95 10 ÿ3 , a ÿ0:59 10 ÿ3 , a 1:80 10 ÿ3 , a 2:92 10 ÿ3 , a 0:98 10 ÿ3 , and a K 2:90 10 ÿ3 . The Coulomb interaction has been shown to yield significant contributions to the elastic K ÿ p scattering amplitude up to kaon laboratory momenta of 100-150 MeV=c [12]. Close to K ÿ p threshold the electromagnetic meson-baryon interactions are thus important and should not be neglected as in previous coupledchannels calculations [3-6,8]. We account for these corrections by adding the quantum mechanical Coulomb scattering amplitude to the strong elastic K ÿ p amplitude, The total elastic cross section is then obtained by performing the integration over the center-of-mass scattering angle. Since this integral is infrared divergent in the presence of the Coulomb ampli- about 6 and 8 keV, which are identified to be K a (2p to 1s) and K complex (3p or higher to 1s), respectively. The pure electromagnetic value of the kaonic hydrogen K a x-ray energy [E EM K a ] is 6.480 6 0.001 keV, which 1997 : The first distinct peak @ KEK

Gas target
KpX imposed in the fit. The effect of varying the yields ratios as well as the fit energy range was studied and included in the systematic error. The two analyses gave consistent results. In both, the fit without kaonic hydrogen contribution gave a 2 =NDOF bration and en range and me Our result i 1s ÿ 1s within 1 of pulsive charac They differ aspects: (1) th times smaller values of 1s a a less repulsiv 2005 : Repulsive shift again @ LNF Repulsive shift G. Beer et al., PRL 94, 212302 (2005)