Hyperfine Interact (2012) 209:133–138DOI 10.1007/s10751-011-0539-6

Kaonic atoms and �(1405)

Slawomir Wycech

Published online: 9 February 2012© Springer Science+Business Media B.V. 2012

Abstract Studies of �(1405) in �π decay channels are briefly presented and therelated uncertainties indicated. The advantages of measurements in the KN channelare stressed. Two methods: studies of upper levels in K-mesic atoms and radiativedecays from the hydrogen atom to �(1405) are discussed.

Keywords Nuclear states of kaons · Exotic atoms

1 Introduction

Understanding of �(1405) is significant for several reasons:

• On the theory side the still unsettled question is the strength of pure quarkcomponents that possibly admix to the KN quasi-bound state [1].

• On the phenomenological level the open question is whether one can generate�(1405) with a short range attraction in the KN channel or is there an additionalsupport from a resonating �π channel. The first option has a long history, theattraction according to the early idea of Dalitz is due to vector meson exchanges.The second option is described by SU(3) chiral effective models and presents acrucial test of the applicability of such models [2–4] in the strange sector.

• The interest in nuclear physics concerns energies and widths of K nuclearstates [5]. These properties are sensitive to the spectrum of �(1405) in the K̄N channel.

Supported by European Hadron 2, LEANNIS project.

S. Wycech (B)National Centre for Nuclear Studies, Hoza 69, 00-681, Warsaw, Polande-mail: wycech@fuw.edu.pl

134 S. Wycech

1.1 Profiles of �(1405) in � π channels

The �(1405) has isospin 0 but may be detected in states of mixed isospin. As aconsequence the positions of resonant peaks generated by �(1405) are differentin different reactions. In the simplest version of theory there might be at leastsix different maxima of cross sections or other measurable quantities. These areindicated below.

The elastic scattering amplitudes in the K̄N system are superimposed by twoisospin amplitudes A0, A1. These are related to channel amplitudes

A(K− p → K− p) = 1

2A0 + 1

2A1

A(K− n → K− n) = A1

where for the simplicity of argument isospin symmetry violations are neglected. Onecan notice that for a single pole in the A0 amplitude the modulus of K− p amplitudedisplays maximum at energy shifted with respect to the pole energy. This shift isgenerated by the background isospin 1 amplitude A1.

The inelastic cross sections are given by the transition amplitudes T

T(K− p → �+ π−) = 1√6

T0 + 1

2T1

T(K− p → �− π+) = 1√6

T0 − 1

2T1

T(K− p → �0 π0) = 1√6

T0

and elastic scattering in decay channels is given by three amplitudes correspondingto isospin 0,1,2,

F(�+ π− → �+ π−) = 1

3F0 + F1

1

2+ F2

1

6

F(�− π+ → �− π+) = 1

3F0 + F1

1

2+ F2

1

6

F(�0 π0 → �0 π0) = 1

3F0 + 2

3F2

Unitarity relates amplitudes A,T and F for each isospin and this relation may bechecked for amplitudes A and T against experiments performed above the K̄Nthreshold. On the other hand, it is known that multichannel potential and/or K-matrix models are unstable against subthreshold extrapolation to the �(1405) region.Hence, the importance of experimental checks in the �,π channels. This may bedone via studies of final state interactions. Very recent experiments allow for asystematization but so far have not been analyzed in terms of models. In a crudeway these are presented in Table 1 and discussed below.

The �(1405) is located close to the K̄ N threshold an thus the �(1405) profilediffers from the lorentzian one. Moreover �(1405) is not an S-matrix eigen-resonance in the multichannel space and may differ in each channel. On top ofthat the experimental data refers to interfering resonant and background isospin 1,2

Kaonic atoms and �(1405) 135

Table 1 Maxima in the �π

spectra, recent experimentsChannel γ p → K+ � π , [6] pp → p K+ � π [7, 8]

Emax E max

�+, π− 1,420 1,400�0, π0 1,415 1,390�−, π+ 1,410 1,400

amplitudes. Assuming a single pole nature of isospin 0 amplitude one expects threedifferent maxima in three different �, π charge channels if the �(1405) formation isinitiated in K̄ N channel. This apparently happens in γ p → K+ � π reactions and theshifted maxima are indicated in Table 1. On the other hand, if the �(1405) formationis initiated in the �,π channel one expects two different maxima. This apparentlyhappens in pp → p K+ � π reactions and the shifted maxima are indicated inTable 1.

Qualitatively one obtains consistency of the various positions of the resonantmaxima in different charge states with the expectations coming from the amplitudesgiven above. At this moment a comparison of specific models with the fresh datais not available. Two points are clear however: (1) the differences in the maximapositions depend on poorly known isospin 1 and unchecked isospin 2 amplitudesand (2) the position of maxima observed in final states—and the shape of �(1405)—depend on the formation mechanism. Moreover, the � π pair is formed with K+—another strongly interacting particle. Thus the final state interactions have to becorrected for K, π interactions which resonate at K∗ state present in some regionsof the available final state phase space.

These uncertainties indicate usefulness of studying �(1405) in the basic K̄Nchannel.

2 Profile of �(1405) in KN channel

To observe the �(1405) in the main KN channel one has to bind either mesonor nucleon or perhaps both particles with the binding energy in the 0–50 MeVregion. Another required condition is that the KN pair should be quasi-free. Suchconditions are met at distant nuclear surfaces and may be materialized in highangular momentum states of K-mesic atoms. Another possibility is the reactionK + N → �(1405) + γ . These two suggestions are discussed below.

• The highest accessible levels in hadronic atoms

Collisions of atomic K̄ mesons with bound nucleons are described by the elasticscattering amplitude

a(E) = AK̄N

(ECM = −ES − EA − ER

), (1)

and the energy in K̄N center of mass ECM is given by a separation energy of thehit nucleon ES, binding energy of the atomic meson EA and a recoil energy of theK̄N pair relative to the nucleus ER. The latter is calculable in terms of nucleon andatomic wave functions. It amounts to about 10 MeV. Thus the region of energies inatomic states which involve predominantly valence nucleons cover the range of −5MeV (Deuteron) to −37 MeV (Helium). In these states the atomic level shift �E

136 S. Wycech

Fig. 1 The Im parts ofeffective KN length as afunction of valence nucleonseparation energy

and width � are conveniently calculated as a sum of meson multiple scattering terms.One has [9]

�E − i�/2 =< T >

[1 − < T DT >

< T >+ < T DT >2

< T >2− < T DT DT >

< T >..

]−1

(2)

where

T(r, r′) = −2π

μa · �i ρ(r)i δ(r − r′), D = − m

2π |r − r′| . (3)

and μ is the reduced K-N and m is the K meson mass, ρ(r)i is the single nucleondensity, the summation extends over all nucleons and the averaging is understood as< T >= ∫ ∫

drdr′ψ(r)∗T(r, r′)ψ(r) where ψ is the atomic wave function.Result (2) is equivalent to that generated by the optical potential of strength V =

− 2πμ

a · �iρ(r)i provided the convergence in the denominator is fast enough. In theknown “upper levels” ≈80% of the complex shift is given just by < T > and in mostcases only < T DT > / < T > is necessary to reach a few % precision. The advantageof (2) is clear in light systems as it may avoid unnecessary double collisions on thesame nucleon.

Measured upper level widths in H,Cd,Ag,In,Cu,Cl,P,Sn,Al,S,Si,Mg,C,Be, are or-dered here with increasing ES. These data handled with (2) allow to extract theIm a(E) parameters, which are plotted in Fig. 1. Calculations were done with Rea = −0.5 f m [10] corresponding to the central Re Voptical =� −60 MeV. (A changeto Re a = −0.7 f m would reduce these numbers by ∼5%. Im a increases for thelargest available separation energies corresponding to subthreshold energy of about26 MeV. It could indicate deeply bound �(1405), but the argument is weak at thismoment, as the experimental errors are large. Precise measurement of upper widthsin Be,C,O and complex level shifts in He, De would allow to separate AK− p(E) andAK−n(E). Such a program was performed with precise measurements of levels inantiprotonic H1,H2,He3,He4 and indicate S and P waves quasi-bound states in theNN system [11]. The same may be obtained with the K− mesons.

•• Atom → Nucleus radiative transitions

The simplest γ -ray radiative transitions from kaonic hydrogen are given in (4)

p K− → γ �. (4)

Kaonic atoms and �(1405) 137

Table 2 Partial widths of the2P states in K-mesic hydrogen

EB(�(1405)) [MeV] �abs [meV] �γ [meV] �γ /�abs (%)

27 1.5 · 10−2 3 0.512 3.9 · 10−2 3 1.3

The first experiment performed in liquid hydrogen [12] gave an upper limit for therate R�γ < 4 · 10−4 per stopped meson. Another experiment in liquid [16] came withR�γ = (0.86 ± 0.07+0.10−0.08) · 10−3. The physics behind these experiments wasbased on reaction p K− → �(1405) → γ �. However, another radiative transition ispossible

p K− → γ �(1405). (5)

which resembles an ordinary atomic radiation. Transition rates are calculable forsome states. Thus the radiative transition from an |2, P > atomic state to the state of�(1405) is found under the assumption that the latter is the NK quasi-bound statedescribed by a wave function ψ�. For the electric radiative transition one finds thewidths given in Table 2. As K− p state is a mixture of isospin 0 and isospin 1 statesan additional factor of 1/2 was included in �γ of Table 2, as well as the Fried Martinfactor.

This γ -transition rate may be compared to the “nuclear” absorption rate �abs

which is obtained from the KN�(1385) coupling [13]. The latter gives the basicwidth of this state. The values in Table 2 are shown for two binding energies of�(1405) characteristic for potential models and chiral effective models. In both casesthe wave function of �(1405) was obtained with the Yamaguchi separable model ofinverse range κ = 3.8/ f m and a strength parameter chosen to fit the binding energy.Once the widths of |2P > states are known the widths of |nP > states are obtained bysimple re-scaling of the atomic wave function at nuclear distances. This consists of thebarrier factor and normalization ψ ∼ rL N(n, L). Normalization factors N scale bothwidths in the same way and one obtains �(n, P) = �(2, P) · 32/3 · (n − 1)(n + 1)n−5.

The atomic cascade of K-meson involves several types of transitions: Auger elec-tron emissions, X-ray emissions, collision induced transitions, the γ -ray emissionsof (4) and the nuclear capture dominated in S waves by the �(1405) and in P-waves by �(1385) formation. The cascade populates many atomic states for sometime. From the |nP > states the γ -ray transitions offer noticeable branching ratios.Thus to estimate the chance for the γ -ray transitions one needs to know the totalprobability of nuclear absorption from all P states. It is commonly assumed thatmost of mesons reach high |n, S > or |n, P > states and are absorbed by the nucleus.In this way the occupation probability of |n, P > states is apparently sizable. Somecascade models [14, 15] estimate these at a few percent level at least in dilute gases.All together one could expect the emission of one γ ray per 103 – 104 mesons stoppedin gas hydrogen.

Measurements would meet a heavy background due to pionic decays. Old exper-iments looking for K− p → γ + X [16, 17] devised ingenious techniques to reduceit which was successful with high energy γ , but in the region of reaction (5) thisbackground was difficult to resolve and removed from the published data (Horvath,private communication). An improvement requires a cleaner beam, tagging of thedecay products and theory of the line shape. The last point is briefly discussed below.

138 S. Wycech

The γ line shape may be determined with the realization that the real final statein (5) consists of the photon and the �,π pair. The full amplitude for the reactionis a product of three factors: the γ emission vertex, the intermediate propagation ofK− p and the transition matrix T(K− p → �,π). The probability of decay involvesa modulus square of the latter summed over channels and the phase space of thefinal pair. By the unitarity condition this yields the absorptive part of the elasticK− p scattering amplitude Im A(K− p → K− p) extrapolated to the �(1405) region.The other ingredients: contribution from the photon phase space and effects of K− ppropagation in the intermediate states are standard. These factors strongly deformthe Lorentzian shape expected for a quasi-bound state. The detailed calculation willbe published elsewhere (Wycech, in preparation), here only an approximate formulafor the line shape is given. One has �γ = ∫

dk S(k) where the spectral density isgiven by

S(k) = Im A(K− p → K− p)(Eatom − k)k · const

[(k + Batom + k2(1/2μKN + 1/2MKN)]2.

(6)

In this expression, k is the photon energy, μKN and MKM are the reduced and totalmasses of the KN system, Batom is the atomic binding and Eatom = MK + MN −Batom. The quantity of interest is Im TKN,KN(Eatom − k) for energies extendingbelow the KN threshold. The resonant peak is strongly deformed on the low k sidebut the related factor given by (6) may be calculated with a good accuracy. Experi-ments would resolve the position of �(1405) and distinguish the two approaches: thephenomenological one from the chiral one.

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