Signature of the (1405) resonance in neutron spectra from the reaction. J. Révai Wigner RCP, Budapest BLTP, JINR, Dubna

Transcription

1 Signature of the (1405) resonance in neutron spectra from K d the reaction J. Révai Wigner RCP, Budapest BLTP, JINR, Dubna

2 Motivation - the Λ(1405) plays a central role in low-enegry Kaon-nuclear physics - strong and sometimes passionate discussions about its structure - below the K p threshold, experimentally ureachable in two-body reactions with stable particles; only in reactions involving n 3 particles - the simplest case : K d K d (1405) ( ) I 0,1 n ( ) p I 1 - a dynamically exact calculation can be performed 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 2

3 Motivation - the Λ(1405) plays a central role in low-enegry Kaon-nuclear physics - strong and sometimes passionate discussions about its structure - below the K p threshold, experimentally ureachable in two-body reactions with stable particles; only in reactions involving n 3 particles - the simplest case : K d K d (1405) ( ) I 0,1 n ( ) p I 1 - a dynamically exact calculation can be performed What a real experiment can tell about the Λ(1405)? 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 3

4 Method Coupled particle channels Faddeev-AGS treatment of the three-body system. K NN N The formalism has been applied to this system by several authors (Bahaui & al (2003), Shevchenko & al. (2007),(2012), Ikeda & Sato (2007),(2012) and, maybe, others...). The main aim of these works was search for quasi-bound states or to produce reliable K d scattering lengths. The present work: K pp Exact calculation of the amplitudes of the break-up processes K pn ABU q pn; f TBU P d; K p K i K d p ABU N; f TBU d; K i n q p P 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 4

5 Exact -- in what sense? In the framework of non-relativistic quantum mechanics, no approximations made in the dynamics, the output corresponds exactly to the input. cm E 50 MeV The process below is certainly basically well described K within this picture. Insisting on some kind of relativization would mean to sacrifice a certain amount of dynamical accuracy... 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 5

6 The on-shell energy relations are: k k k EnEEn 2 2 q pn Ecm mn m m 2 2 n, 2 2 q pn p K m m K n m 2 2 pn K, pn 2 P K E m m m K 2 d n p K, d p 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 6

7 The on-shell amplitude for a given neutron energy depends on E t and : n, f A( E, t, ) A ( q, p, ) n f BU n f E n p 2E N n n, q 2( E E ) n n cos( p, q ) The neutron spectra of different possible processes are then proportional to d q p PE (, t, )~ (2 ) AE (, t, ) 4 n 2 n f n, K, d n f dq dp den PK N n t The inclusive neutron spectrum (when no other particles are detected) is given by 1 PE ( ) dtpe (, t, ) n n f f 1 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 7

8 The input: separable s-wave interactions We have used two sets of phenomenological potentials, taken from N.V. Shevchenko One- versus two-pole KN potential: K d scattering length PRC 85,035203(2011), and a more recent one K d KN Near-threshold scattering and properties of kaonic deuterium NPA ,50(2012) These potentials reproduce all known experimental data on the lowenergy KN system (the first one being fitted to the older KEK data on the kaonic hydrogen 1s level shift, while the latter one reproduces the latest SIDDHARTA values). 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 8

9 (1405) Here are their resonance parameters (pole positions) in MeV KEK SIDDHARTA 1-pole i ( i) i ( ,8 i) 2-pole i ( i) i ( i) Since the main issue of the work is to study the appearance of subthreshold resonances of different type in a 3-body reaction, we kept both interactions, not only the most advanced one. NN potential: two-term separable potential with repulsion, reproduces the deuteron and the singlet- and triplet s-wave phase shifts 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 9

10 N 3 IN 2, 2, SN 1 I N N interaction: 1, reproducing the scarce experimental data. In 1 state two-channel. N 2 N interaction neglected. L 0 The calculation at present is restricted to, and to incident kaon energies cm E up to (kaon LAB momenta < 250 MeV/c). K 50 MeV 1 3 (I 2 I 2 (I For physical masses and mixed) we have 12 unknown functions, while for averaged masses only) Numerical method: expansion of the unknown functions on cubic spline basis. Smaller matrices, no problem with logarithmic singularities they are integrated with known functions. Also no interpolation of the solutions needed, when the break-up amplitudes are calculated. 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 10

11 Results K d A by-product: effect of the physical masses on the scattering length (fm): averaged KEK SIDDHARTA KEK physical SIDDHARTA 1-pole i - 1,47 + 1,22 i i - 1,50 + 1,23 i 2-pole i ,23 i i i a few percent effect mainly in the real part. Now insignificant, may be of some use, if precise atom level shifts will be available. K d 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 11

12 We have calculated the inclusive neutron spectra for incident kaon energies both below and above the deuteron break up threshold in the range E cm K 050 MeV. Overall shape strong peak near the origin with no sign of the. Above break-up: a cusp at, when the system is at its threshold and additional neutrons from, yielding a structureless peak from to. Kinematical reason: to E n see the peak, should exceed the energy of the incident kaon by the amount of energy, which separates the pole position from the threshold, while in the deuteron the neutron energy distribution is dominated by the low-energy part. Esmaili, Akaishi and Yamazaki (EAY) (PRC 83, ) proposed a method to more or less eliminate the disturbing kinematical effects in order to reveal the dynamical ones: they suggest to consider the DEViation spectrum: P DEV E E n n 0 P( En) P ( E ) nonres n P( E n ) Eth KN K d K pn E E n th (1405) KN 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 12

13 Let s see how this idea can be realized in our case: K d n Apart from the full break-up amplitude A q p ; T P ; BU N f BU K d i we can define two approximate ones: A q p ; t P ; sing BU N f, KN K d i the s.c. single scattering amplitude, in which the full break-up operator is replaced by the two-body -matrix and the Born amplitude: A q p t t,kn ; V P ; Born BU N f, KN K d i T BU Kinematics in initial and final states: deuteron wave function and coordinate transformation 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 13

14 Thus we have three amplitudes with the properties: A BU three-body dynamics + three-body kinematics sing A BU two-body dynamics + three-body kinematics Born A BU three-body kinematics Born sing P / P P / P and we expect that the DEV spectra and will display (reveal) three- and two-body dynamics, respectively. BU BU BU Born BU Our main results are displayed in the following pictures, where for a given potential and a given incident kaon energy we plotted four sing ( n), DEV( n), DEV( n) 0 0 P E P E P E quantities:, and for comparison, the two-body elastic cross section corresponding to. E n 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 14

15 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 15

16 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 16

17 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 17

18 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 18

19 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 19

20 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 20

21 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 21

22 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 22

23 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 23

24 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 24

25 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 25

26 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 26

27 E n It is expected, that the dependence of the born spectrum is basically determined by the initial and final states, while the details of the VKN, potential influence it only weekly. This expectation is important, if the DEV spectrum method is to be applied for extracting information about from an experimentally measured neutron spectrum. Therefore we calculated P not with our realistic nonres ( E ) KN n interactions, but with the simplest possible separable potential: (1405) q V q 1 1 I I born KN 2 I 2, KN 2 I 2 q ( ) q ( ) KN KN and took I0 I1 I0 I1 I0 I1 1;, KN, KN KN KN born 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 27

28 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 28

29 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 29

30 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 30

31 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 31

32 We also asked the question, under which conditions could the resonance P( E n ) be observed in the direct spectra. We modified two of the interaction parameters for one of the potentials (KEK 1-pole) ( and (1405) I 0 KN, ) in such a way, that the position of the remained at its original place, while its width could be made smaller. KN I 0 KN, KN The results: 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 32

33 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 33

34 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 34

35 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 35

36 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 36

37 Conclusions P( E ) (1405) little chance to see in the inclusive spectra n the in the considered energy range; reasons: large width, kinematical effects DEV spectrum method eliminates the kinematical effects and reveals (in most cases) the desired maxima in the neutron spectra shape and position are significantly changed/shifted with respect to the original 2-body resonance; separate problem, how to deduce the parameters of the initial (1405) from a measured DEV spectrum; not discussed here KN 3 of the considered potentials (KEK1, KEK2 and SIDDHARTA 2) yield neutron DEV spectra with a bump structure, which can be related to their (1405) pole positions; in the case of SIDDHARTA 1 no maxima are seen in the DEV spectra, while in single scattering approximation the resonance peak is reproduced; probable reason: extreme closeness of the resonance to the KN threshold combined with its rather large width 4 Oct. 2012, Barcelona J. Révai, Signature of the Λ(1405) 37