Unresolved issues in the search for eta-mesic nuclei Neelima G. Kelkar Dept. de Física, Universidad de los Andes, Bogotá, Colombia  Knowing an unstable state when you see one S-matrix poles Cross section bumps Phase shift jumps, Time delay etc  Fun and frustration in finding eta mesic nuclei Reaction mechanisms of eta production Analysis of final state interaction  Where do we stand?

What is a resonance? What is a quasibound state? What is a virtual state? What is a quasivirtual state? Locating unstable states

S-matrix poles in the complex energy and momentum plane The energy is given by E = p 2 /2μ, where μ is the reduced mass of the system. Physical sheet  Unphysical sheet  Bound, Quasibound states width Resonances are defined as positive energy states on the unphysical sheet Resonances and Quasibound state poles lead to exponential decay

Cross section bumps not a sufficient condition for the existence of a resonance H. Ohanian and C. G. Ginsburg, Am. J. Phys. 42, 301 (1974). Argand diagrams anticlockwise loop in the Argand diagram of the complex scattering amplitude (often used in locating hadronic resonances) - alone cannot guarantee the existence of a resonance N. Masuda, Phys. Rev. D, 2565 (1970); P. D. B. Collins, R. C. Johnson and G. G. Ross, Phys. Rev. 176, 1952 (1968). Inverse correspondence (unstable state)  (pole of an S-matrix)? L. Fonda, G. C. Ghirardi and G. L. Shaw, Phys, Rev. D 8, 353 (1973); G. Calucci and G. C. Ghirardi, Phys. Rev. 169, 1339 (1968); G. Calucci, L. Fonda and G. C. Ghirardi, Phys. Rev. 166, 1719 (1968).

Collision times (time delay) An intuitive picture (delay due to the creation and propagation of an unstable state): Time delay in a resonant (R) scattering process is much larger than in a non- resonant (NR) scattering process A + B  A + B

E.P. Wigner, Phys. Rev. 98, 145 (1955) The first term has a peak at and the second one at The interaction has delayed the radial wave packet by Multichannel scattering

A Pedagogic example of time delay – the deuteron The S-matrix for a neutron-proton system constructed from a square well potential which reproduces the correct binding energy of the deuteron is given as a function of l as,

What do we expect? Bound state never decays, time delay must be infinite at the real negative energy where the bound state occurs Virtual state also occurs at a real negative energy. However the wave function is not normalizable, state is unphysical, hence, infinite negative time delay Quasibound (or unstable bound) state – complex energy Bound  real part of energy is negative, finite imaginary part gives width … positive finite time delay at real negative energy n-p system has one bound state at MeV – the deuteron n-p system has one virtual state at -0.1 MeV We put a small imaginary potential and get a fictitious quasibound state too.

A more realistic example from the unstable states of the eta mesons and nuclei N. G. Kelkar, K. P. Khemchandani, B. K. Jain, J. Phys. G 32, 1157 (2006)

Dwell time delay (rather than phase time delay) With the scattering phase shift close to threshold energies Phase time delay of Wigner: for s-waves (l=0) becomes singular! Dwell time delay (difference of the time spent in a region with and without interaction) can be shown to be related to phase time delay as: N. G. Kelkar, PRL 99, (2007) Close to threshold, the real scattering phase shift for s-waves

Searching for ETA MESIC NUCLEI

The η meson is a pseudoscalar meson (spin 0, parity negative) with a mass around 547 MeV. It is an isoscalar meson. Lifetime? very short, decays in s Eta beams difficult to make! Flavour wave functions

Early experimental searches … At Brookhaven National Laboratory (BNL) R. E. Chrien et al., PRL 60, 2595 (1988) concluded negatively on the existence of 15 η O formed from the (π +,p) reaction on 16 O and similarly for lithium, carbon and aluminium nuclei. At Los Alamos Meson Physics Facility (LAMPF) J.D. Johnson et al., PRC 47, 2571 (1993) the double charge exchange reaction π + 18 O  π - 18 Ne was investigated based on an idea of Haider and Liu, PRC 36, 1636 (1987) where the charge exchange could proceed through π + n  π 0 p  π - p or π + n  η p  π - p The η could be in the continuum or in an intermediate bound state with the nucleus. -weak affirmation of an eta mesic state was found

More recently … At the Mainz Microtron facility (MAMI) M. Pfeiffer et al., PRL 92, found η-mesic 3 He with a binding energy of -4.4±4.2 MeV and a width of 25.6 ± 6.1MeV via photoproduction -this work was criticized and an experiment with better statistics was planned and repeated by F. Pheron et al., Phys. Lett. B 709, 21 (2012) - an unambiguous conclusion was however not reached! Time delay in elastic η 3 He  η 3 He scattering Small scattering length favours a broad quasibound state around -5 MeV

Similar conclusions for η 4 He  η 4 He from time delay

Bremsstrahlung induced eta –mesic nuclei production was claimed by G. A. Sokol et al., Phys. Atomic Nuclei 71, 509 (2008) γ + 12 C  p(n) + 11 η B ( 11 η C)  π + + n + X a lowering of the mass of the N* resonance in the π + n spectrum for a relative angle between the particles of was taken as an indication of the formation of an η – mesic nucleus. … and the search continues …. one of the most recent efforts “Search for η-mesic 4 He with WASA-at-COSY” P. Adlarson et al., PRC 87, (2013) W. Krzemien, P. Moskal, J. Smyrski and M. Skurzok, EPJ Web of Conferences 66, (2014) … for latest references and a detailed review: N. G. Kelkar, K. P. Khemchandani, N. J. Upadhyay, B. K. Jain, Rep. Prog. Phys. 76, (2013)

η- nucleon scattering amplitude (η- nucleon interaction) should in principle be deduced from η – nucleon elastic scattering With a lifetime of s, eta beams are not available! η– nucleon and η – nucleus interaction must be deduced from eta meson producing reactions! π N  η N, γ N  η N p p  p p η, p n  d η, p d  3 He η, p d  p d η, d d  4 He η, p 6 Li  η 7 Be γ 3 He  η 3 He etc. What do we need?  Reaction mechanism for η production  Model for the final state interactions Theoretical investigations

Mass of the η meson ~ 547 MeV, mass of pion ~ 140 MeV Large momentum transfer to the nucleus Phase space alone cannot explain the data near threshold

Laget and Lecolley recognized the need for the one-, two- and three-body graphs in meson production. Applying to the p d  3He η they found that the one- and two-body graphs underestimate the cross sections by 2-3 orders of magnitude! (PRL 61, 2069 (1988)). Faeldt and Wilkin found good agreement with the threshold data on the p d  3 He η reaction (NPA 587, 769 (1995)) using the three body mechanism – two step model: Motivated by the success of the two step model, the final state interaction between η- 3 He was included explicitly using few body equations K. P. Khemchandani, N. G. Kelkar, B. K. Jain, NPA 708, 312 (2002)

 The two step model with an input η N interaction corresponding to a scattering length of a ηN = (0.88, 0.41) worked well!  The off-shell rescattering found essential for reproducing the data Rescattering of an off-shell η which is brought on-shell due to the final state interaction with the nucleus was indeed found to be important later for the p + 6 Li  η + 7 Be reaction too N. J. Upadhyay, N. G. Kelkar, B. K. Jain, NPA 824, 70 (2009).

Isotropic angular distributions near threshold are well reproduced in the two step model – however the model fails at high energies! Forward peaks in the angular distribution cannot be reproduced! Restricting the intermediate pion to forward angles K. P. Khemchandani, N. G. Kelkar, B. K. Jain, PRC 76, (2007) K. P. Khemchandani, N. G. Kelkar, B. K. Jain, PRC 68, (2003)

A similar situation indeed exists in the calculation of the p d  3 H Λ K + reaction V. I. Komarov, A. V. Lado, Yu Uzikov, J. Phys. G 21, L69 (1995)

and the p d  3He X reaction … L. A. Kondratyuk and Yu N. Uzikov, arXiv:nucl-th/ K. Schoenning et al., PRC79, (2009) p d  3 He ω reaction at 1450 MeV beam energy

η– nucleus final state interaction and the need for few body equations Scattering length approximation: Energy dependence of the reaction is determined by the on-shell scattering amplitude of the final state and approximating the s-wave phase shift near threshold - squared amplitude in the absence of FSI A proper description of FSI  η can be produced off-shell and brought on-shell due to FSI The off-shellness enters by expressing the final state wave function as a solution of the Lippmann – Schwinger equation The half-off-shell η – nucleus T – matrix is generated by solving few-body equations for the η – nucleus system.

An η N t-matrix which gives a scattering length of (0.88, 0.41) fm is used as input for the few body T-matrix for η 3 He N. G. Kelkar, K. P. Khemchandani, N. J. Upadhyay, B. K. Jain Rep. Prog. Phys. 76, (2013) Scattering length of (-2.31, 2.57) taken from C. Wilkin, PRC47, R938 (1993) Any conclusions regarding the sign or magnitude of the η – nucleus scattering length based on fits to data using the scattering length approximation with arbitrary multiplicative factors can be misleading!

Where do we stand?

 A good knowledge of the η N interaction is crucial for the interpretation of data on η meson production on nuclei and the theoretical prediction of η mesic nuclei Phenomenological and theoretical studies involving meson-baryon coupled channels and the relevant baryon resonances obtain a wide range of scattering lengths  Theoretical studies (chiral perturbation theory) predict mostly small real part of the scattering length and phenomenological ones predict large values.  One of the most elaborate phenomenological calculation, however, predicts (0.3, 0.18) fm (J. Durand et al., PRC78, (2008)) which is curiously close to the very first prediction of (0.28, 0.19) fm of Bhalerao and Liu, PRL 54, 865 (1985).  A small scattering length would favour quasibound states of light eta-mesic nuclei and heavy eta-mesic nuclei with lower binding energies.  Experimental searches can be broadly divided into direct searches (difficult to perform) and indirect searches involving η production (depend on theoretical models for interpretation)  Experiments planned at JPARC, MAMI, COSY facilities

Table taken from N. G. Kelkar, K. P. Khemchandani, N. J. Upadhyay, B. K. Jain Rep. Prog. Phys. 76, (2013)

Thank you! Dziҿkujҿ!