Unresolved issues in the search for eta-mesic nuclei

Neelima G. KelkarDept. 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 planeThe energy is given by E = p2/2μ, 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 resonanceH. 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 resonanceN. 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 squarewell 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 energyBound 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 -2.22 MeV – the deuteronn-p system has one virtual state at -0.1 MeVWe put a small imaginary potential and get a fictitious quasiboundstate 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, 210403 (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 10-18 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

ηO formed from the (π+,p) reaction on 16O 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+ 18O π- 18Ne was investigated based on an idea of Haider and Liu, PRC 36, 1636 (1987) where the charge exchange could proceed through

π+ n π0 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, 252001 found η-mesic 3He with a binding energyof -4.4±4.2 MeV and a width of 25.6 ± 6.1MeVvia photoproduction- this work was criticized and an experiment with better statistics wasplanned and repeated byF. Pheron et al., Phys. Lett. B 709, 21 (2012)- an unambiguous conclusion was however not reached!

Time delay in elastic η 3He η 3He scattering

Small scattering length favoursa broad quasibound state around -5 MeV

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

Bremsstrahlung induced eta –mesic nuclei production was claimed byG. A. Sokol et al., Phys. Atomic Nuclei 71, 509 (2008)

γ + 12C p(n) + 11ηB (11

ηC) π+ + n + X

a lowering of the mass of the N* resonance in the π+ n spectrum for a relativeangle between the particles of 1800 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 4He with WASA-at-COSY”P. Adlarson et al., PRC 87, 035204 (2013)

W. Krzemien, P. Moskal, J. Smyrski and M. Skurzok, EPJ Web of Conferences 66, 09009 (2014) … for latest references

and a detailed review: N. G. Kelkar, K. P. Khemchandani, N. J. Upadhyay, B. K. Jain, Rep. Prog. Phys. 76, 066301 (2013)

η- nucleon scattering amplitude (η- nucleon interaction) shouldin principle be deduced from η – nucleon elastic scattering

With a lifetime of 10-18s, eta beams are not available!

η– nucleon and η – nucleus interaction must be deduced from etameson producing reactions!

π N η N, γ N η N

p p p p η, p n d η, p d 3He η, p d p d η,

d d 4He η, p 6Li η 7Be

γ 3He η 3He 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 explainthe 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-bodygraphs 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 thep d 3He 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-3He was included explicitly using few body equationsK. 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 aηN = (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 withthe nucleus was indeed found to be important later for the p + 6Li η + 7Be reaction tooN. 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 distributioncannot be reproduced!

Restricting the intermediate pion to forward angles

K. P. Khemchandani, N. G. Kelkar, B. K. Jain, PRC 76, 069801 (2007)

K. P. Khemchandani, N. G. Kelkar, B. K. Jain, PRC 68, 064601 (2003)

A similar situation indeed exists in the calculation of the p d 3HΛ 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/9510010

K. Schoenning et al., PRC79, 044002 (2009)

p d 3He 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-shellscattering 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 3He

N. G. Kelkar, K. P. Khemchandani, N. J. Upadhyay, B. K. JainRep. Prog. Phys. 76, 066301(2013)

Scattering length of (-2.31, 2.57) taken fromC. 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, 025204 (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. JainRep. Prog. Phys. 76, 066301(2013)

Thank you!

Dziҿkujҿ!