Brief remarks. m 2hyp,i + p2π,i + 2π + p 2π,i = m 2 hyp,i + p2 π,i + E π,i (2) m K + m A =

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1 1 Brief remarks In FINUDA the strangeness-exchange reaction is used to produce Λ- hypernuclei with stopped K s: K stop + Z A Z Λ A + π (1) Thanks to the energy conservation, we can write for each bound state of the hypernucleus: m K + m A = m m 2hyp,i + p2π,i + 2π + p 2π,i = m 2 hyp,i + p2 π,i + E π,i (2) where m K is the K mass, m A is the target nucleus ground state mass, m hyp,i is the mass of the particular hypernucleus level, m π is the pion mass, p π,i is the pion momentum for the produced hypernucleus level and, E π,i is the corresponding pion total energy. So, measuring the pion momentum with the experimental apparatus, we can get the hypernucleus mass in the specific level: m hyp,i = (m K + m A E π,i ) 2 p 2 π,i (3) to be compared with existing data from π spectroscopic measurements. In general, in literature, the quantity m hyp m A is reported, instead of m hyp : m hyp,i m A = (m K + m A E π,i ) 2 p 2 π,i m A (4) Other quantites that are quoted in literature are the Λ binding energies, B Λ,i and the hypernucleus excitation energies Ex i = B Λ,g.s. B Λ,i. Starting from the relation: m hyp,i = m A 1n + m Λ B Λ,i (5) where m A 1n indicates the mass of the hypernuclear core in its ground state, and considering the hypernucleus mass m hyp,i determined above, (3), for each bound state, we have: B Λ,i = m A 1n + m Λ m hyp,i (6) Ex i = B Λ,g.s. B Λ,i (7) Therefore, to evaluate B Λ,i we need to know the quantities m A, m A 1n, m Λ, m K and m π. Obviously: m π = MeV m K = MeV

2 2 m Λ = MeV To evaluate the masses of the target nuclei, m A, and of their cores, m A 1n, we can use the AME2003 evaluation and the tables therein (G. Audi, A.H. Wapstra and C. Thibault, Nucl. Phys. A 729 (2003), ). The.pdf file is also available from this page. Nuclear masses can be evaluated following two different methods. Following the explanation of Table I in the AME2003 file, we have: m N (A, Z) = m A (A, Z) Z m e + B e (Z) (8) where m N (A, Z) is the nuclear mass of the species with A nucleons and Z protons, m A (A, Z) is its atomic mass (column 9 of table I), m e = MeV is the electron mass and B e (Z) the total binding energy of the Z electrons, negligible for our purposes. Equivalently: m N (A, Z) = Zm p + (A Z)m n A B.E. + B e (Z) (9) where m p = MeV is the proton mass, m n = MeV is the neutron mass, B.E. is the binding energy per nucleon (column 7 of table I) and B e (Z) the total binding energy of the Z electrons. In the following table masses of the five different targets (plus 28 Si) used in the first FINUDA data taking and of their nuclear cores are reported; they have been evaluated following equation (9). Normally six decimal digits are listed, to be rounded off at the end of further calculations; we remember to this purpose that the FINUDA goal resolution on hypernuclear energy levels is E 0.7 MeV FWHM. target m A (MeV) m A 1n (MeV) 6 Li ( 5 Li) Li ( 6 Li) C ( 11 C) Al ( 26 Al) Si ( 27 Si) V ( 50 V )

3 Another interesting quantity is the expected momentum of the formation pion for the different known hypernuclear levels of each target. It can be evaluated from the energy conservation (2): 3 p π = [(mk + m A ) 2 + m 2 hyp ] 2 m2 π m 2 hyp (10) 2(m K + m A ) where m hyp is given by equation (5). In the following tables the expected values of p π are reported (with several decimal digits, as discussed above) for the present FINUDA targets. For each target the used B Λ values are indicated as well as their source. 7 Li B Λ from H. Noumi et al., Nucl. Phys. A 691 (2001), 123 (KEK E336 experiment, E 2 MeV FWHM) C B Λ from T. Nagae et al., Nucl. Phys. A 691 (2001), 76 (KEK E369 experiment, E 1.45 MeV FWHM) Al: no precise B Λ value is given in literature for 27 Λ Al.

4 4 28 Si B Λ from Hasegawa et al., Phys. Rev. C 53 (1996), (KEK E140a experiment, E MeV FWHM) V B Λ from T. Nagae et al., Nucl. Phys. A 691 (2001), 76 (KEK E369 experiment, E 1.95 MeV FWHM) To complete the discussion, it is necessary to note that the (temptative) quantum numbers (J P ) that in literature are assigned to the different hypernuclear levels of each target are related to the quantum numbers of the corresponding core (A 1 neutrons) levels. Information concerning nuclear structure can be obtained from the following sites: wwwndc.tokai.jaeri.go.jp In the last site, from the Nuclear Structure & Decay Data section it is possible to access:

5 5 the AME2003 atomic evaluation files; the Evaluated Nuclear Structure Data File (ENSDF); the Experimental Unevaluated Nuclear Data List (XUNDL). From the ENSDF database it is possible to retrieve the adopted scheme of energy levels and gamma transitions for each nuclide, as postscript file, html table or in the ENSDF format. From the XUNDL database it is possible to download experimental, unevaluated, nuclear structure data compiled in the ENSDF format for nuclides with A 21..ps file of adopted schemes for 5 Li, 6 Li, 11 C, 26 Al, 27 Si and 50 V are available also from this page.