1. Probing Beta Decay Matrix Elements through Heavy Ion Charge Exchange Reactions
(1)( 2)
(3)
(2)
(4)
Bellone J.I. , Lenske H. , Colonna M. , J.A. Lay , within the NUMEN collaboration
Dipartimento di Fisica e Astronomia, Università degli studi di Catania
INFN/LNS, via S. Sofia, 62, CT – 95123, Catania , Italia
Institut für Theoretische Physik, Justus-Liebig-Universität Giessen, D Giessen, Germania
Departamento de FAMA, Universidad de Sevilla, Apartado 1065, E Sevilla, Spagna

2. J. Barea, J. Kotila, F. Iachello, Phys. Rev. Lett. 109 042501(2012)
Experimental observable → half – life :
● Dirac vs Majorana neutrino masses: 0νββ
elementary particle physics factor
Beyond Standard Model Physics?
PF factor
0νββ NME
calculated through different nuclear structure models
→ values differ of about a factor of 3
J. Barea, J. Kotila, F. Iachello, Phys. Rev. Lett (2012)
DCEX reactions
same initial and final nuclei involved
same Gamow-Teller, Fermi and rank-2 tensor operators, but combined through different coefficients
F. Cappuzzello, M. Cavallaro, C. Agodi, M. Bondì, D. Carbone, A.Cunsolo, A. Foti, Eur. Phys. J. A (2015), 51

3. First steps toward DCEX Cross Section Factorization
● 0νββ strength from DCEX Cross Section measurements → DCEX Cross Section factorization
● DCEX reaction ≈ sequence of 2 Single Charge Exchange (SCEX) processes
analogy with 2νββ
→ DCEX = sequence of 2 uncorrelated SCEX processes ν A B n p e - a b p n π A B
DCEX
analogy with 0νββ
→ DCEX = sequence of 2 correlated SCEX processes n A B p e
ν = ν - a b p n π A B

4. Zero – range approximation
CEX Cross Section
(CEX)
Direct reactions are supposed to be dominated by elastic scattering processes, while the inelastic ones
can be treated as perturbations → DWBA
R. H. Bassel, R. M. Drisko, G. R. Satchler, The Distorted Wave Theory of Direct Nuclear Reactions
DWBA → Transition matrix element can be written in terms of distorted waves and internal
coordinate – depending nuclear wave functions
→ calculations done in momentum space
Zero – range approximation

5. → Nuclear Transition Matrix Element given by
Reaction Kernels
Distortion Factor
Transition Form Factor
radial transition density
~ β / ββ decay strength

6. → Direct Reactions separation ansatz : Reaction Kernel Gaussian shaped
for small momentum transfer (q << 1/σ)
monopole component of j (qρ) multipole expansion dominates αβ
not exact factorization
H. Lenske, J. I. Bellone, M. Colonna, J. A. Lay , Heavy Ion Single Charge Exchange and Beta decay Matrix Elements, submitted.

7. Black Disk Approximation ( BDA )
|V | << |W| → χ(r) ~ 0 dove |W| ≠ 0 e χ(r) = PW dove |W| = 0
→ analytical determination of Distortion Factor
Distortion Factor
H. Lenske, J. I. Bellone, M. Colonna, J. A. Lay , Heavy Ion Single Charge Exchange and Beta decay Matrix Elements, submitted.

8. ● Is BDA a good approximation?
Let’s check the effects of Real and Imaginary part of the Optical Potential
on SCEX Cross Section for Heavy Ions ( HIDEX code, by H. Lenske)
F. Cappuzzello et al., Nucl. Phys. A 739 (2004) 30-56 40
Ca ( O, F) K 18 18 40
(1 state for both K and F ) + 40 18
@ 270 MeV
H. Lenske, J. I. Bellone, M. Colonna, J. A. Lay , Heavy Ion Single Charge Exchange and Beta decay Matrix Elements, submitted.

9. ● Is BDA a good approximation?
Let’s check the effects of Real and Imaginary part of the Optical Potential
on SCEX Cross Section for Heavy Ions ( HIDEX code, by H. Lenske)
F. Cappuzzello et al., Nucl. Phys. A 739 (2004) 30-56 40
Ca ( O, F) K 18 18 40 + 40 18
@ 270 MeV
(1 state for both K and F )

10. SCEX Transition Matrix Element vs momentum transfer
H. Lenske, J. I. Bellone, M. Colonna, J. A. Lay, Heavy Ion Single Charge Exchange and Beta decay Matrix Elements, submitted.

11. Ca + O → K + F → Ar + Ne DCEX - 2νββ DCEX Reaction Kernel40 18
DCEX - 2νββ
@ 270 MeV
DCEX Reaction Kernel
DCEX Distortion Factor
free propagator
under particular conditions,
can be expressed in terms of SCEX Reaction Kernels product

12. → Direct Reactions separation ansatz : Reaction Kernel Gaussian shaped
off – shell angular integration
easily performed +
Pole Approximaxion +
Single state dominance (SSD) is assumed
considered state at 2.29 MeV for K , studied through experiments on
Ca ( n , p ) and Ar ( p , n ) reactions
and
F g.s. ( 1 ), studied through experiments on O ( p , n ) nuclear reaction and Ne β - decay 40 + 18

13. + + separation ansatz SSD Ca + O → K + F → Ar + Ne Pole Approximaxion40 18

14. SUMMARY and CONCLUSIONS
♣ Direct Reactions separation ansatz : Reaction Kernel Gaussian shaped
enable both SCEX and DCEX Cross Section factorization, for Heavy Ions at low energy
♣ Factorization “exact” for q = 0 (both for SCEX and DCEX), but it works well up to q ≈ 25 MeV
(for SCEX processes);
♣ Distortion Factor N analytical determination in BDA and behaves like ≈ 1/A
(both for SCEX and DCEX processes);
♣ “Work in progress” :
(DCEX ≈ sequence of 2 uncorrelated SCEX processes)
code development for DCEX factorized Heavy Ion Cross Section
for reactions under study within the NUMEN collaboration.
♣ Next (main goal):
analogy DCEX → 0νββ
(DCEX ≈ sequence of 2 correlated SCEX processes) D αβ

15. THANK YOU
FOR YOUR ATTENTION

17. M.A. Franey, W. G. Love, Phys. Rev. C 31 (1985) 488
Optical potential
→ Double folding approach both for Real and Imaginary part
in HIDEX code
Heavy ion reactions are strongly absorptive processes → elastic scattering and peripheral inelastic reactions are mainly sensitive to the nuclear surface regions of the interacting nuclei
→ Impulse approximation
Real and Imaginary NN optical potentials →
M.A. Franey, W. G. Love, Phys. Rev. C 31 (1985) 488
Coulomb interaction included in the evaluation of distorted waves and calculated by doubly folding Coulomb potential for a point-like charge with target and projectile charge densities.
\rho_1 e \rho_2 = one-body local nuclear densities  Fermi distribution

18. A. K. Kerman, H. McManus, R. M. Thaler, Ann. Phys. 8 (1959) 551-635
Nuclear potential
→ Central term
→ Tensorial term
with
→ Spin – Orbit term
V (r) = sum of 2 (T) or 3 (C) Yukawians, with strengths and ranges chosen to represent the
long – range tail (1.414 fm) of the One Pion Exchange Potential (OPEP) and medium and
short-range parts, which corresponds to σ (0.40 fm) and ω, ρ and δ (0.25 fm) meson
exchange, respectively.
where
c,t,LS NN
A. K. Kerman, H. McManus, R. M. Thaler, Ann. Phys. 8 (1959)

19. Transition Form Factor
Radial Transition Density
(for each excitation energy ω)
~ β decay strength
- calculated through HIDEX code (H. Lenske), using QRPA approach, starting from
1QP nuclear energy levels, g.s. calculated in HFB Mean Field Theory approach
and excited states using a Wood – Saxon nuclear potential, with parameters set in order to fit
experimental single – particle energies.
- normalized in order to reproduce the transition amplitude belonging to GT multiple operator

20. (residual nucleus) belonging to the same SU(4) S, T multiplet
● DCEX reaction rates are expected to be small → yeld increased using projectile (target) and ejectile
(residual nucleus) belonging to the same SU(4) S, T multiplet 18
O projectile ( 0 , g.s.) → + 18
Ne ejectile (0 , g.s.) +
lightest non-radiactive T = 1 isotope;
easily produced with high intensities.
nearly 100% GT sum rule strength 3(N-Z) is exhausted by the F g.s. involved in the T = 1, intermediate transition;
not possible the reversed reaction because it should require a radioactive beam and the
( Ne, O) is characterized by smaller beta decay strength. + 18 20 20 40
Ca target (0 , g.s.) → + 40
Ar residual nucleus (0 , g.s.) +
NOT members of the same T multiplet → GT transition NOT super allowed
BUT GT transition is mainly distributed in the 1 excited state
of K involved in the intermediate channel
(above all 2.73 MeV state, followed by the states at 2.33 and 4.4 MeV) + 40
NMEs invariant under time reversal → possibility of studying a given reaction by esploring it’s time – reversal counterpart , i.e. the opposite reaction, in order to obtain a better (signal/bg) ratio.

21. Why are we interested on DCEX Cross Section? B p e
ν = ν - a b p n π B
First attempts to determine NMEs from nuclear reactions
were done using ( π , π ) DCEX reactions,
→ but processes described by different kind of operators
→ no information about 0νββ NMEs
Analogies between Heavy Ion DCEX reactions and 0νββ :
❶ same initial and final nuclei involved;
❷ same Gamow-Teller, Fermi and rank-2 tensor operators, but combined through different coefficients;
❸ large linear momentum (~ 100 MeV/c) involved in the intermediate off-shell stastes;
❹ non – local processes, characterized by 2 vertices localized in the same pair of valence nucleons;
❺ same nuclear medium involved, so in medium effects are expected to influence
system in both cases (possibility to extract information about g quenching);
❻ off-shell propagator . p p π π p n n A
W. R. Gibbs, M. Elghossain, W. B. Kaufmann, Phys. Rev. C 48 (1993) 1546
(es. Nucl. Phys. A 355 (1981) , E. Oset, D. Strottman  regarding pion induced SCEX and nuclear structure). A
F. Cappuzzello, M. Cavallaro, C. Agodi, M. Bondì, D. Carbone, A.Cunsolo, A. Foti, Eur. Phys. J. A (2015), 51

22. decay amplitude is proportional to the effective Majorana mass
0ν ββ
Not allowed by SM
allowed if and m ≠ 0 ν
● Neutrinos involved in EW porcesses are flavour eigenstates → they do not have definite masses, but their masses are combinations of mass eigenstates
decay amplitude is proportional to the effective Majorana mass
neglegible with respect to the average neutrino energy and momentum
if CP is conseved is real

23. Doppio Decadimento Beta
Neutrino: particella di Dirac o di Majorana?
(processo del 2° ordine)
Spettro di energia continuo, analogamente al decadimento beta singolo
processo a 4 leptoni
processo a 2
leptoni
proibito nell’ambito del
Modello Standard
possibile solo se il neutrino
ha massa di Majorana
Un nucleo che può decadere può anche decadere , sebbene con vite medie diverse.

24. effective Majoron coupling constant
● light neutrino exchange
● sterile neutrino exchange
sterile neutrino
mass
● heavy neutrino exchange
● light/heavy neutrino exchange with Majororn emission
effective Majoron coupling constant

25. ● SCEX Cross Section factorization is possible only making some approximations:
Eikonal approximation → ok for E : U (x + Δx) – U (x) ~ const. per Δx ~ λ
Strong absorption (or Black Disk) limit
→ ok for describing heavy nuclei;
moreover this approximation allows to simplify distortion factor. b
i/f
i/f
for (p, n) light ion reactions
T. N. Taddeucci et al., Nucl. Phys. A 469 (1987) 125 – 172
for (p, n) heavy ion reactions
F. Osterfeld, Rev. Mod. Phys., vol. 64, 2 (1992)
|V | << |W| and χ ~ 0 in the range where |W| ≠ 0 and equale to PW elsewhere