This is a sel -a chi ed e sion o an o iginal a icle. This e sion
may di e om he o iginal in pagina ion and ypog aphic de ails.
Au ho (s):
Ti le:
Yea :
Ve sion:
Copy igh :
Righ s:
Righ s u l:
Please ci e he o iginal e sion:
CC BY 3.0
h ps://c ea i ecommons.o g/licenses/by/3.0/
Double cha ge exchange eac ions as a p obe o neu inoless double be a decay
nuclea ma ix elemen s
© Au ho s, 2020
Published e sion
San opin o, E.; Fe e i, J.; Ga cía-Tecocoa zi, H.; Magana Vse olodo na, R.
San opin o, E., Fe e i, J., Ga cía-Tecocoa zi, H., & Magana Vse olodo na, R. (2020). Double
cha ge exchange eac ions as a p obe o neu inoless double be a decay nuclea ma ix
elemen s. In L. Acos a, P. Amado -Valenzuela, & D. J. Ma ín-Lámba i (Eds.), SNP '20 : XLIII
Symposium on Nuclea Physics. Ins i u e o Physics. Jou nal o Physics : Con e ence Se ies, 1610.
h ps://doi.o g/10.1088/1742-6596/1610/1/012013
2020
Jou nal o Physics: Con e ence Se ies
PAPER • OPEN ACCESS
Double cha ge exchange eac ions as a p obe o neu inoless double
be a decay nuclea ma ix elemen s
To ci e his a icle: E. San opin o e al 2020 J. Phys.: Con . Se . 1610 012013
View he a icle online o upda es and enhancemen s.
This con en was downloaded om IP add ess 130.234.21.0 on 26/11/2020 a 13:15
Con en om his wo k may be used unde he e ms o heC ea i e Commons A ibu ion 3.0 licence. Any u he dis ibu ion
o his wo k mus main ain a ibu ion o he au ho (s) and he i le o he wo k, jou nal ci a ion and DOI.
Published unde licence by IOP Publishing L d
XLIII Symposium on Nuclea Physics 2020
Jou nal o Physics: Con e ence Se ies 1610 (2020) 012013
IOP Publishing
doi:10.1088/1742-6596/1610/1/012013
1
Double cha ge exchange eac ions as a p obe o
neu inoless double be a decay nuclea ma ix
elemen s
E. San opin o1, J. Fe e i2, H. Ga c´ıa-Tecocoa zi3, R. Magana
Vse olodo na1and wi hin he NUMEN p ojec .
E-mail: [email p o ec ed]
1INFN, Sezione di Geno a, ia Dodecaneso 33, Geno a 16146, I aly
2Depa men o Physics, Uni e si y o Jy ¨askyl¨a, P.O. Box 35 (YFL), 40014 Jy ¨askyl¨a,
Finland
3Depa men o Physics, Uni e si y o La Pla a (UNLP), 49 y 115 cc. 67, 1900 La Pla a,
A gen ina
Abs ac . The o malism o desc ibe hea y-ion double cha ge exchange (DCE) p ocesses in
he eikonal and small-momen um ans e app oxima ions in oduced in Phys. Re . C 98,
061601(R) (2018) is b ie ly discussed. I is also shown ha , unde he p e ious app oxima ions,
he hea y-ion DCE c oss-sec ion can be ac o ized in e ms o a eac ion and a nuclea pa .
A double cha ge exchange e ec i e po en ial is explici ly de i ed in he closu e app oxima ion
and also o he i s ime he explici o m o he DCE nuclea ma ix elemen s, ha a e o he
o m o double Gamow-Telle and double Fe mi. The ecen hypo hesis o a linea co ela ion
be ween double Gamow-Telle neu inoless double be a decay and DCE nuclea ma ix elemen s
is con i med hanks o he i s explici de i a ion o DCE nuclea ma ix elemen s, and by means
o mic oscopic IBM2 calcula ions.
1. In oduc ion
Neu inoless double be a (0νββ) decays ank as one o he mos in e es ing Beyond he S anda d
Model p ocesses. Thei expe imen al obse a ion would imply ha he conse a ion o he lep on
numbe is iola ed and ha he neu inos a e Majo ana- ype pa icles. Mo eo e , i may also
p o ide a mean o measu e he neu ino mass. The e a e se e al ongoing expe imen al sea ches,
including EXO [1], Cuo e [2], KamLAND-Zen [3] and Ge da [4]. None o hem has p o ided
indica ions o 0νββ decays ye .
The e ha e been s ong heo e ical e o s o p o ide guidelines o he expe imen alis s.
Howe e , he e a e s ong disc epancies among he esul s in he di e en app oaches [5, 6,
7, 8, 9, 10]. In pa icula , he Nuclea Ma ix Elemen s (NMEs) compu ed wi hin he di e en
nuclea models disag ee by mo e han a ac o o wo. Fu he mo e, hese esul s may equi e
addi ional eno maliza ion o quenching [11]. A possible mean o o e come hese di icul ies is
o use hea y-ion double-cha ge exchange (DCE) p ocesses o pu cons ain s on neu inoless
0νββ nuclea ma ix elemen s.
Se e al hea y-ion DCE expe imen s a e ongoing a RNCP Osaka [12, 13], RIBF RIKEN [14],
and LNS INFN [15, 16, 17]. The i s wo o hem make use o high-ene gy hea y-ion double-
XLIII Symposium on Nuclea Physics 2020
Jou nal o Physics: Con e ence Se ies 1610 (2020) 012013
IOP Publishing
doi:10.1088/1742-6596/1610/1/012013
2
cha ge exchange p ocesses in o de o s udy mul i-spin-isospin lip exci a ion modes, such as
a high-ene gy double Gamow-Telle gian esonance (DGT-GR) [12], ha has been p edic ed
h ee decades ago [18, 19]. The expe imen a LNS-INFN is aiming o ex ac in o ma ion o
pu cons ain s on some o he nuclea ma ix elemen s ele an o 0νββ decays [15, 16, 17].
All hese expe imen s ha e igge ed a s ong heo e ical in e es [20, 21, 22, 23], including he
e o in he imp o emen o he eac ion pa [24, 25, 26].
In he p esen con ibu ion, we b ie ly discuss some aspec s o he hea y-ion DCE o malism
o Re . [21]. The e, o he i s ime he o malism o desc ibe hea y-ion DCE was explici ly
de eloped by making use o he eikonal app oxima ion.
2. DCE po en ial
In DCE eac ions, wo pai s o nucleons – one om he a ge and one om he p ojec ile –
in e ac . In pa icula , wo p o ons (neu ons) a e con e ed in o wo neu ons (p o ons) in he
a ge , and wo neu ons (p o ons) a e con e ed in o wo p o ons (neu ons) in he p ojec ile,
wi h he mass numbe o he a ge , A, and he p ojec ile, a, bo h emaining unchanged. A
de i a ion o a DCE e ec i e po en ial, desc ibing bo h long- and sho - ange in e ac ions,
was ca ied ou in Re . [21] by conside ing he one-pion-exchange and sho - ange-in e ac ion
diag ams depic ed in Fig. 1.
Figu e 1. Leading diag ams in a double-cha ge-exchange p ocess. F om le o igh , hey
ep esen a double-pion-exchange in e ac ion, a double con ac e m and a mixed one-pion-
exchange plus con ac e m. Figu e aken om Re . [21]; APS CopyRigh .
By making use o he closu e app oxima ion, which consis s in a e aging o e he in e media e
nuclea s a es [27, 28], one ob ains [21]
VDCE(~q1, ~q2) = 4
3 π
mπ4(~σP1·~q1)(~σT1·~q1)
ω1(ω1+¯
EP)~τP1 ·~τT1 (~σP2·~q2)(~σT2·~q2)
ω2(ω2+¯
EP)(ω2+¯
ET)~τP2 ·~τT2
+ 2 c2
T
¯
EF
P+¯
EF
T
+c2
GT(~σP1·~σT1)(~σP2·~σT2)
¯
EGT
P+¯
EGT
T
+cTcGT(~σP2·~σT2)
¯
EGT
P+¯
EF
T
+cTcGT(~σP1·~σT1)
¯
EF
P+¯
EGT
T
×(~τP1 ·~τT1)(~τP2 ·~τT2) +  π
mπ2(~σP1·~q1)(~σT1·~q1)
ω1(ω1+¯
EP)(ω1+¯
ET)~τP1 ·~τT1
×cT(~τP2 ·~τT2) + cGT(~σP2 ·~σ2)(~τP2 ·~τT2)+ 1 ↔2i,
(1)
whe e  π
mπ2≃400 MeV· m3[29], he alues o he pa ame e s cGT = 217 MeV m3and
cT= 151 MeV m3a e aken om he li e a u e [29], and ωi=q~q2
i+m2
π. The labels P1,
P2, T1 and T2 s and o he nucleons wi hin he p ojec ile (P1 and P2) and he a ge (T1
and T2) in ol ed in he DCE p ocess. The p ojec ile and a ge closu e ene gies a e gi en by
¯
Eα
p=hEa
n−Ea
iiαand ¯
Eα
=hEA
n−EA
iiα, espec i ely, and he supe sc ip α=GT o F, indica es
he ype o ene gy exci a ion. The i s line o Eq. (1) co esponds o he double-pion-exchange
con ibu ion ( i s diag am o Fig. 1), he second o he double-con ac e m (second diag am
o Fig. 1), inally he hi d line o he mixed pion-exchange con ac - e m ( hi d diag am o Fig.
1).
XLIII Symposium on Nuclea Physics 2020
Jou nal o Physics: Con e ence Se ies 1610 (2020) 012013
IOP Publishing
doi:10.1088/1742-6596/1610/1/012013
3
3. DCE c oss-sec ion and mic oscopic IBM2 nuclea ma ix elemen s
The p e ious o malism can be used o compu e he DCE nuclea ma ix elemen s wi hin
he mic oscopic In e ac ing Boson Model (IBM2) [30] and he DCE c oss-sec ions in he low-
momen um ans e limi by making use o he dis o ed-wa e Bo n app oxima ion (DWBA).
I one conside s ansi ions be ween 0+and 0+g ound s a es, he di e en ial c oss sec ion is
gi en by [21]
dσ
dΩ=k
k0µ
4π2¯h22
|Ti |2,(2)
whe e µis he educed mass o he a ge -p ojec ile sys em, kand k0a e he incoming and
ou going momen a, and Ti is he T-ma ix o he eac ion. Ti can be calcula ed by means o
he Dis o ed Wa e Bo n App oxima ion (DWBA),
Ti =DΨ−
~
k0Φ VΨ+
~
kΦiE=1
(2π)3/2Zd~
Rei(χ(b)−~
Q·~
R)Mi (~m),(3)
whe e one also uses he eikonal app oxima ion o he c.m. sca e ing. In he p e ious equa ion,
Ψ+
~
k,~
k0a e he wa e unc ions which desc ibe he c.m. mo ion o he a ge and p ojec ile ions,
Φi, he in insic wa e unc ions o he nuclei be o e and a e he in e ac ion, which can be
w i en as he p oduc o p ojec ile and a ge nucleon wa e unc ions. In he pa icula case o
0+
i→0+
ansi ions o he a ge , one ge s [21]
Mi (m)→2" MDGT
T→T0MDGT
P→P0
¯
EGT
P+¯
EGT
T!+ MDF
T→T0MDF
P→P0
¯
EF
P+¯
EF
T!# ,(4)
whe e MDGT and MDF a e Double-Gamow-Telle (DGT) and Double-Fe mi (DF) nuclea
ma ix elemen s, espec i ely, o he p ojec ile/ a ge (A = P, T). The p e ious ma ix elemen s
a e de ined as [21]
MDGT
A→A0=cGT DΦ(A0)
J0X
n,n0
[~σn×~σn0](0)~τn~τn0Φ(A)
JE(5)
and
MDF
A→A0=c DΦ(A0)
J0X
n,n0
~τn~τn0Φ(A)
JE,(6)
whe e he sum uns o e he nucleons (n, n0) in ol ed in he p ocess. They a e calcula ed in he
mic oscopic In e ac ing Boson Model (IBM2) [21, 30]. Finally, he c oss sec ion o Eq. (2) can
be w i en in he eikonal app oxima ion and low-momen um ans e limi as
dσ
dΩ→k
k0µ
4π2¯h22
2F(θ) MDGT
T→T0MDGT
P→P0
¯
EGT
P+¯
EGT
T
+MDF
T→T0MDF
P→P0
¯
EF
P+¯
EF
T!
2
,(7)
whe e F(θ) is an angula dis ibu ion [21, Eq. (14)].
4. Linea co ela ion be ween DCE and 0νββ nuclea ma ix elemen s
In Re . [20], he au ho s discussed he possible eme gence o a linea co ela ion be ween DGT
DCE and 0νββ nuclea ma ix elemen s by means o a la ge-scale shell-model calcula ion. In
Re . [21], he p e ious hypo hesis was con i med by making use o a di e en nuclea model,
he mic oscopic IBM2. Mo eo e , as also discussed in he p e ious sec ions, he exis ence o a
p ocedu e o ac o ize he DCE c oss sec ions in e ms o eac ion and nuclea pa s was explici ly

XLIII Symposium on Nuclea Physics 2020
Jou nal o Physics: Con e ence Se ies 1610 (2020) 012013
IOP Publishing
doi:10.1088/1742-6596/1610/1/012013
4
demons a ed in he eikonal app oxima ion. Mos impo an , a mic oscopic desc ip ion o DCE
p ocesses was de eloped wi h he de i a ion o a DCE po en ial in he closu e app oxima ion
[21]. Thanks o his, one may hink o use he p esen and o hcoming expe imen al da a
o hea y-ion DCE c oss-sec ions o place an uppe limi on 0νββ NMEs in e ms o he DCE
expe imen al da a a e y o wa d angles, whe eas in he case o la ge sca e ing angles, he
nuclea pa is expec ed o be a con olu ion o beam and a ge NMEs.
The eme gence o he linea co ela ion be ween DCE and 0νββ nuclea ma ix elemen s can
be shown by calcula ing he p e ious DCE and 0νββ NMEs wi hin he same nuclea model and
o a su icien ly la ge se o nuclei. Then, a simple linea eg ession analysis can be conduc ed
and a eg ession line can be d awn. I one plo s he mic oscopic IBM2 esul s [21, 31] o he
116Cd →116Sn, 128Te →128Xe, 82Se →82K , and 76Ge →76Se DGT DCE and 0νββ NMEs,
one ob ains he clea eg ession line shown in Fig. 2.
Figu e 2. Co ela ion be ween calcula ed DCE-DGT NMEs [21] and 0νββ-DGT NMEs [31].
The o ange squa e, g een iangle, ed s a , and blue ci cle s and o 116Cd →116Sn, 128Te →
128Xe, 82Se→82K and 76Ge →76Se da a, espec i ely. Figu e om Re . [21]; APS CopyRigh .
Finally, i is wo h o no e ha he slopes o ou cu es in Fig. 2 and [21, Figs. 3] and o
hose epo ed in [20, Figs. 4] a e qui e di e en . The main eason o his misma ch esides
in he p ocedu es used in Re . [20] and [21] o ex ac he o m o he DCE po en ial, which
is needed o calcula e he DCE NMEs. In he i s case, he au ho s did no de i e he DCE
ma ix elemen s explici ly, bu hey made a guess on he o m only o he DGT DCE nuclea
ma ix elemen s ; see [20, Eq. (6)]. In he second case, he au ho s de i ed he DCE po en ial o
Eq. (1) and [21, Eq. (3)] explici ly om he diag ams o Figs. 1. Because o his, in he esul s
o [20, Figs. 4] one can app ecia e he eme gence o a linea co ela ion be ween DGT DCE and
0νββ nuclea ma ix elemen s, bu he slopes o he cu es a e “ andom”. On he con a y, he
slopes o he cu es in Fig. 2 and [21, Figs. 3] a e he “physical” ones.
5. Conclusion
The o malism o calcula e he c oss-sec ions and Nuclea Ma ix Elemen s (NMEs) o hea y-ion
Double Cha ge Exchange (DCE) p ocesses o Re . [21] was b ie ly desc ibed. I was also shown
ha he hea y-ion DCE c oss-sec ion can be ac o ized in e ms o a eac ion and a nuclea
pa and ha he e is a linea co ela ion be ween DCE NME’s and neu inoless NME’s [21].
This will make i possible o ex ac he 0νββ NMEs om expe imen al measu emen s o DCE
c oss-sec ions.
XLIII Symposium on Nuclea Physics 2020
Jou nal o Physics: Con e ence Se ies 1610 (2020) 012013
IOP Publishing
doi:10.1088/1742-6596/1610/1/012013
5
The nex s ep will be o compu e he spec oscopic ampli udes and adial ansi ion densi ies
in he mic oscopic IBM scheme. This will open he possibili y o gi e p edic ions wi h
mic oscopic IBM models in he ield o cha ge exchange eac ions.
[1] J. B. Albe e al. (EXO-200 Collabo a ion), Na u e 510, 229 (2014).
[2] K. Al onso e al. (CUORE Collabo a ion), Phys. Re . Le . 115, 102502 (2015).
[3] A. Gando e al. (KamLAND-Zen Collabo a ion), Phys. Re . Le . 117, 082503 (2016).
[4] M. Agos ini e al. (GERDA Collabo a ion), Na u e 544, 47 (2017).
[5] J. Suhonen and O. Ci i a ese, Nucl. Phys. A 847, 207 (2010).
[6] A. Me oni, S. T. Pe co and F. ˇ
Simko ic, JHEP 1302, 025 (2013).
[7] R. A. Sen’ko and M. Ho oi, Phys. Re . C 88, 064312 (2013).
[8] F. ˇ
Simko ic, V. Rodin, A. Faessle and P. Vogel, Phys. Re . C 87, 045501 (2013).
[9] M. T. Mus onen and J. Engel, Phys. Re . C 87, 064302 (2013).
[10] J. Ba ea, J. Ko ila and F. Iachello, Phys. Re . C 91, 034304 (2015).
[11] J. Engel and J. Men´endez, Rep . P og. Phys. 80, 046301 (2017).
[12] M. Takaki e al., JPS Con . P oc. 6, 020038 (2015).
[13] M. Takaki e al., CNS Ann. Rep. 94, 9 (2014).
[14] T. Uesaka e al., RIKEN RIBF NP-PAC, NP1512-RIBF141 (2015).
[15] F. Cappuzzello e al., EPJ Web Con . 117, 10003 (2016).
[16] F. Cappuzzello, M. Ca alla o, C. Agodi, M. Bondi, D. Ca bone, A. Cunsolo and A. Fo i, Eu . Phys. J. A
51, 145 (2015).
[17] F. Cappuzzello e al., Eu . Phys. J. A 54, 72 (2018).
[18] P. Vogel, M. E icson, and J. D. Ve gados, Phys. Le . B212, 259 (1988).
[19] N. Aue bach, L. Zamick, and D. C. Zheng, Ann. Phys. (N.Y.) 192, 77 (1989).
[20] N. Shimizu, J. Men´endez and K. Yako, Phys. Re . Le . 120, 142502 (2018).
[21] E. San opin o, H. Ga c´ıa-Tecocoa zi, R.I. Maga˜na Vse olodo na and J. Fe e i, Phys. Re . C 98, 061601
(R) (2018).
[22] V. dos S. Fe ei a, A. R. Samana, F. K mpo i´c and M. Chiappa ini, Phys. Re . C 101, 044314 (2020).
[23] J. I. Bellone, S. Bu ello, M. Colonna, J. A. Lay and H. Lenske, Phys. Le . B 807,135528 (2020).
[24] H. Lenske, J. I. Bellone, M. Colonna, and J. A. Lay, Phys. Re . C 98, 044620 (2018); H. Lenske, J. Phys.
Con . Se . 1056, 012030 (2018).
[25] J. I. Bellone, M. Colonna, H. Lenske and J. A. Lay, J. Phys. Con . Se . 1056, 012004 (2018).
[26] H. Lenske, F. Cappuzzello, M. Ca alla o and M. Colonna, P og. Pa . Nucl. Phys. 109, 103716 (2019).
[27] W. C. Hax on and G. J. S ephenson, P og. Pa . Nucl. Phys. 12, 409 (1984).
[28] S. M. Bilenky and C. Giun i, In . J. Mod. Phys. A 30, 1530001 (2015).
[29] G. F. Be sch and H. Esbensen, Rep. P og. Phys. 50, 607 (1987).
[30] A. A ima, T. Oh suka, F. Iachello and I. Talmi, Phys. Le . 66B, 205 (1977).
[31] J. Ba ea, J. Ko ila and F. Iachello, Phys. Re . C 87, 014315 (2013).