Neutral-Current Neutrino-Nucleus Scattering off Xe Isotopes
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Neu al-Cu en Neu ino-Nucleus Sca e ing o Xe Iso opes
© 2018 P. Pi inen e al. The publica ion o his a icle was unded by SCOAP3.
Published e sion
Pi inen, Pekka; Suhonen, Jouni; Yd e o s, E.
Pi inen, P., Suhonen, J., & Yd e o s, E. (2018). Neu al-Cu en Neu ino-Nucleus Sca e ing o
Xe Iso opes. Ad ances in High Ene gy Physics, 2018, A icle 9163586.
h ps://doi.o g/10.1155/2018/9163586
2018
Resea ch A icle
Neu al-Cu en Neu ino-Nucleus Sca e ing o Xe Iso opes
P. Pi inen ,1J. Suhonen ,1and E. Yd e o s2
1Uni e si y o Jy askyla, Depa men o Physics, P.O. Box 35, 40014, Finland
2Ins i u o Tecnol´
ogico de Ae on´
au ica, DCTA, 12228-900 S˜
ao Jos´
e dos Campos, B azil
Co espondence should be add essed o P. Pi inen; pekka.a.pi inen@s uden .jyu. i
Recei ed 26 Ap il 2018; Accep ed 18 Augus 2018; Published 4 Oc obe 2018
Academic Edi o : A hanasios Ha zikou elis
Copy igh © 2018 P. Pi inen e al. This is an open access a icle dis ibu ed unde he C ea i e Commons A ibu ion License,
which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed. The
publica ion o his a icle was unded by SCOAP3.
La ge liquid xenon de ec o s aiming o da k ma e di ec de ec ion will soon become iable ools also o in es iga ing neu ino
physics. In o ma ion on he e ec s o nuclea s uc u e in neu ino-nucleus sca e ing can be impo an in dis inguishing neu ino
backg ounds in such de ec o s. We pe o m calcula ions o di e en ial and o al c oss sec ions o neu al-cu en neu ino
sca e ing o he mos abundan xenon iso opes. The nuclea -s uc u e calcula ions a e made in he nuclea shell model o
elas ic sca e ing and also in he quasipa icle andom-phase app oxima ion (QRPA) and mic oscopic quasipa icle-phonon model
(MQPM) o bo h elas ic and inelas ic sca e ing. Using sui able neu ino ene gy dis ibu ions, we compu e es ima es o o al
a e aged c oss sec ions o 8B sola neu inos and supe no a neu inos.
1. In oduc ion
When he idea o neu inos was i s sugges ed by Pauli
in1930,i was hough ha heywouldne e beobse ed
expe imen ally. Only wo decades la e in e ac ion o neu-
inos wi h ma e was de ec ed in he amous Cowan-
Reines expe imen [1]. Mo e ecen ly, de ec ion and esea ch
o neu inos ha e become mo e and mo e o an e e yday
commodi y, and a ious mo e e sa ile ways o examine
in e ac ions o he li le neu al one ha e eme ged and a e
being es ed in labo a o ies all o e he wo ld.
Cohe en elas ic neu ino-nucleus sca e ing (CE]NS) is
a p ocess whe e he neu ino in e ac s wi h he a ge nucleus
as a whole ins ead o a single nucleon. Al hough CE]NS
has been p edic ed since he 1970s [2], i was disco e ed
only e y ecen ly by he COHERENT collabo a ion [3].
Due o he cohe en enhancemen , his expe imen had he
ema kable ea u e o de ec ing neu inos wi h a compac
14.6 kg de ec o ins ead o a massi e de ec o olume which
is used in con en ional neu ino expe imen s. Cohe en
neu ino-nucleus sca e ing is on one hand an impo an
po en ial sou ce o in o ma ion o beyond-s anda d-model
physics [4–11], bu on he o he hand i may also hinde new
disco e ies as i will s a dis u bing da k ma e de ec o s in
he nea u u e.
A g ea expe imen al e o has been pu in o di ec ly
de ec ing da k ma e in he pas ew decades (see [12] o
a e iew). The nex -gene a ion de ec o s a e expec ed o
be sensi i e enough o p obe c oss sec ions low enough
o s a obse ing CE]NS as an i educible backg ound
[13, 14]. Sola neu inos, a mosphe ic neu inos, and di use
supe no a backg ound neu inos p o ide a na u al sou ce
o backg ound neu inos, which o ob ious easons canno
be shielded agains . As he e a e unce ain ies in he luxes
o each o he a o emen ioned neu ino ypes, he sensi i i y
o WIMP (weakly in e ac ing massi e pa icle) de ec ion is
basically limi ed o he magni ude o his unce ain y. To
make ma e s wo se, i has been shown ha o some speci ic
WIMP masses and c oss sec ions he ecoil spec a o CE]NS
e y closely mimic ha o sca e ing WIMPs [14].
I is he e o e o u mos impo ance o de ise a way o go
h ough his neu ino loo . One po en ial way o achie ing
his is ha ing di ec ional sensi i i y in he de ec o [15, 16]. As
sola and a mosphe ic neu inos ha e a dis inc sou ce wi hin
he sola sys em, i is expec ed ha hei ecoil di ec ion
would be di e en o ha o WIMPs, which a e ypically
Hindawi
Ad ances in High Ene gy Physics
Volume 2018, A icle ID 9163586, 11 pages
h ps://doi.o g/10.1155/2018/9163586
2Ad ances in High Ene gy Physics
assumed o be g a i a ionally bound in a halo spanning he
galaxy. Also a ising om he di e en o igin o neu inos
and WIMPs is he idea o using iming in o ma ion o
disc imina e be ween neu ino and WIMP induced e en s
in a de ec o [17]. Due o he mo ion o he Ea h a ound
he Sun, i is expec ed ha he sola neu ino lux peaks
a ound Janua y, bu he WIMP lux peaks in June when
he eloci ies o he Sun and Ea h a e he mos in phase.
The ecoil spec a o WIMPs and neu inos could also be
dis inguished i he WIMP-nucleus in e ac ion happens ia
a nons anda d ope a o eme ging in he e ec i e ield heo y
amewo k [18, 19].
Some o he leading da k ma e expe imen s use a
liquid xenon a ge [20–24], which allows o easy scalabili y
o la ge de ec o olumes. I is expec ed ha he xenon
de ec o s a e he i s o hi he neu ino loo . In his a icle
we compu e c oss sec ions o elas ic and inelas ic neu ino-
nucleus sca e ing o he mos abundan xenon iso opes.
Fo he cohe en sca e ing we use he quasipa icle andom-
phase app oxima ion (QRPA) amewo k and he nuclea
shell model o model he nuclea s uc u e and we compa e
he esul s be ween he wo models. The wa e unc ions o he
s a es o odd-mass xenon iso opes a e ob ained by using he
mic oscopic quasipa icle-phonon model (MQPM) on op o
a QRPA calcula ion. Inelas ic sca e ing is compu ed in he
QRPA/MQPM o malism. In ou calcula ions we conside 8B
sola neu inos and supe no a neu inos.
A simila QRPA calcula ion has been made in [25]
o 136Xe, whe e bo h cha ged-cu en and neu al-cu en
inelas ic sca e ing was examined. Simila compu a ions o
neu al-cu en neu ino-nucleus sca e ing c oss sec ions
ha e been made be o e o he s able cadmium iso opes in
[26] and o molybdenum iso opes in [27]. Bo h calcula-
ions used he QRPA/MQPM app oach. To ou knowledge
his a icle p esen s he i s calcula ion o neu al-cu en
neu ino-nucleus sca e ing wi hin a comple e mic oscopic
nuclea amewo k o Xe iso opes o he han 136Xe.
This a icle is o ganized as ollows. In Sec ion 2 we ou line
he o malism used o compu e neu al-cu en neu ino-
nucleus sca e ing. In Sec ion 3 we summa ize he nuclea -
s uc u e calcula ions made o he a ge xenon iso opes.
In Sec ion 4 we discuss he esul s o ou c oss-sec ion
calcula ions and in Sec ion 5 conclusions a e d awn.
2. Neu al-Cu en
Neu ino-Nucleus Sca e ing
In his sec ion we summa ize he o malism used o compu e
neu al-cu en neu ino-nucleus sca e ing p ocesses. We
examine s anda d-model eac ions media ed by he neu al
𝑍0boson, namely, he p ocesses
]+(𝐴,𝑍)→ ]+(𝐴,𝑍),(1)
]+(𝐴,𝑍)→ ]+(𝐴,𝑍)∗,(2)
i.e., he elas ic and inelas ic sca e ing o neu inos o a
nucleus (wi h 𝐴nucleons and 𝑍p o ons), espec i ely. In
he elas ic p ocess he ini ial and inal s a es o he a ge
]
]
q
k
(A, Z)
(A, Z)
(∗)
Z0p
k
p
Figu e 1: A diag am o he neu al-cu en sca e ing p ocess. The
ou momen a o he in ol ed pa icles a e labeled in he igu e.
nucleus a e he same, while in he inelas ic p ocess exci a ion
o he a ge nucleus akes place. The kinema ics o he
sca e ingp ocessisillus a edinFigu e1.Welabel he ou
momen a o he incoming and ou going neu ino as 𝑘and
𝑘
, espec i ely. The momen a o he a ge nucleus be o e
and a e in e ac ing wi h he neu ino a e 𝑝and 𝑝
.The
momen um ans e o he nucleus is e e ed o as 𝑞=
𝑘
−𝑘=𝑝
−𝑝
. The neu ino kine ic ene gy be o e and
a e sca e ing is 𝐸and 𝐸.
The neu al-cu en neu ino-nucleus sca e ing di e en-
ial c oss sec ion o an exci ed s a e o ene gy 𝐸ex can be
w i en as [28]
𝑑2𝜎
𝑑Ω𝑑𝐸ex =𝐺2
Fk𝐸
𝜋(2𝐽+1)(∑
≥0𝜎
CL +∑
≥1𝜎
T), (3)
which comp ises he Coulomb-longi udinal (𝜎
CL)and ans-
e se (𝜎
T)pa s. They a e de ined as
𝜎
CL =(1+cos 𝜃)⟨𝐽M(𝑞)𝐽⟩2
+(1+cos 𝜃−2𝑏sin2𝜃)⟨𝐽L(𝑞)𝐽⟩2
+𝐸ex
𝑞(1+cos 𝜃)
×2Re {⟨𝐽M(𝑞)𝐽⟩∗⟨𝐽L(𝑞)𝐽⟩},
(4)
and 𝜎
T=(1−cos 𝜃+𝑏sin2𝜃)
⋅[⟨𝐽Tmag
(𝑞)𝐽⟩2+⟨𝐽Tel
(𝑞)𝐽⟩2]
∓𝐸+𝐸
𝑞(1−cos 𝜃)×2
⋅Re {⟨𝐽Tmag
(𝑞)𝐽⟩⟨𝐽Tel
𝐽⟩∗},
(5)
whe e he minus sign is aken o neu ino sca e ing and he
plus sign o an ineu ino sca e ing. 𝐽and 𝐽a e he ini ial
and inal s a e angula momen a o he nucleus. We use he
abb e ia ion
𝑏=𝐸k𝐸k
𝑞2,(6)
Ad ances in High Ene gy Physics 3
and 𝑞is he magni ude o he h ee-momen um ans e .
The o malism and a ious di e en ope a o s in ol ed a e
discussed in de ail in [28, 29].
To compu e he a e aged c oss sec ion ⟨𝜎⟩, we need o
old he compu ed c oss sec ions wi h he ene gy dis ibu ion
o he incoming neu inos. We ake he supe no a neu ino
spec um o be o a wo-pa ame e Fe mi-Di ac cha ac e
𝑓FD (𝐸)= 1
𝐹2(𝛼])𝑇]
(𝐸/𝑇])2
1+𝑒𝑘/(]−]),(7)
whe e 𝛼]is he so-called pinching pa ame e and 𝑇]is he
neu ino empe a u e. The no maliza ion ac o 𝐹2(𝛼])is
de ined by he o mula
𝐹(𝛼])=∫𝑥𝑑𝑥
1+𝑒−],(8)
and he empe a u e and mean ene gy o neu inos a e ela ed
by ⟨𝐸]⟩
𝑇]=𝐹3(𝛼])
𝐹2(𝛼]).(9)
We also examine sola neu inos om 8B be a decay. We use
an 8B neu ino ene gy spec um om [30].
3. Nuclea S uc u e o he Ta ge Nuclei
In his sec ion we ou line he nuclea -s uc u e
calcula ions pe o med o he in es iga ed nuclei
128,129,130,131,132,134,136Xe. We ha e pe o med compu a ions
in he quasipa icle andom-phase app oxima ion (QRPA),
mic oscopic quasipa icle-phonon model (MQPM), and he
nuclea shell model.
3.1. QRPA/MQPM Calcula ions. The nuclea s uc u e o
e en-e en Xe iso opes was compu ed by using he cha ge-
conse ing QRPA amewo k. The QRPA is based on a
BCS calcula ion [31], whe e quasipa icle c ea ion and anni-
hila ion ope a o s a e de ined ia he Bogoliubo -Vala in
ans o ma ion as 𝑎†
=𝑢𝑐†
+V
𝑐,
𝑎=𝑢
𝑐−V𝑐†
,(10)
wi h he egula pa icle c ea ion and annihila ion ope a o s
𝑐†
and
𝑐de ined in [32]. He e 𝛼con ains he quan um
numbe s (𝑎,𝑚)wi h 𝑎=(𝑛,𝑙,𝑗). The exci ed s a es wi h
espec o he QRPA acuum a e c ea ed wi h he phonon
c ea ion ope a o
𝑄†
=∑
N (𝐽)(𝑋
[𝑎†
𝑎†
]𝜔𝜔+𝑌
[
𝑎
𝑎]𝜔𝜔)(11)
o an exci ed s a e 𝜔=(𝐽
,𝑀,𝜋,𝑘),whe e𝑘is a
numbe labeling he exci ed s a es o gi en 𝐽.In heabo e
equa ion
N (𝐽)=√1+𝛿 (−1)𝜔
1+𝛿 ,(12)
and 𝑋
and 𝑌
a e ampli udes desc ibing he wa e unc ion
ha a e sol ed om he QRPA equa ion
[AB
−B∗−A∗][X
Y]=𝐸[X
Y], (13)
whe e he ma ix A is he basic Tamm-Danko ma ix and B
is he so-called co ela ion ma ix, bo h de ined in de ail in
[32].
We pe o m he QRPA calcula ions using la ge model
spaces consis ing o he en i e 0𝑠–0𝑑,1𝑝–0𝑓–0𝑔,2𝑠–1𝑑–0ℎ,
and 1𝑓–2𝑝majo shells, adding also he 0𝑖13/2 and 0𝑖11/2
o bi als. The single-pa icle bases a e cons uc ed by sol ing
he Sch ¨
odinge equa ion o a Coulomb-co ec ed Woods-
Saxon po en ial. We use he Woods-Saxon pa ame e s gi en
in [33]. We make an excep ion o 136Xe, adop ing he se o
adjus ed alues o single-pa icle ene gies om [25]. Due o
he neu on-magic na u e o 136Xe, adjus ed single-pa icle
ene gies a e necessa y o ge ag eemen wi h expe imen al
ene gy le els. The Bonn one-boson exchange po en ial [34]
was used o es ima e he esidual wo-body in e ac ion.
The QRPA o malism in ol es se e al pa ame e s ha
ha e o be ixed by i ing obse ables o expe imen al da a.
In he BCS calcula ion we i he p o on and neu on pai ing
s eng hs 𝐺p
pai and 𝐺n
pai so ha he lowes quasipa icle
ene gy ma ches he empi ical pai ing gap gi en by he h ee-
poin o mula [35]:
Δp(𝐴,𝑍)=1
4(−1)+1 [𝑆p(𝐴+1,𝑍+1)−2𝑆p(𝐴,𝑍)
+𝑆p(𝐴−1,𝑍−1)],
Δn(𝐴,𝑍)=1
4(−1)−+1 [𝑆n(𝐴+1,𝑍)−2𝑆n(𝐴,𝑍)
+𝑆n(𝐴−1,𝑍)].
(14)
I should be no ed ha o he neu on-magic 136Xe his
p ocedu e canno be done o he neu on pai ing s eng h.
We ha e ins ead used a ba e alue o 𝐺pai =1.0 o 136Xe.
The pa icle-pa icle and pa icle-hole e ms o he wo-
body ma ix elemen s a e scaled by s eng h pa ame e s 𝐺pp
and 𝐺ph, espec i ely. The ene gies o he compu ed QRPA
s a es a e qui e sensi i e o hese model pa ame e s. We i
he lowes exci ed s a es o each 𝐽sepa a ely o expe imen al
alues om [36] by al e ing he alues o 𝐺pp and 𝐺ph.The
alues used o he model pa ame e s a e gi en in Table 1.
The QRPA p ocess is known o p oduce s a es ha a e
spu ious, namely, he i s exci ed 0+s a e and he i s 1−
s a e. The i s 0+s a e has been deemed spu ious in [26, 37].
The i s 1−s a eisspu iousdue ocen e -o -massmo ion
as desc ibed in [32]. We ha e i ed he ene gies o hese
s a es o ze o, i possible, by using he model pa ame e s
𝐺pp and 𝐺ph, and subsequen ly he s a es ha e been omi ed
om calcula ions o he e en-mass iso opes and also om
he MQPM calcula ions o he odd-mass iso opes. The
con ibu ions o hese spu ious s a es o he o al neu ino-
nucleus sca e ing c oss sec ion would be iny in any case.
4Ad ances in High Ene gy Physics
Table 1: Model pa ame e s used in he BCS and QRPA calcula ions. Fo each nucleus (column 1) he alues o 𝐺pp and 𝐺ph (column 2) a e
gi en o he impo an 𝐽phonons in columns 3 o 9.
Nucleus 𝐺0
+1−2+3−4+5−6+
128Xe pp 0.796 1.000 1.000 1.000 1.000 1.000 1.000
ph 0.298 0.500 0.527 0.500 0.652 0.883 0.934
130Xe pp 0.730 1.000 1.000 1.000 1.000 1.000 1.000
ph 0.303 0.500 0.531 0.500 0.581 0.833 0.788
132Xe pp 0.653 1.000 1.000 1.000 1.000 1.000 1.000
ph 0.319 0.500 0.533 0.500 0.436 0.933 1.000
134Xe pp 0.500 1.000 1.000 1.000 1.000 1.000 1.000
ph 0.370 0.500 0.511 0.500 0.596 1.000 0.891
136Xe pp 0.843 1.000 1.000 1.000 1.000 1.000 1.000
ph 0.100 0.500 0.583 0.500 0.700 0.747 0.891
Table 2: The alence-space unca ions made in he shell-model calcula ions. The i s column labels he Xe iso ope; he ollowing i e columns
gi e he minimum/maximum numbe o neu ons on he single-pa icle o bi als 0𝑔7/2,1𝑑5/2,1𝑑3/2,2𝑠1/2,and1ℎ11/2, espec i ely.
Nucleus 0𝑔7/2 1𝑑5/2 1𝑑3/2 2𝑠1/2 1ℎ11/2
128Xe 8/8 6/6 0/4 0/2 4/12
129Xe 8/8 6/6 0/4 0/2 4/12
130Xe 8/8 4/6 0/4 0/2 0/12
131Xe 8/8 6/6 0/4 0/2 0/12
132Xe 0/8 0/8 0/4 0/2 0/12
134Xe 0/8 0/8 0/4 0/2 0/12
136Xe 0/8 0/8 0/4 0/2 0/12
Odd-mass xenon iso opes 129,131Xe a e hen compu ed by
using he MQPM o malism, in which we use a combina ion
o one- and h ee-quasipa icle s a es by coupling a quasi-
pa icle wi h a QRPA phonon o o m he h ee-quasipa icle
con igu a ions. The MQPM basic exci a ion can be w i en in
e ms o quasipa icle and QRPA-phonon c ea ion ope a o s
as [38]
Γ†
(𝑗𝑚)=∑
𝐶
𝑎†
+∑
,𝐷
[𝑎†
𝑄†
] .(15)
The ampli udes 𝐶and 𝐷a e compu ed by sol ing he MQPM
equa ions o mo ion. The de ailed desc ip ion o he p ocess
can be ound in [38]. No addi ional model pa ame e s a e
equi ed o he MQPM calcula ion aside o he pa ame e s
i ed o he BCS/QRPA calcula ion desc ibed abo e. We do
he MQPM calcula ions o 129Xe and 131Xe using 130Xe and
132Xe as e e ence nuclei, espec i ely. We selec all QRPA
phonons o 𝐽≤6wi h an ene gy less han 10 MeV o be used
in he calcula ion.
3.2. Shell-Model Calcula ions. We pe o m shell-model cal-
cula ions o Xe iso opes using he shell-model code
NuShellX@MSU [39]. We use he 0𝑔7/2,1𝑑5/2,1𝑑3/2,2𝑠1/2,
and 0ℎ11/2 alence space and he SN100PN in e ac ion [40].
The single-pa icle ene gies associa ed wi h he a o emen-
ioned o bi als in he SN100PN in e ac ion a e 0.8072, 1.5623,
3.3160, 3.2238, and 3.6051 MeV, espec i ely, o p o ons, and
−10.6089,−10.2893,−8.7167,−8.6944,and−8.8152MeV o
neu ons.
The ma ix dimension in he shell-model calcula ion
inc eases apidly when mo ing away om he 𝑁=82shell
closu e o 136Xe. Fo 132,134,136Xe we we e able o do a ull
calcula ion wi h no unca ions, bu o 128−131Xe we had o
pu es ic ions on he neu on alence space. The unca ions
made o each iso ope a e shown in de ail in Table 2. Fo he
iso opes 128−131Xe we assume a comple ely illed 0𝑔7/2 o bi al
and o 128,129,131Xe we also assume he 1𝑑5/2 o bi al o be ull.
These should be easonable app oxima ions when aiming o
desc ibe he g ound s a e and low-lying exci ed s a es in he
xenon nuclei. The o bi als 0𝑔7/2 and 1𝑑5/2 ha e he lowes
single-pa icle ene gies and he exci a ions a e likely o ake
place om highe o bi als when he neu on numbe o he
nuclei is qui e la ge.
The compu ed ene gy le els o he e en-mass xenon
iso opes a e gi en in Figu e 2 and he odd-mass iso opes
in Figu e 3. Fo he e en-mass iso opes he expe imen al
ene gy spec a a e e y well ep oduced by he shell-model
calcula ions. The accu acy is somewha diminished when
mo ing o lowe masses om he closed neu on majo shell
o 136Xe, bu a decen co espondence be ween expe imen al
and heo e ical le els can be ound. Fo he odd-mass iso-
opes he si ua ion is mo e complex, bu he posi i e-pa i y
s a es a e well ep oduced by he calcula ions. Howe e , he
nega i e-pa i y s a es 11/2−and 9/2−a e compu ed o be
much lowe han in he expe imen al spec um. This e ec
has been obse ed in ea lie calcula ions using he SN100PN
in e ac ion in his mass egion [41]. The expe imen al da a o
he xenon iso opes was ob ained om [36].
Ad ances in High Ene gy Physics 5
EXP SM EXP SM EXP SM EXP SM EXP SM
0.0
0.5
1.0
1.5
2.0
Ene gy (MeV)
128Xe 130Xe 132 Xe 134Xe 136Xe
0+0+0+0+0+0+0+
0+
0+
0+
0+
0+
0+
0+
0+0+0+
2+
2+
2+
2+
2+
2+2+
2+
2+
2+
2+2+
2+
2+
2+
0/2+
2+
2+
2+
2+
2+
2+
4+
4+
4+
4+
4+
4+
4+4+
4+
4+
4+
4+
3+7−
3+
3+
3+
3+
3+
4+
4+
4+
4+
4+6+
6+
6+
6+
6+6+
2+
2+
Figu e 2: Expe imen al and shell-model ene gy spec a o e en-mass xenon iso opes. A maximum o eigh lowes ene gy le els a e shown
o each iso ope. F om le o igh : 128Xe, 130Xe, 132Xe, 134Xe, and 136Xe.
EXP SM EXP SM
0.0
0.1
0.2
0.3
0.4
Ene gy (MeV)
129 Xe 131Xe
5/2+
5/2+
5/2+
5/2+
5/2+
3/2+
3/2+
3/2+
3/2+
3/2+
3/2+
3/2+
3/2+
7/2+
7/2−
11/2−
11/2−
11/2−11/2−
9/2−
9/2−
9/2−
9/2−
1/2+
1/2+
1/2+
1/2+
1/2+
Figu e 3: Expe imen al and shell-model ene gy spec a o odd-
mass xenon iso opes 129Xe (le ) and 131Xe ( igh ).
To gi e a u he measu e o accu acy o ou calcula ion,
we compu ed he g ound s a e magne ic momen s o 129Xe
and 131Xe. Fo 129Xe he expe imen al magne ic momen o
he 1/2+g ound s a e is 𝜇exp =−0.7779763(84)𝜇Nwhile he
shell-model calcula ed alue is 𝜇sm = −1.360𝜇N.Fo 131Xe
3/2+g ound s a e he numbe s a e 𝜇exp = +0.691862(4)𝜇N
and 𝜇sm = +1.059𝜇N o expe imen and shell model,
espec i ely. The sign o he magne ic momen in bo h cases
is co ec , bu he magni ude o bo h o ou calcula ed alues
is somewha la ge han ha o he expe imen al ones.
4. Neu ino Sca e ing Resul s
In his sec ion we p esen he esul s o ou calcula ions
o neu ino-nucleus sca e ing c oss sec ions by me hods
desc ibed in Sec ion 2. We ha e compu ed o al c oss sec ions
o cohe en and inelas ic neu ino-nucleus sca e ing as a
unc ion o he neu ino ene gy and also a e aged o al c oss
sec ions o sola 8B neu inos and supe no a neu inos
sca e ing o he mos abundan xenon iso opes. In he
ollowing calcula ions o a e aged supe no a neu ino c oss
sec ions we ha e used wo di e en neu ino empe a u es
co esponding o di e en neu ino la o s. We ollow he
choices o [26, 37] and ha e he elec on neu inos desc ibed
by pa ame e s 𝛼=3.0,⟨𝐸]⟩ = 11.5MeV, and 𝑇]=
2.88MeV, and he muon and au neu inos by 𝛼=3.0,
⟨𝐸]⟩ = 16.3MeV, and 𝑇]=4.08MeV. Whene e we e e
o supe no a neu inos in he ollowing ex hese pa ame e
alues a e used in he calcula ions.
4.1. Cohe en Elas ic Sca e ing. In Table 3 we p esen he
o al c oss sec ion o cohe en neu ino-nucleus sca e ing
o he a ge xenon iso opes as a unc ion o neu ino ene gy.
In Table 3 we only show calcula ions in he nuclea shell
model, bu he alues o he QRPA/MQPM o malism a e
e y simila , which is e lec ed on he o al a e aged c oss
sec ions shown la e . The c oss sec ions ise apidly o small
neu ino ene gies and s a o sa u a e when app oaching 100
MeV. The c oss sec ions a e la ge o he highe -𝐴iso opes,
ollowing he 𝑁2cohe en enhancemen .
We p esen he o al a e aged c oss sec ion o supe no a
neu inos as well as sola 8B neu inos in Table 4. Resul s
o cohe en sca e ing a e shown o he shell model and
QRPA/MQPM calcula ions. The esul s be ween he shell
model and quasipa icle app oaches a e e y simila . Some
small di e ences can be obse ed in he esul s o he odd-
mass iso opes, bu hose a e s ill no e y signi ican . The
c oss sec ions o he supe no a neu inos a e la ge han
o 8B neu inos by oughly a ac o o 3 o 5 depending on
he neu ino la o . This is due o he a e age ene gy o he
supe no a neu inos being la ge a 11.5 MeV o 16.3 MeV,
while he 8Bspec umpeaksa a ound7MeV.
6Ad ances in High Ene gy Physics
Table 3: Cohe en elas ic neu al-cu en sca e ing c oss sec ion o neu inos sca e ing o xenon a ge s as a unc ion o neu ino ene gy.
The c oss sec ions o each iso ope a e gi en in uni s o cm2in columns 2-8 as a unc ion o he neu ino ene gy (column 1). The compu a ions
we e made in he nuclea shell model.
𝐸]𝜎(cm2)
(MeV)128Xe 129Xe 130Xe 131Xe 132Xe 134Xe 136Xe
55.16×10−40 5.31×10−40 5.46×10−40 5.61×10−40 5.76×10−40 6.08×10−40 6.40×10−40
10 2.02×10−39 2.08×10−39 2.14×10−39 2.20×10−39 2.26×10−39 2.38×10−39 2.50×10−39
20 7.44×10−39 7.65×10−39 7.86×10−39 8.07×10−39 8.29×10−39 8.73×10−39 9.19×10−39
30 1.47×10−38 1.51×10−38 1.55×10−38 1.59×10−38 1.63×10−38 1.71×10−38 1.80×10−38
40 2.19×10−38 2.25×10−38 2.31×10−38 2.37×10−38 2.43×10−38 2.55×10−38 2.67×10−38
50 2.80×10−38 2.88×10−38 2.94×10−38 3.02×10−38 3.09×10−38 3.24×10−38 3.39×10−38
60 3.25×10−38 3.33×10−38 3.40×10−38 3.49×10−38 3.57×10−38 3.73×10−38 3.91×10−38
70 3.55×10−38 3.64×10−38 3.72×10−38 3.81×10−38 3.89×10−38 4.07×10−38 4.25×10−38
80 3.75×10−38 3.84×10−38 3.92×10−38 4.02×10−38 4.10×10−38 4.28×10−38 4.48×10−38
Table 4: To al a e aged c oss sec ion o 8B sola neu inos and elec on and muon/ au supe no a neu inos (SN]e/SN]x)sca e ingo
xenon a ge s. The esul s a e shown o calcula ions in he nuclea shell model (SM) and he QRPA/MQPM o malisms. C oss sec ions o
cohe en sca e ing a e gi en in uni s o 10−39 cm2and o inelas ic sca e ing in 10−43 cm2.
⟨𝜎⟩coh,8B⟨𝜎⟩coh,SN]e⟨𝜎⟩coh,SN]x⟨𝜎⟩inel,8B⟨𝜎⟩inel,SN]e⟨𝜎⟩inel,SN]x
Nucleus Model (10−39 cm2)(10
−39 cm2)(10
−39 cm2)(10
−43 cm2)(10
−43 cm2)(10
−43 cm2)
128Xe SM 1.064 3.051 5.692 - - -
QRPA 1.065 3.052 5.696 1.567 38.10 152.0
129XeSM1.0953.1385.853---
MQPM 1.105 3.166 5.903 2.208 45.11 173.4
130Xe SM 1.125 3.223 6.008 - - -
QRPA 1.126 3.225 6.013 1.564 40.94 161.0
131Xe SM 1.157 3.313 6.173 - - -
MQPM 1.167 3.336 6.215 3.699 54.14 195.4
132Xe SM 1.188 3.401 6.335 - - -
QRPA 1.189 3.403 6.339 2.341 48.21 180.4
134Xe SM 1.253 3.585 6.671 - - -
QRPA 1.253 3.585 6.673 3.107 56.10 201.7
136XeSM1.3203.7737.016---
QRPA 1.320 3.773 7.016 2.102 53.43 200.5
4.2. Inelas ic Sca e ing. Due o he limi a ions o he shell
model in desc ibing high-lying exci ed s a es, we compu e
inelas ic sca e ing p ope ies using only he QRPA/MQPM
o malism, which is known o depic well he collec i e
p ope ies o exci ed nuclea s a es. The o al c oss sec ion as a
unc ion o neu ino ene gy is gi en in Table 5 o each xenon
iso ope. Fo smalle neu ino ene gies, 0 o 30 MeV, he c oss
sec ions a e sligh ly la ge o he odd-mass iso opes han o
hei neighbo ing iso opes. The ene gies o sola neu inos
i comple ely in o his ange, which leads o he a e aged
c oss sec ions o sola neu inos o be la ge o he odd-mass
iso opes.
The o al a e aged inelas ic c oss sec ions a e lis ed in
Table4.Theinelas icsca e ingc osssec ionsa esomeo de s
o magni ude smalle han he cohe en c oss sec ions, as
expec ed. He e he c oss sec ions o he supe no a neu inos
a e an o de o magni ude o wo la ge han o 8Bsola
neu inos, again due o he supe no a neu inos ha ing
on a e age a highe ene gy. The e ec o neu ino ene gy
appea s mo e p onounced in inelas ic sca e ing han in
cohe en sca e ing, howe e . The c oss sec ions o he odd-
mass iso opes a e again sligh ly la ge han hose o he
neighbo ing iso opes.
We can compa e ou inelas ic sca e ing esul s wi h hose
calcula ed in [26] o Cd iso opes using he same supe no a
neu ino pa ame e s. The esul s o Cd iso opes in [26] in
hecaseo elec onneu ino ange om4.38×10−42 cm2 o
106Cd o 4.96×10−42 cm2 o 111Cd, wi h a gene al dec easing
end wi h inc easing mass numbe o e en-mass nuclei.
Ou esul s o Xeiso opesinTable4a e e ysimila in
magni ude, bu he end is a he ising han dec easing wi h
inc easing mass numbe . This could be a shell e ec , as adding
neu ons o Cd iso opes akes he nucleus u he away om
a closed majo shell, bu o he xenon nuclei i ge s close o
a shell closu e. Same conclusions can be made o he o he
neu ino la o s.
We show he con ibu ions om di e en mul ipole
channels o he o al a e aged c oss sec ions in Figu e 4
Ad ances in High Ene gy Physics 7
Table 5: Inelas ic neu al-cu en sca e ing c oss sec ion o neu inos sca e ing o xenon a ge s as a unc ion o neu ino ene gy. The
c oss sec ions a e gi en in uni s o cm2. The compu a ions we e made in he QRPA/MQPM o malism.
𝐸]𝜎(cm2)
(MeV)128Xe 129Xe 130Xe 131Xe 132Xe 134Xe 136Xe
51.71×10−45 2.10×10−45 1.29×10−46 2.74×10−44 7.74×10−45 1.28×10−44 2.00×10−47
10 3.56×10−43 5.27×10−43 3.54×10−43 8.49×10−43 5.49×10−43 7.57×10−43 4.75×10−43
20 1.41×10−41 1.71×10−41 1.53×10−41 2.00×10−41 1.78×10−41 2.06×10−41 2.02×10−41
30 6.50×10−41 7.45×10−41 6.85×10−41 8.18×10−41 7.53×10−41 8.29×10−41 8.41×10−41
40 1.85×10−40 1.94×10−40 1.91×10−40 2.05×10−40 2.02×10−40 2.16×10−40 2.20×10−40
50 3.99×10−40 3.85×10−40 4.05×10−40 3.95×10−40 4.20×10−40 4.38×10−40 4.47×10−40
60 7.17×10−40 6.41×10−40 7.20×10−40 6.45×10−40 7.35×10−40 7.55×10−40 7.66×10−40
70 1.14×10−39 9.51×10−40 1.14×10−39 9.44×10−40 1.15×10−39 1.16×10−39 1.18×10−39
80 1.66×10−39 1.30×10−39 1.64×10−39 1.28×10−39 1.65×10−39 1.66×10−39 1.67×10−39
0−0+1−1+2−2+3−3+4−4+
0−0+1−1+2−2+3−3+4−4+0−0+1−1+2−2+3−3+4−4+
0−0+1−1+2−2+3−3+4−4+
0.0
0.5
1.0
1.5
2.0
2.5
Vec o
Axial- ec o
In e e ence
128 Xe
0.0
0.5
1.0
1.5
2.0
Vec o
Axial- ec o
In e e ence
129 Xe
0.0
1.0
2.0
3.0
4.0
Vec o
Axial- ec o
In e e ence
134 Xe
0.0
1.0
2.0
3.0
Vec o
Axial- ec o
In e e ence
131 Xe
⟨⟩(10−42 cＧ2)
⟨⟩(10−42 cＧ2)⟨⟩(10−42 cＧ2)
⟨⟩(10−42 cＧ2)
Figu e 4: The con ibu ions o mul ipole channels 𝐽≤4 o he o al a e aged c oss sec ion o inelas ic sca e ing o supe no a elec on
neu inos. Ba plo s a e shown o a ep esen a i e sample o 128Xe ( op le ), 129Xe ( op igh ), 134Xe (bo om le ), and 131Xe (bo om igh ).
A di ision o ec o , axial- ec o , and in e e ence pa s o he in e ac ion is shown. C oss sec ions a e gi en in uni s o 10−42 cm2.
o supe no a elec on neu inos and Figu e 5 o sola
neu inos. I is e iden ha he mos dominan con ibu ion
comes om an axial- ec o 1+mul ipole ansi ion in all
cases bu one. Smalle , ye s ill impo an con ibu ions
a ise om he axial- ec o 1−and 2−channels o highe
neu ino ene gies. This is cha ac e is ic beha io o neu al-
cu en sca e ing, which has been obse ed in [26] o Cd
iso opes and in [27] o Mo iso opes. The con ibu ions ge
mo e e enly dis ibu ed among he di e en mul ipoles wi h
inc easing neu ino ene gy.
8Ad ances in High Ene gy Physics
0−0+1−1+2−2+3−3+4−4+
0−0+1−1+2−2+3−3+4−4+
0−0+1−1+2−2+3−3+4−4+0−0+1−1+2−2+3−3+4−4+
134 Xe 131 Xe
0.00
0.05
0.10
0.15
Vec o
Axial- ec o
In e e ence
128Xe
0.00
0.05
0.10
0.15
Vec o
Axial- ec o
In e e ence
129Xe
0.00
0.05
0.10
0.15
0.20
0.25
Vec o
Axial- ec o
In e e ence
0.00
0.05
0.10
0.15
0.20
0.25
Vec o
Axial- ec o
In e e ence
⟨⟩(10−42 cＧ2)
⟨⟩(10−42 cＧ2)⟨⟩(10−42 cＧ2)
⟨⟩(10−42 cＧ2)
Figu e 5: The con ibu ions o mul ipole channels 𝐽≤4 o he o al a e aged c oss sec ion o inelas ic sca e ing o sola 8Bneu inos.Ba
plo s a e shown o a ep esen a i e sample o 128Xe ( op le ), 129Xe ( op igh ), 134Xe (bo om le ), and 131Xe (bo om igh ). A di ision o
ec o , axial- ec o , and in e e ence pa s o he in e ac ion is shown. C oss sec ions a e gi en in uni s o 10−42 cm2.
Fo he odd-mass nuclei ou calcula ions also show a sig-
ni ican con ibu ion om a ec o 0+channel, and o sola
neu inos sca e ing o 129Xe his channel in ac becomes he
s onges . Fo he e en-mass iso opes his channel is mo e
supp essed, bu i becomes mo e signi ican o he lowe
ene gy sola neu inos. Simila la ge 0+con ibu ions we e
obse ed in [26] o Cd iso opes. This is p oblema ic as, in
p inciple, he 0+con ibu ion is expec ed o be small because
i anishes a he limi 𝑞→0. The pa icle-numbe iola ion
o he quasipa icle amewo k can be an explana ion o he
la ge compu ed 0+con ibu ion. A de ailed examina ion on
he o igins o he 0+anomaly will be conduc ed in a la e
s udy. A his ime one should ega d he 0+con ibu ions
wi h cau ion as hey a e p obably a leas pa ially spu ious.
In Figu es 6 and 7 we show he domina ing con ibu-
ions o he inelas ic sca e ing c oss sec ion om a ious
inal s a es o 128Xe and 131Xe, espec i ely. We no ice ha
he majo con ibu ions a e e y simila o he sola and
supe no a elec on neu inos o he e en-mass 128Xe, whe e
he leading con ibu ions come om 1+s a es a 8.4 MeV,
5.0 MeV, and 6.7 MeV. Fo sola neu inos he e is also a
no able con ibu ion om a 0+s a e a 2.4 MeV. The si ua ion
is e y much di e en o he odd-mass 131Xe, whe e o
supe no a neu inos he e is a pile-up o 5/2+,3/2+,and
1/2+s a es a oughly 8 MeV gi ing la ge con ibu ions o
he o al c oss sec ion in addi ion o he la ge con ibu ions
om lowe -lying 5/2+and 3/2+s a es. Howe e , o sola 8B
neu inos his peak a 8 MeV is much smalle , and he leading
con ibu ions a e mo e localized o he 5/2+s a e a 1.8 MeV
and he 3/2+s a e a 2.9 MeV. I is in e es ing ha a ela i ely
small change in he a e age neu ino ene gy can lead o he
highe -lying s a es o gi e much la ge con ibu ions o he
o al c oss sec ion.
Following he discussion on he anomalously la ge 0+
mul ipole con ibu ion in 129Xe we show he dominan inal
s a es o neu inos sca e ing o 129Xe in Figu e 8. As
expec ed om he la ge 0+mul ipole, he la ges con ibu-
ions he e come om 1/2+s a es a ene gies o oughly 2−3
MeV. Some hing in he nuclea -s uc u e calcula ion seems
o a o he 0+mul ipole ansi ion o 1/2+ inal s a es o e
he 1+mul ipole ansi ion o 3/2+s a es. O he wise simila
conclusions can be made o 129Xe as o 131Xe abo e abou