Physics Le e s B 843 (2023) 138029
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5D Elko spino field non-minimally coupled o nonme ici y in (Q)
g a i y
F.M. Belchio a, A.R.P. Mo ei a a, R.V. Malu a,b,∗, C.A.S. Almeida a
aUni e sidade Fede al do Cea á (UFC), Depa amen o de Física, Campus do Pici, Fo aleza -CE, C.P. 6030, 60455-760 - B azil
bDepa amen o de Física Teó ica and IFIC, Cen o Mix o Uni e sidad de Valencia -CSIC. Uni e sidad de Valencia, Bu jasso -46100, Valencia, Spain
a i c l e i n o a b s a c
A icle his o y:
Recei ed 14 Feb ua y 2023
Recei ed in e ised o m 23 May 2023
Accep ed 13 June 2023
A ailable online 16 June 2023
Edi o : A. Ringwald
Keywo ds:
Elko field
Da k ma e
Thick b ane
Symme ic elepa allel g a i y
This pape aims o in es iga e he localiza ion o he fi e-dimensional spino field known as Elko
(dual-helici y eigenspino s o he cha ge conjuga ion ope a o ) by employing a Yukawa-like geome ical
coupling in which he Elko field is non-minimally coupled o nonme ici y scala Q. We adop he
b anewo ld scena ios in which he fi s -o de o malism wi h sine-Go don and linea supe po en ials
is employed o ob ain he wa p ac o s. A linea unc ion suppo s he ze o-mode apping wi hin
he geome ic coupling, leading o he same e ec i e po en ial as he scala field. Mo eo e , an exo ic
e m mus be added o ob ain eal- alued massi e modes. Such modes a e in es iga ed h ough he
Sch ödinge -like app oach.
©2023 The Au ho (s). Published by Else ie B.V. This is an open access a icle unde he CC BY license
(h p://c ea i ecommons .o g /licenses /by /4 .0/). Funded by SCOAP3.
1. In oduc ion
The e is a consensus ha gene al ela i i y (GR) needs o be
modified o explain physics a bo h quan um and galac ic scales.
In his sense, modified g a i y heo ies ha e gained conside able
in e es in he li e a u e o e he pas yea s. Such heo ies can be
cons uc ed by adding geome ical in a ian , such as a gene alized
unc ion o cu a u e scala R[1–3], ene gy-momen um enso T
[4–6], Gauss-Bonne e m G[7–9], among o he s, in o Eins ein-
Hilbe ac ion. In pa icula , (R)g a i y has been in es iga ed as
a possible mechanism o explain he ecen ly obse ed cosmic ac-
cele a ion and could desc ibe da k ma e [10–13].
I is s ill possible o cons uc a modified g a i y whe ein he
g a i a ional dynamics is encoded by o sion scala T a he han
cu a u e. This heo y is known as elepa allel equi alen o gen-
e al ela i i y (TEGR) [14–16]. Like he GR, i can be di ec ly ex-
ended by assuming a gene al unc ion o o sion scala , esul -
ing in (T)g a i y, which has been in ensi ely wo ked in he
li e a u e [17–23]. Ano he cu a u e- ee modified g a i y is he
symme ic elepa allel equi alen o gene al ela i i y (STEGR) [24],
whose g a i a ional ac ion depends on nonme ici y scala Q. Such
a g a i y and i s di ec ex ension (Q)g a i y ha e gained high-
*Co esponding au ho .
E-mail add esses: belchio @fisica.u c.b (F.M. Belchio ), allan.mo ei a@fisica.u c.b
(A.R.P. Mo ei a), . .malu @fisica.u c.b (R.V. Malu ), ca los@fisica.u c.b
(C.A.S. Almeida).
ligh in se e al con ex s, such as wo mhole, black hole, da k ma -
e , and cosmology [25–31].
On he o he hand, se e al kinds o opological de ec s can build
a hick b ane model, whe e ou Uni e se is a memb ane embed-
ded in a wa ped fi e-dimensional space ime [32–44]. This wo k
aims o in es iga e he Elko field localiza ion in b anewo lds mod-
els assuming a (Q)g a i y [45,46]. The Elko field is a spino o
hal spin wi h mass dimension one ( ou -dimensional space ime),
which is he eigenspino o he cha ge conjuga ion [47–52]. This
field has been employed as he fi s e mionic field o desc ibe
da k ma e , besides being ega ded as a da k field since i does
no in e ac wi h he elec omagne ic field. In his sense, only he
g a i on and Higgs field can in e ac wi h he Elko field. In Re s.
[53,54] he Casimi ene gy was analyzed o he Elko field, con-
fi ming i s e mionic na u e heo e ically wi h a epulsi e Casimi
o ce. Recen ly, i has been p oposed some a emp s o de ec he
Elko field a he La ge Had on Collide (LHC) [55,56].
The confinemen o he Elko spino in Randall-Sund um scena -
ios was add essed o he fi s ime in [57], showing ha he Elko
field, like o he fields, equi es a sui able coupling o be apped
on a fi e-dimensional b anewo ld. In his wo k, Yukawa-like cou-
pling be ween he Elko spino and he backg ound scala field was
p oposed. O he wo k we e p oposed by conside ing Yukawa-like
[58,59] and dila on-like coupling [60,61]. In addi ion, a geome ic
coupling wi h he Ricci scala was conside ed in Re . [62]. Howe e ,
he e is a common issue in hese wo ks: a complex alue po en-
ial, which makes s udying massi e and esonan modes difficul .
h ps://doi.o g/10.1016/j.physle b.2023.138029
0370-2693/©2023 The Au ho (s). Published by Else ie B.V. This is an open access a icle unde he CC BY license (h p://c ea i ecommons .o g /licenses /by /4 .0/). Funded by
SCOAP3.
F.M. Belchio , A.R.P. Mo ei a, R.V. Malu e al. Physics Le e s B 843 (2023) 138029
In he con ex o s ing-like b ane, such an issue can be o e come
by adding an exo ic e m, as shown in Re . [63]. Ou in es iga ion
will conside a geome ical coupling be ween he Elko field and
he nonme ici y scala h ough a Yukawa-like in e ac ion in 5D
hick b anewo ld. This coupling leads o no malizable ze o-mode
besides p o iding eal- alued massi e modes [46].
This pape is o ganized as ollows: In sec ion 2, he main
concep s o (Q)g a i y a e b iefly discussed. Nex , he fi e-
dimensional hick b ane is s udied in sec ion 3, and he fi s -o de
o malism is in oduced o ob ain analy ical solu ions. In sec ion
4, a Yukawa-like coupling is p oposed o s udy he localiza ion o
Elko field ze o-mode as well as i s esona es massi e modes. The
conclusion o his p esen wo k and u u e pe spec i es a e dis-
cussed in sec ion 5.
2. (Q)g a i y
A b ie e iew o symme ic elepa allel g a i y as well as he
equa ions o mo ion o (Q)g a i y, is p esen ed in his sec ion.
Le us ini ially dis inguish essen ial concep s o GR and STEGR. An
impo an ea u e o Riemannian geome y is he me ici y condi-
ion gi en by
∇MgNP =0,(1)
whe e gNP is he me ic and ∇Mis he co a ian de i a i e wi h
he Le i-Ci i a PMN as affine connec ion. Such condi ion is
sha ed by GR and i s di ec modifica ion as (R)g a i y. We
deno e he bulk coo dina e indices by capi al La in index M=
0, ..., D −1.
On he o he hand, since STEGR is based on non iemannian
geome y, he ela ion (1)is no longe sa isfied, leading o non-
anishing nonme ici y enso [24]
QMNP =∇MgNP,(2)
which has he ollowing independen aces
QM=gNP QMNP,(3)

QM=gNPQNMP.(4)
The mo e gene al connec ion

PMN o STEGR is hen defined as

PMN =PMN +LPMN,(5)
whe e LPMN is defined as he dis o ion enso , which is w i en
in nonme ici y enso e ms as [24]
LPMN =1
2gPQ(QPMN −QMPN −QNPM). (6)
A his poin , i is con enien o in oduce a mo e gene al enso
ha con ains he nonme ici y, i s independen aces, and dis o -
ion enso . Such enso is known as nonme ici y conjuga e gi en
by
PPMN =−1
2LPMN +1
4(QP−
QP)gMN −1
8(δP
MQN+δP
NQM).
(7)
Besides, i s con ac ion wi h nonme ici y enso p o ides he non-
me ici y scala Q=QPMNPPMN. The Ricci scala is w i en as
R =Q+B, whe e Bis a bounda y e m. Such esul shows ha
STEGR is equi alen o GR since he bounda y e m anishes when
in eg a ed in he ac ion.
The (Q)g a i y ep esen s a di ec ex ension o STEGR. The
fi e-dimensional g a i a ional ac ion o his g a i y is assumed as
[45,46]
S=d5x√−g1
2 (Q)+Lm,(8)
whe e Lm ep esen s he ma e Lag angian o be defined in he
nex sec ion. The a ia ion o ac ion (8)wi h espec o he me ic
gi es he ollowing equa ion
2
√−g∇K(√−g QPKMN)−1
2gMN
+ Q(PMKLQNKL −2QLKMPKNL)=TMN,(9)
whe e TMN is he ene gy-momen um enso . Fu he mo e, i we
a y he ac ion (8)wi h espec o he connec ion, one ge s
∇M∇N(√−g QPKMN)=0.(10)
He e, we se ≡ (Q)and Q≡∂ (Q)/∂ Q o simplici y.
3. Thick b ane scena ios and fi s -o de o malism
This sec ion is dedica ed o cons uc ing he b anewo ld scena -
ios in (Q)g a i y. Le us conside a single scala field as a ma e
sou ce ha gene a es he hick b ane. Thus, he ma e Lag angian
is gi en by
Lm=−1
2∂Mφ∂Mφ−V(φ). (11)
The ene gy-momen um enso associa ed o his Lag angian eads
TMN =∂Mφ∂Nφ+gMNLm.(12)
Le us now use he ansa z o a gene ic Randall-Sund um-like
me ic as [32,33]
ds2=e2Aημνdxμdxν+dy2,(13)
whe e ημν is he Minkowski me ic, e2Ais he wa p ac o , and
y ep esen he ex a dimension. He e, he coinciden gauge, i.e.,

PMN =0is also conside ed. Then, o his me ic he nonme ic-
i y scala is Q=12A2, while he scala field and g a i a ional
equa ions ead
φ +4Aφ=Vφ,(14)
A 
Q+ QA =−φ2
3,(15)
12 QA2−
2=φ2
2−V.(16)
In he con ex o b anewo lds and opological s uc u es, ana-
ly ical solu ions o equa ions o mo ion can be ob ained by em-
ploying he so-called fi s -o de o malism [34–36,39,42], which is
in oduced h ough he ollowing assump ion
A=−αW(φ), (17)
whe e W(φ) is he supe po en ial.
We assume he unc ion (Q)as being (Q) =Q+kQn,
which ep esen s a good gene aliza ion o STEGR, whe e he pa-
ame e s kand n ep esen he de ia ion om he usual heo y.
F om field equa ions (15) and (16), we ob ain
φ=3α[1+nCn(αW)2n−2]Wφ,(18)
V(φ) =9α2
21+nCn(αW)2n−22W2
φ
−6[1+Cn(αW)2n−2](αW)2,(19)
whe e Cn=12n−1k(2n −1). One w i es he ene gy densi y ρ(y) =
−e2ALmin e ms o he supe po en ial as
2
F.M. Belchio , A.R.P. Mo ei a, R.V. Malu e al. Physics Le e s B 843 (2023) 138029
ρ(y)=e2A(y)(9α2[1+nCn(αW)2n−2]2W2
φ
−6[1+Cn(αW)2n−2](αW)2). (20)
Thus, one can comple ely de e mine he hick b ane sys em
wi h a specific supe po en ial choice. In he sequel, we choose
wo kinds o supe po en ial o ob ain he analy ical exp ession o
scala field solu ion, wa p ac o , po en ial, and ene gy densi y. The
fi s supe po en ial is he sine-Go don one, o which we conside
n =1. The second supe po en ial is linea one o n =2.
3.1. Sine-Go don supe po en ial
Ou fi s example is he sine-Go don, whose supe po en ial is
gi en by
W(φ) =β2sinφ
β.(21)
He e, we make n =1. The scala field solu ion and he po en ial
ead
φ(y)=βa csin{ anh[3αy(1+k)]},(22)
V(φ) =3
2(1+k)α2β23(1+k)cos2φ
β−4β2sin2φ
β.(23)
Subs i u ing he solu ion (22)in o (17), we ob ain he ollowing
exp ession o he wa p ac o
A(y)=β2
3(1+k)lnsech[3α(1+k)y].(24)
We can finally w i e he ene gy densi y as being
ρ(y)=3(1+k)α2β2[cosh(3αy(1+k))]−2β2
3(1+k)
×3(1+k)−(3(1+k)+2β2) anh2(3αy(1+k)).
(25)
In Fig. 1, i is plo ed he beha io o he scala field solu ion
φ, po en ial V(φ), he wa p ac o e2Aand he ene gy densi y ρ.
As we can see, he scala field exhibi s a kink-like beha io , as
expec ed. Besides, he po en ial has an oscilla ing beha io , and he
ene gy densi y eels he a ia ion o he pa ame e k, ending o
become mo e localized as he pa ame e is inc eased.
3.2. Linea supe po en ial
We can conside linea example wi h he ollowing supe po en-
ial
W(φ) =βφ. (26)
In his case, we ake n =2so ha he scala field o linea supe -
po en ial is
φ(y)= an(18√2kα2β2y)
6√2kαβ,(27)
and he po en ial
V(φ) =9
2α2β21+72k(αβφ)22
−6(αβφ)2[1+36k(αβφ)2].
(28)
Besides, he wa p ac o is
A(y)=ln[cos(18√2kα2β2y)]
216kα2β2,(29)
which b ings us o an ene gy densi y o he o m
ρ(y)=cos(18√2kα2β2y)1
108kα2β2
×9α2β2(1+ an2(18√2kα2β2y)2
−1
24k an2(18√2kα2β2y)2+ an2(18√2kα2β2y).
(30)
F om Fig. 2, we see ha he scala field solu ion o polynomial
supe po en ial is also kink-like. Fu he mo e, he influence o he
pa ame e kon he solu ion o he scala field φand on he po en-
ial V(φ) is qui e e iden , di ec ly a ec ing he ene gy densi y. I is
in e es ing o no e ha he ene gy densi y becomes less localized
as we choose a smalle alue o k.
Assuming he hick b ane cons uc ed in (Q)g a i y, he s a-
bili y unde small enso pe uba ions and g a i y localiza ion was
add essed in [45], while he localiza ion o Di ac e mions wi h ge-
ome ical coupling was ea ed in [46]. Ahead we p oceed o s udy
he apping o he 5D Elko field.
4. Elko field localiza ion
Due o i s p ope ies, he Elko spino could be a na u al candi-
da e o da k ma e and dese es an in es iga ion abou i s local-
iza ion on b anewo ld.
4.1. 4D Elko spino and da k ma e
Fi s ly, we in oduce some ea u es o Elko spino in ou -
dimensional space- ime. The Elko in e ac s weakly wi h ma e
fields and adia ion as a mass dimension one spino . Besides, he e
is obse a ional e idence ha sugges s ha da k ma e is sel -
in e ac ing. The Elko in e ac ion is es ic ed o g a i on and Higgs
field. The e m o sel -in e ac ion o a 4D Elko field can be accom-
plished by [51]
gλ[λ(x)λ(x)]2,(31)
whe e gλ ep esen s he dimensionless coupling cons an and λ(x)
is he Elko quan um field
λ(x)=d4k
(2π)4
1
√2ωk
βaβ(k)λA
β(k)e−ikx +a†
β(k)λA
β(k)eikx.
(32)
The spin one-hal eingenspino λS/A
β(k)mus sa is y he ela ion
CλS/A
β(k) =±λS/A
β(k), whe e Cis he cha ge conjuga ion ope a o ,
λSsel conjuga e (posi i e), λAan i-sel conjuga e (nega i e), and
β=({+, −}, {−, +}) ep esen s he helici y. Also, his spino and
i s dual sa is y he ollowing o hono mali y ela ions
λS/A
β(k)λS/A
β(k)=±2mδββ ,
λS
β(k)λA
β(k)=λA
β(k)λS
β(k)=0.(33)
Since he Elko field is e mionic one, he c ea ion and annihila ion
ope a o s mus sa is y an i-commu a ion ela ions w i en below
{aβ(k‘),aβ(k)}={a†
β(k‘),a†
β(k)}=0,
{a†
β(k‘),aβ(k)}=δ(3)(k‘−k)δββ .(34)
3
F.M. Belchio , A.R.P. Mo ei a, R.V. Malu e al. Physics Le e s B 843 (2023) 138029
Fig. 1. Fo he sine-Go don supe po en ial wi h α=β=1. (a) Scala field. (b) Po en ial. (c) Wa p ac o . (d) Ene gy densi y.
Fig. 2. Fo he linea supe po en ial wi h α=β=1. (a) Scala field. (b) Po en ial. (c) Wa p ac o . (d) Ene gy densi y.
4
F.M. Belchio , A.R.P. Mo ei a, R.V. Malu e al. Physics Le e s B 843 (2023) 138029
Fig. 3. The shape o he po en ial and o he ze o-mode wi h α=β=1. (a) and (b) sine-Go don supe po en ial. (c) and (d) linea supe po en ial.
As we said be o e, he 4D Elko field can also be coupled o
Higgs h ough he Yukawa-like e m
gφλ[φ†(x)φ(x)][λ(x)λ(x)],(35)
whe e gφλ ep esen s he dimensionless coupling cons an be-
ween Elko and Higgs. Fo a mo e de ailed e iew o he Elko field,
we sugges o he eade he Re s. [47,48,50,51].
4.2. 5D Elko field localiza ion wi h Yukawa-like coupling
Now, ou in e es is o e i y whe he he Elko field is apped
on he hick b ane, which was buil in he p e ious sec ion. Fo
his pu pose, a nonminimal coupling be ween he Elko field and
nonme ici y scala is assumed. This is accomplished by a Yukawa-
like in e ac ion (simila o (35)). Fi s ly, he con o mal coo dina e
dz =e−Ady is in oduced, so ha he me ic (13)is now w i en
as ollows
ds2=e2A(ημνdxμdxν+dz2). (36)
In he fi e-dimensional space ime, he ac ion o he Elko field
wi h a Yukawa-like coupling is gi en by [57,62]
S=d5x√−g1
4(DMλDMλ+DMλDMλ) +G(Q)λλ,(37)
whe e G(Q)is a sui able unc ion o nonme ici y scala employed
o ob ain a no malizable mode o he Elko field. F om ac ion (37),
we obse e ha he 5D Elko spino has mass dimension 3/2 unlike
he 4D Elko, which has mass dimension one. In addi ion, we define
he co a ian de i a i e in symme ic elepa allel g a i y as DM=
∂M+M, whe e Mis he o sion- ee spin connec ion defined by
M=1
4ωab
Mγaγb,(38)
whe e γa ep esen s he Di ac ma ix in fla space ime. To ob ain
ωab
M, we use he Ca an equa ion dθa+ωa
bθb=0wi h he o sion-
ee condi ion. Recall ha θa=ha
MdMxand ωa
b=ωa
bMdMx, whe e
ha
Mis he ielbein ha sa is y he ela ion gMN =ηabha
Mhb
N. We
should poin ou ha he spin connec ion is p ese ed since we
deal wi h anishing o sion. Thus, o he me ic (13), we ha e he
only non anishing connec ion μ=1
2˙
Aγμγ4. He e, he do s ands
o he de i a i e wi h espec o he con o mal coo dina e, i.e.,
d/dz.
The co esponding equa ion o mo ion ob ained om he ac ion
(37)is
DM(√−gDMλ) −2√−gG(Q)λ =0.(39)
I can be shown ha using he me ic (36)and spin connec ion
(38), he abo e equa ion becomes
λ−˙
Aγ4γμ∂μ−˙
A2λ+∂2
4λ+3˙
A∂4λ−2e2AGλ=0,(40)
whe e  =ημν ∂μ∂ν.
Simila o he solu ion (32), i is possible o adop a Kaluza-
Klein decomposi ion o he Elko field as
λ±(xμ,z)=
n,β
χn(z)[λA
n,β (x)+λS
n,β (x)],(41)
implying he ollowing equa ion o χ(z)
¨
χ+3˙
A˙
χ−˙
A2+im ˙
A+2e2AGχ=−m2χ.(42)
To ob ain (42), we assume ha he 4D Elko spino sa isfies he
ollowing ela ions
γμ∂μλA
n,±(x)=∓iλA
n,∓(x), γμ∂μλS
n,±(x)=±iλS
n,±(x), (43)
γ4λA
n,±(x)=±λS
n,∓(x), γ4λS
n,±(x)=∓λA
n,∓(x), (44)
λA
n,±(x)=m2λA
n,±(x), λS
n,±(x)=m2λS
n,±(x), (45)
5

F.M. Belchio , A.R.P. Mo ei a, R.V. Malu e al. Physics Le e s B 843 (2023) 138029
Fig. 4. Theshapeo he ela i ep obabili yando hemassi emodeswi hα=β=1 o sine-Go don supe po en ial. (a) k=1. (b) m2=3.235. (c) m2=2.293.
whe e m ep esen s he 4D Elko spino mass.
The nex s ep is o ans o m he equa ion (42)in o he
Sch ödinge -like o m. Fo his pu pose, we mus make he ol-
lowing change χ(z) =e−3
2A(z)ψ(z) ha implies in
−¨
ψ+Vψ=m2ψ, (46)
whe e he e ec i e po en ial eads
V=3
2¨
A+13
4˙
A2+im ˙
A+2e2AG.(47)
I is con enien o ake he unc ion Gas G(Q) =cQ o in es-
iga e ze o-mode localiza ion (m =0). Then he po en ial (47)is
ew i en as
V=3
2¨
A+13
4+24c˙
A2+im ˙
A.(48)
He e, cis a pa ame e o be chosen in a way ha allows he local-
iza ion o he Elko field ze o-mode on he b ane.
4.2.1. Ze o-mode
To ob ain a ac o ized po en ial, we mus ha e c=−1
24 , so ha
(48) o m =0 educes o
V=3
2¨
A+9
4˙
A2.(49)
I is wo h men ioning ha he abo e po en ial is he same as
he scala field. Thus, he ze o-mode akes he simple o m
ψ0(z)=N0e3
2A(z).(50)
In Fig. 3, we plo he beha io o he e ec i e po en ial and he
ze o-mode o he sine-Go don ype supe po en ial (Fig. 3a and b),
and o he linea supe po en ial (Fig. 3c and d). In bo h cases,
when we inc ease he alue o he kpa ame e , he po en ial well
in ensifies, making he ze o-modes mo e localized.
4.2.2. Massi e modes
I would be conside ably ha d o s udy he massi e spec um
since he e ec i e po en ial (48)is complex. Inspi ed by 6D Elko
on s ing-like b ane [63], we add he exo ic e m −ime−A
4√3√Q o
he unc ion G(Q)in o de o emo e he imagina y pa o he
e ec i e po en ial. Then, o massi e modes, he e ec i e po en ial
also assumes he o m (49). As his e ec i e po en ial is ac o ized,
we can undoub edly say ha he massi e spec um has no achy-
onic modes.
No ice ha Eq. (46)can only be sol ed nume ically, e en
elimina ing he imagina y pa . To do his, we use he in e po-
la ion me hod and assume he bounda y condi ions: ψe en(0) =
1, ˙
ψe en(0) =0 o e en modes and ψodd(0) =0, ˙
ψodd(0) =1 o
odd modes [66]. We choose he bounda y condi ions on accoun
o he beha io o he e ec i e po en ials V(z)(Fig. 3a and c)
which a e o e en unc ions, ensu ing ha he solu ions will be
wa e unc ions e en ψe en o odd ψodd. As expec ed, we see om
(Fig. 4b and c) and (Fig. 5b and c) ha he massi e eigen unc ions
ha e di e en beha io s nea he o igin bu all exhibi pe iodic be-
ha io a om he o igin.
Fu he mo e, we can analyze he esonan modes, which a e he
massi e modes ha exhibi a espec i ely la ge ampli ude nea he
b ane [48,64]. To iden i y he esonance modes, i is necessa y o
calcula e he ela i e p obabili y P(m)o finding a pa icle wi h a
espec i e mass min a na ow band 2zb[22,65,66]
P(m)=zb
−zb|ψ(z)|2dz
zmax
−zmax |ψ(z)|2dz.(51)
He e, zmax ep esen s he limi o he domain. I is in e es ing o
no e ha he choice o pa ame e zbdoes no change he posi ions
o he esonance peaks. Howe e , wi h he smalle alue o he zb
pa ame e , i becomes easie o iden i y he esonance peaks.
A his poin , we mus highligh ha he beha io o he po-
en ial shown in Fig. 3would commonly no suppo esonances.
6
F.M. Belchio , A.R.P. Mo ei a, R.V. Malu e al. Physics Le e s B 843 (2023) 138029
Fig. 5. Theshapeo he ela i ep obabili yando hemassi emodeswi hα=β=1 o linea supe po en ial. (a) k=−0.02. (b) m2=4.186. (c) m2=3.344.
Howe e , when analyzing he ela i e p obabili ies we find eso-
nan modes o he e en solu ions (Fig. 4b and Fig. 5b). Fu he -
mo e, he ela i e p obabili ies gi e us a comple e iew o he
beha io o he massi e modes. As we can obse e, o he Sino-
Go don supe po en ial, he fi s peak o he ela i e p obabili y
ep esen s a massi e mode ha p esen s g ea e ampli ude close
o he b ane (Fig. 4a). The same goes o he linea supe po en ial
(Fig. 5a).
5. Final ema ks and pe spec i es
In his pape , we ha e shown ha he Elko field can be con-
fined on a hick b ane in (Q)modified symme ic elepa allel
g a i y h ough a Yukawa-like in e ac ion be ween he field and
he nonme ici y scala . Such a coupling allows us o ob ain a
no malizable ze o-mode, besides p o iding a eal- alued e ec i e
po en ial ha enables us o s udy massi e and esona es modes
h ough he Sch ödinge app oach.
Employing he fi s -o de o malism, we ha e buil he b ane
sys em whe e i has been used he unc ion (Q) =Q+kQn. Fo
n =1, we ha e conside ed a Sine-Go don supe po en ial, whe eas
a linea one o n =2. In bo h cases, only he Elko ze o-mode was
shown o be confined on b ane. We ha e plo ed he beha io o
e ec i e po en ial and ze o-mode, o e ing he same esul s as he
scala field. We ha e also plo ed he massi e and esona es modes
o n =1 and n =2. Th ough Fig. 4(a) and Fig. 5(a), i is possible o
see mo e clea ly he exis ence o esonan modes, bu hese modes
exis only in e en solu ions.
Ou esul s ep esen a gene aliza ion o wo ks whe e only he
massless modes we e analyzed. Besides, i is wo h emphasizing
ha all p e ious wo ks on he Elko field localiza ion we e ca ied
ou in he con ex o gene al ela i i y. Fo he fi s ime, he influ-
ence o nonme ici y on he apping o Elko spino on b anewo ld
is in es iga ed.
Fo u u e wo ks, we could s udy he localiza ion o he Elko
field by conside ing a dila on-like geome ical coupling in which
he unc ion G(Q)would be di ec ly in oduced in he Elko kine ic
e m. Wi h such a coupling, an exo ic e m migh no be necessa y
o ob ain eal- alued massi e and esona es modes. Fu he mo e,
his in es iga ion could be ex ended o highe codimensions.
Decla a ion o compe ing in e es
The au ho s decla e ha hey ha e no known compe ing finan-
cial in e es s o pe sonal ela ionships ha could ha e appea ed o
influence he wo k epo ed in his pape .
Da a a ailabili y
No da a was used o he esea ch desc ibed in he a icle.
Acknowledgemen s
The au ho s hank he Fundac¸ão Cea ense de Apoio ao Desen-
ol imen o Cien ífico e Tecnológico (FUNCAP), he Coo denac¸ão de
Ape eic¸oamen o de Pessoal de Ní el Supe io (CAPES), and he
Conselho Nacional de Desen ol imen o Cien ífico e Tecnológico
(CNPq), G an s no. 200879/2022-7 (RVM) and no. 309553/2021-0
(CASA) o financial suppo . R. V. Malu acknowledges he De-
pa amen de Física Teò ica de la Uni e si a de València o he
kind hospi ali y. The au ho s also hank he anonymous e e ees
o hei aluable commen s and sugges ions.
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