5D Elko spinor field non-minimally coupled to nonmetricity in f(Q) gravity

Physics Le e s B 843 (2023) 138029 Con en s lis s a ailable a ScienceDi ec Physics Le e s B jou nal homepage: www.else ie .com/loca e/physle b 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. Re e ences [1] S. Capozziello, V.F. Ca done, A. T oisi, Phys. Re . D 71 (2005) 043503. [2] D. Bazeia, L. Losano, R. Menezes, G.J. Olmo, D. Rubie a-Ga cia, Eu . Phys. J. C 75 (12) (2015) 569. [3] B.M. Gu, B. Guo, H. Yu, Y.X. Liu, Phys. Re . D 92 (2) (2015) 024011. [4] R. My zakulo , Eu . Phys. J. C 72 (2012) 2203. 7 F.M. Belchio , A.R.P. Mo ei a, R.V. Malu e al. Physics Le e s B 843 (2023) 138029 [5] D. Deb, F. Rahaman, S. Ray, B.K. Guha, Phys. Re . 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