BPS submodels of the Skyrme model

Physics Le e s B 769 (2017) 362–367 Con en s lis s a ailable a ScienceDi ec Physics Le e s B www.else ie .com/loca e/physle b BPS submodels o he Sky me model C. Adam a,∗, J. Sanchez-Guillen a, A. We eszczynski b aDepa amen o de Física de Pa ículas, Uni e sidad de San iago de Compos ela and Ins i u o Galego de Física de Al as Ene xias (IGFAE), E-15782 San iago de Compos ela, Spain bIns i u e o Physics, Jagiellonian Uni e si y, Lojasiewicza 11, K aków, Poland a i c l e i n o a b s a c A icle his o y: Recei ed 23 Ma ch 2017 Accep ed 3 Ap il 2017 A ailable online 5 Ap il 2017 Edi o : A. Ringwald We show ha he s anda d Sky me model wi hou pion mass e m can be exp essed as a sum o wo BPS submodels, i.e., o wo models whose s a ic field equa ions, independen ly, can be educed o fi s o de equa ions. Fu he , hese fi s o de (BPS) equa ions ha e non i ial solu ions, a leas locally. These wo submodels, howe e , canno ha e common solu ions. Ou findings also shed some ligh on he a ional map app oxima ion. Finally, we conside ce ain gene alisa ions o he BPS submodels. ©2017 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 Sky me model [1] is a nonlinea field heo y suppo ing opological soli on solu ions (“Sky mions”), which finds i s main applica ion as a low-ene gy e ec i e heo y o QCD [2]. In he analysis o Sky me models (and ela ed heo ies suppo ing opo- logical soli ons), wo impo an concep s a e opological ene gy bounds and he ela ed no ion o Bogomolnyi equa ions [3,4]. In- deed, some imes i is possible o educe he s a ic field equa ions o fi s -o de equa ions (“Bogomolnyi equa ions”) such ha he co esponding solu ions sa u a e he bound. Wi hin he se o gen- e alised Sky me models (bu all based on he same Sky me field U∈SU(2)), wo cases o BPS Sky me models sa is ying a Bogomol- nyi equa ion a e known. The fi s one is he “BPS Sky me model” consis ing o a e m sex ic in fi s de i a i es and a po en ial [5]. In his model, Bogomolnyi solu ions exis o a bi a y opological de- g ee (“ba yon numbe ”) B. The second one consis s o he Sky me e m (qua ic in fi s de i a i es) and a pa icula po en ial [6]. In his second case, howe e , only he |B| =1solu ions (and, o cou se, he i ial acuum solu ion) sa u a e he bound and obey he co esponding Bogomolnyi equa ion. Highe Bconfigu a ions a e unbound in his model and u n in o ligh ly bound Sky mions once u he e ms a e added wi h sufficien ly small coupling con- s an s [7–9]. Bo h hese BPS models a e genuine Sky me models in he sense ha hey can be ound wi hin he se o gene alised Sky me model by an app op ia e choice o coupling cons an s. In he p esen pape we wan o conside a sligh ly di e en ype o BPS submodels. We will find ha he s anda d Sky me *Co esponding au ho . E-mail add ess: adam@ paxp1.usc.es (C. Adam). model (consis ing o he sigma model e m and he Sky me e m), oo, may be w i en as a sum o wo BPS submodels, whe e each o he wo submodels sepa a ely leads o a Bogomolnyi equa ion and non i ial solu ions (a leas , locally). These submodels, how- e e , a e no Sky me models on hei own, i.e., i is no possible o ge jus one o hese submodels by a choice o coupling cons an s wi hin he se o gene alised Sky me models. In o he wo ds, he wo submodels a e mul iplied by he same coupling cons an s, and elimina ing one au oma ically elimina es he o he . Fu he , he wo submodels ha e no common solu ions (excep o he i ial acuum solu ion). This mus , o cou se, be expec ed, because i is known ha , al hough he s anda d Sky me model has a opo- logical ene gy bound ( he Sky me–Faddee bound [1,10]) and a co esponding Bogomolnyi equa ion, his fi s -o de equa ion is oo es ic i e and only allows o he i ial solu ion. Despi e hese impedimen s, ne e heless, he exis ence o u he BPS submod- els is o in e es and sheds new ligh on se e al issues wi hin he Sky me model. The gene alised Sky me model we wan o conside consis s o ou e ms. The sigma model (Di ichle ) e m L2=1 2T (LμLμ), (1.1) he Sky me (qua ic) e m L4=1 16T ([Lμ,Lν]2), (1.2) and he sex ic e m, which is jus he ba yonic cu en squa ed e m L6=λ2π4BμBμ,Bμ=1 24π2μνρσ T LνLρLσ,(1.3) h p://dx.doi.o g/10.1016/j.physle b.2017.04.003 0370-2693/©2017 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. C. Adam e al. / Physics Le e s B 769 (2017) 362–367 363 whe e Lμ≡U†∂μUis he le in a ian cu en . The las e m, wi hou de i a i es, is he po en ial L0(U) =−μ2U( U)which p o ides a mass o pionic exci a ions. He e, we assume ha uni s o ene gy and leng h ha e been chosen such ha he coupling con- s an s o he Di ichle and Sky me e m a e scaled away (Sky me uni s). λand μa e, hus, dimensionless coupling cons an s con- olling he ela i e s eng hs o he co esponding e ms. The o iginal ma ix SU(2)Sky me field Ucan be decomposed in o a eal scala ξand complex scala uby U=exp(iξ τ· n)=cos ξ1+isin ξ τ· n(1.4)  n=1 1+|u|2(u+u∗), −i(u−u∗), 1−|u|2.(1.5) Fo opologically non i ial configu a ions he ull a ge space S3 has o be co e ed, which means ha ξshould ake alues in he ull in e al [0, π]and ushould ake alues in he ull complex plane C. Each e m o he gene alized Sky me model can hen be ew i en in his new a ge space coo dina es as L2=L(1) 2+L(2) 2,L(1) 2≡4sin 2ξuμ¯ uμ (1+|u|2)2,L(2) 2≡ξμξμ (1.6) L4=L(1) 4+L(2) 4, L(1) 4≡4sin 2ξξμξμuμ¯ uμ (1+|u|2)2−ξμ¯ uμξμuν (1+|u|2)2 L(2) 4≡4sin 4ξ(uμ¯ uμ)2−u2 μ¯ u2 ν (1+|u|2)4(1.7) L6=λ2sin4ξ (1+|u|2)4iμνρσ ξνuρ¯ uσ2,L0=−μ2U(ξ). (1.8) Fo la e con enience we ha e di ided he quad a ic and qua ic e ms in o wo pa s. Ob iously, wi hin he Sky me model con ex , i is no possible o elimina e, e.g., L(1) 2wi hou elimina ing, a he same ime, also L(2) 2. A well-known example o a Sky me heo y wi h he BPS p op- e y is he BPS Sky me model [5], and i will be use ul o e iew i b iefly. I is defined as LBPS =L6+L0(1.9) which a e he field decomposi ion eads LBPS =λ2sin4ξ (1+|u|2)4iμνρσ ξνuρ¯ uσ2−μ2U(ξ). (1.10) The Bogomolnyi equa ion o s a ic field configu a ions is λsin2ξ (1+|u|2)2iijkξiuj¯ uk=±μ√U.(1.11) I is s aigh o wa d o no ice ha he e a e o he possibili ies o dis ibu e he de i a i es o he fields and hei con ac ions wi h he epsilon symbol o o m new fi s o de equa ions wi h he same field con en . The e o e, hey can be ea ed as new Bogo- molnyi equa ions o he pe inen Sky me-like models. This will be analyzed in he es o he pape . 2. The fi s BPS submodel L(1) 2and L(1) 4can be combined in o he ollowing exp ession L(1)=4sin 2ξuμ¯ uμ (1+|u|2)2+(ξμuμ)(ξν¯ uν)−ξ2 μ(uν¯ uν) (1+|u|2)2. (2.1) The ene gy o he s a ic case eads E(1)=4d3xsin2ξ (1+|u|2)2ui¯ ui+(iijkξjuk)(−iimnξm¯ un)(2.2) =4d3xsin2ξ (1+|u|2)2(ui±iijkξjuk)(¯ ui∓iimnξm¯ un) ∓8d3xisin2ξ (1+|u|2)2ijkξiuj¯ uk ≥8d3xisin2ξ (1+|u|2)2ijkξiuj¯ uk=8π2d3xB0=8π2|B| whe e he bound is sa u a ed o solu ions o he ollowing Bogo- molnyi equa ion ui±iijkξjuk=0 (2.3) and i s complex conjuga ion. No e ha his equa ion implies some cons ain s o he fields, namely uiξi=¯ uiξi=0,u2 i=¯ u2 j=0.(2.4) I is in e es ing o conside he pa icula solu ions p o ided by he ansa z in sphe ical pola coo dina es ξ=ξ( ), u =u(θ, ϕ). This ansa z au oma ically sa isfies he fi s cons ain uiξi=0. To sim- pli y he second cons ain , i is use ul o use he s e eog aphic p ojec ion om he uni sphe e spanned by (θ, ϕ) o he plane spanned by (x, y), say, whe e z≡x +iy = an(θ/2)eiϕ, because he me ic on he uni sphe e is con o mally fla in he coo dina es (x, y), such ha he cons ain u2 i=0 simplifies o uzu¯ z=0, i.e., umus be ei he holomo phic o an i-holomo phic in he complex coo dina e z. The sign choice in he Bogomolnyi equa ion ( oge he wi h he bounda y condi ions imposed on ξ) de e mine whe he u is holomo phic o an i-holomo phic. I is ins uc i e o inse he ansa z in sphe ical pola coo di- na es di ec ly in o he ene gy unc ional. In a fi s s ep we ge ijkξjukimnξm¯ un=ξ2 jui¯ ui, such ha bo h e ms a e p opo ional o ui¯ ui. Nex , using he me ic in hese coo dina es ds2=d 2+ 2ds2 S2,ds2 S2=4 1+x2+y2(dx2+dy2)(2.5) we ge o he olume elemen dR3=d 2dS2,dS2=2i (1+z¯ z)2dzd¯ z(2.6) and, u he , ui¯ ui≡gijui¯ uj=(1+z¯ z)2 2 2(uz¯ u¯ z+u¯ z¯ uz). (2.7) As a esul , he ene gy unc ional ac o ises, E(1)=2E(1) ξE(1) u(2.8) whe e E(1) u=dS2 (1+z¯ z)2 (1+u¯ u)2(uz¯ u¯ z+u¯ z¯ uz)(2.9) is jus he CP(1) (non-linea sigma) model on S2. I s fini e en- e gy solu ions a e p o ided by all holomo phic (posi i e wind- ing numbe ) and an i-holomo phic (nega i e winding numbe ) a- ional unc ions. These solu ions sa u a e he Bogomolnyi bound E(1) u≥4π|N|, whe e he winding numbe Nis gi en by he deg ee o he a ional map. 364 C. Adam e al. / Physics Le e s B 769 (2017) 362–367 The second ene gy unc ional is E(1) ξ=d sin2ξ(1+ξ2)(2.10) and allows o an almos i ial Bogomolnyi bound, E(1) ξ=d sin2ξ(1∓ξ)2∓2ξ≥2d sin2ξξ =2 π  0 sin2ξdξ=π(2.11) whe e we used he bounda y condi ions ξ(0) =π, ξ(∞) =0. The co esponding Bogomolnyi equa ion is sin2ξ(ξ2±1)2=0,(2.12) and he solu ion wi h he igh bounda y condi ions is he com- pac on ξ( )=π− o 0 ≤ ≤π 0 o >π.(2.13) The fi s de i a i e o he compac on is no con inuous a he com- pac on bounda y =π, bu he ene gy densi y is con inuous, and bo h he Bogomolnyi equa ion and he ull second-o de Eule – Lag ange (EL) equa ion hold e e ywhe e in space o his solu ion, owing o he p esence o he ac o sin2ξ. I may be checked eas- ily ha he ba yon numbe Bis equal o he winding numbe N o his ansa z. Finally, he ene gy is E(1)=2 ·π·4π|N| =8π2|B|, as i mus be. In pa icula , i ollows ha his submodel has BPS solu ions o a bi a y a ional maps u =R(z) =p(z)/q(z)(whe e pand q a e polynomials wi hou common di iso ). This is in e es ing, be- cause a ional maps ha e been employed o cons uc app oxima e solu ions o he ull s anda d Sky me model L2+L4[11]. Ra io- nal maps canno be genuine solu ions o his model, because he a ional map ansa z is incompa ible wi h he EL equa ion esul - ing om he e m L(2) 4(excep o |B| ≤1). Inse ing he a ional map ansa z di ec ly in o he co esponding ene gy unc ional, ne - e heless, defines a es ic ed a ia ional p oblem. The ene gy now depends on he pa icula a ional map, and minimisa ion leads o a ional maps wi h in e es ing disc e e symme ies (e.g. he sym- me ies o pla onic solids) o |B| ≥3. Fu he , hese symme ies ag ee wi h he symme ies o he ull nume ical solu ions [12], and also he ene gies o he co esponding a ional map app oxima- ions a e a he close o he ene gies o he nume ically calcula ed Sky mions. He e we may conclude ha he a he good quali y o he a ional map app oxima ion may be unde s ood om he ac he model has a BPS submodel which is exac ly sol ed by a ional maps, and he e is only one e m (L(2) 4) which p e en s a ional maps om being exac solu ions. 3. The second BPS submodel L(2) 2and L(2) 4can be combined in o he exp ession L(2)=ξμξμ+4sin 4ξ(uμ¯ uμ)2−u2 μ¯ u2 ν (1+|u|2)4.(3.1) The ene gy unc ional o s a ic configu a ions eads E(2)=d3xξ2 i+4sin 4ξ1 (1+|u|2)4(iijkuj¯ uk)2(3.2) =d3xξi∓2isin2ξ (1+|u|)2ijkuj¯ uk2 ±4d3xisin2ξ (1+|u|)2ijkξiuj¯ uk ≥4d3xisin2ξ (1+|u|)2ijkξiuj¯ uk=4π2d3xB0=4π2|B|. The bound is sa u a ed o solu ions o he Bogomolnyi equa ions ξi∓2isin2ξ (1+|u|)2ijkuj¯ uk=0.(3.3) Obse e ha he Bogomolnyi equa ions lead o some cons ains o he fields, uiξi=¯ uiξi=0.(3.4) Assuming he ansa z ξ=ξ( ), u =g(θ)eimϕ, we ge o u he solu- ion u= an θ 2eimϕ,(3.5) which, o |m| >1, has a conical singula i y along he zaxis. The ene gy densi y and winding numbe densi y, on he o he hand, a e smoo h. Fu he , mis equal o he ba yon numbe , B =m, o genuine Sky mion configu a ions (i.e., whe e ξobeys he co e- sponding bounda y condi ions). I is in e es ing o no e ha he BPS Sky me model leads o he same solu ion o u his ansa z. The wo BPS equa ions (1.11) and (3.3) a e, in ac , e y simila o his ansa z (iden ical o u, di e en o ξ( )). The esul ing equa- ion o ξ eads dξ d =±m 2sin2ξ. (3.6) Choosing he minus sign (a nega i e slope o ξ( )), he solu ion is ξ=a cco ms0−1 (3.7) whe e s0is an in eg a ion cons an . A →0we ha e co ξ→ −∞, i.e., ξ→π, as desi ed. In he limi →∞, howe e , co ξ does no app oach ∞, i.e., ξdoes no app oach 0. Ins ead, he p ofile unc ion ξ akes alues only in he in e al π≥ξ( ) ≥ a cco ms0>0. The local solu ion o he BPS equa ion, he e o e, canno be ex ended o a solu ion on he ull a ge space, i.e., o a Sky mion. Ins ead, i leads o a “ ac ional” Sky mion, whe e bo h he ba yon numbe and he BPS ene gy may ake a bi a y ac- ional alues, defined by he choice o he in eg a ion cons an s0. 4. Some gene alisa ions We now wan o conside some gene alisa ions o he wo BPS submodels, by mul iplying each e m by a ce ain coupling unc- ion. 4.1. The dila on-YM like model We conside he ollowing model ˜ L(2)=h2(ξ, u¯ u)ξμξμ+g2 2(ξ, u¯ u)sin4ξ(uμ¯ uμ)2−u2 μ¯ u2 ν (1+|u|2)4. (4.1) The ene gy in eg al o s a ic configu a ions eads C. Adam e al. / Physics Le e s B 769 (2017) 362–367 365 ˜ E(2)=d3xh2ξ2 i+g2 2sin4ξ1 (1+|u|2)4(iijkuj¯ uk)2(4.2) =d3xh2ξi∓ig2sin2ξ (1+|u|)2ijkuj¯ uk2 ±2d3xig2h2sin2ξ (1+|u|)2ijkξiuj¯ uk ≥2d3xig2h2sin2ξ (1+|u|)2ijkξiuj¯ uk =2π2d3xg2h2B0=2π2|B|g2h2S3 whe e ·S3is he a ge space a e age o he a ge space unc ion inse ed be ween he b acke s. The bound is sa u a ed o solu ions o he Bogomolnyi equa ions ξi∓ig2sin2ξ (1+|u|)2ijkuj¯ uk=0.(4.3) No e, ha his Bogomolnyi equa ion is iden ical o he one in he dila on – SU(2)Yang–Mills model desc ibing a magne ic monopole [13–16]. Fu he , i leads o he cons ain s (3.4), again. An example can be p o ided by a pa icula choice o g2and h2, g2=1 cos2ξ 2 ,h2=1.(4.4) Then, using he ansa z in sphe ical pola coo dina es, again, he opologically non i ial solu ions a e again (3.5) o u, whe eas ξ= ξ( )obeys dξ d =±2m 2sin2ξ 2.(4.5) Imposing he ollowing bounda y condi ions: ξ( =0) =0 and ξ( =∞) =π, he pe inen solu ion is co ξ 2=m ⇒ξ=2 a cco m .(4.6) He e, g2h2S3=2 π π  0 dξsin2ξ·1 cos2ξ 2=4 (4.7) and he ene gy and opological cha ge a e ˜ E(2)=8π2m,B=m.(4.8) 4.2. The holomo phic map like model Now, we define he model ˜ L(1)=h1uμ¯ uμ (1+|u|2)2+g2 1sin4ξ(ξμuμ)(ξν¯ uν)−ξ2 μ(uν¯ uν) (1+|u|2)2. (4.9) The ene gy o he s a ic case eads ˜ E(1)=d3xh1ui¯ ui (1+|u|2)2 +g2 1 sin4ξ (1+|u|2)2(iijkξjuk)(−iimnξm¯ un) (4.10) =d3xh1 (1+|u|2)2(ui±ig1sin2ξijkξjuk) ×(¯ ui∓ig1sin2ξimnξm¯ un) ∓2d3xig1h1sin2ξ (1+|u|2)2ijkξiuj¯ uk ≥2d3xig1h1sin2ξ (1+|u|2)2ijkξiuj¯ uk =4π2d3xg1h1B0=4π2|B|g1h1S3 whe e he bound is sa u a ed o solu ions o he ollowing Bogo- molnyi equa ion ui±ig1sin2ξijkξjuk=0 (4.11) and i s complex conjuga ion. No e ha his o mula implies he cons ain s (2.4) o he complex and eal fields. An example o BPS Sky mions o his ype can be ound o h1=sin2ξ, g1=1 sin3ξ.(4.12) We assume ha ξ=ξ(z)and u =u(x, y)in ca esian coo dina es (x, y, z). Then, he scala obeys g2sin2ξξz=−1⇒ξz=−sin ξ(4.13) which is he sine-Go don kink equa ion and he e o e ξ=2a c ane−z.(4.14) This means ha he complex scala obeys ui∓iijuj=0 (4.15) whe e (i, j) ∈{1, 2}and he e o e i is jus a holomo phic (an i- holomo phic) unc ion in x +iy. Fo example, u =ρmeimϕ, when cylind ical coo dina es (ρ, ϕ, z)a e used. So, finally we ge holo- mo phic 2-dimensional soli ons loca ed on a sine-Go don b ane wi h co-dimension one, and ˜ E(1)=4π2|B|1 sin ξS3=16π|B|,B=m.(4.16) 5. Summa y We ound he in e es ing esul ha he gene alized Sky me model can, in ac , be exp essed as a sum o h ee BPS submod- els, LSk =L2+L4+L6+L0 =L(1) 2+L(1) 4+L(2) 2+L(2) 4+(L6+L0) ≡L(1) BPS +L(2) BPS +LBPS.(5.1) In compa ison wi h he BPS Sky me model, howe e , he new BPS submodels e eal some impo an di e ences. 1. The new BPS submodels canno be ob ained as a ce ain limi (pa icula alues o he model pa ame e s) o he ull Sky me model. Each o hem consis s o wo e ms – one eme ging om he Di ichle pa and one om he Sky me pa . 2. The new BPS submodels a e comple ely independen o he pion mass. The po en ial pa o he gene alized Sky me model only con ibu es o he BPS Sky me model. 366 C. Adam e al. / Physics Le e s B 769 (2017) 362–367 3. While he BPS Sky me model possesses well-beha ed opolog- ically non i ial solu ions i.e., BPS Sky mions, whose compac o non-compac na u e depends on he po en ial, he o he BPS submodels p o ide solu ions wi h di e en cha ac e is- ics. The second BPS submodel leads o ac ional Sky mions, i.e., o solu ions wi h a non-in ege ba yon numbe . The fi s BPS submodel has compac Sky mions as solu ions. Ob iously, he wo submodels canno ha e common solu ions, because his would lead o a BPS solu ion o he ull s anda d Sky me model L2+L4, which a e known no o exis . 4. One cha ac e is ic ea u e o he solu ions o he wo submod- els is he opposing e ec which hey ha e on he Sky mion size. Indeed, he model L(1)leads o fini e size Sky mions (compac ons) whose size, in addi ion, is independen o he ba yon numbe . The model L(2), on he o he hand, leads o ac ional Sky mions, i.e., “Sky mions” o “mo e han infini e size” which do no e en fi in he infini e in e al ∈[0, ∞]. The size o he Sky mions o he ull model L2+L4is a comp omise be ween hese wo ex emes, i.e., Sky mions wi h infini e size, which decay algeb aically o →∞. 5. The submodel L(1)leads o Sky mions wi h a shell-like s uc- u e, whe e he ene gy densi y is ze o bo h a he cen e =0 and ou side he compac on bounda y and akes i s maximum alue a some nonze o adius. This eflec s a known beha iou o he Sky mions o he ull model L2+L4, which ha e a shell-like s uc u e, as well [12]. As men ioned al eady, he submodel L(1)is sol ed by a bi a y a ional maps u(z) o an ansa z ξ( )and u(z)in sphe ical pola coo dina es, and he same “ansa z” p o ides a he good app oxi- ma ions o solu ions o he ull s anda d Sky me model L2+L4. He e we jus wan o ema k ha inse ing he a ional map ansa z in o he s a ic ene gy unc ional o he gene alised Sky me model (5.1) leads exac ly o he same es ic ed a ia ional p oblem o he a ional map u =R(z). Tha is o say, he es ic ed ene gy unc ional is Egen =4πd  2ξ2+μ2 2U(ξ) +2Bsin2ξ(ξ2+1) +Isin4ξ 21+λ2 4ξ2 (5.2) whe e he a ional map R(z)mus minimize he unc ional [11] I=1 4πdS21+z¯ z 1+R¯ R4 Rz¯ R¯ z2(5.3) exac ly as in he a ional map app oxima ion o he model L2+L4. This ac was poin ed ou ecen ly in [9] and used he e o a de ailed s udy o he B =4Sky mion ( he helium nucleus) wi hin he gene alised Sky me model. In addi ion o in oducing he p ope BPS submodels o he s anda d (massless) Sky me model, we also conside ed some gen- e alisa ions, whe e each e m in he submodels is mul iplied by a field-dependen coupling unc ion. I he coupling unc ions only depend on he p ofile unc ion ξ, hen he gene alisa ions based on L(1)con inue o be o he holomo phic ype, i.e., he ufield has a CP(1)-model ene gy densi y in a sepa a ion-o - a iable ansa z. The second model, L(2), gene alises o a field heo y which sha es i s Bogomolnyi equa ions wi h he ones in he dila on – SU(2) Yang–Mills model o a magne ic monopole [13–16], whe e ξplays he ole o he dila on. Bo h gene alised models lead o genuine opological soli ons obeying he equi ed bounda y condi ions o app op ia e choices o he coupling unc ions. The e exis s a lowe dimensional coun e pa o he Sky me model, he baby Sky me model [17] (he e in he complex field o - mula ion) Lbaby =λ2L2+λ4L4+λ0L0 =λ2 uμ¯ uμ (1+|u|2)2−λ4 (uμ¯ uμ)2−u2 μ¯ u2 ν (1+|u|2)4−λ0U(5.4) whe e one can dis inguish wo BPS submodels. The CP1(sigma) model LCP1≡L2=uμ¯ uμ (1+|u|2)2(5.5) and he baby BPS Sky me model [18] LBPS ≡L4+λ0L0=(uμ¯ uμ)2−u2 μ¯ u2 ν (1+|u|2)4−λ0U.(5.6) Bo h BPS submodels a e genuine baby Sky me models. Fu he , hey a e ue BPS heo ies, i.e., he e exis Bogomolnyi equa ions and solu ions which sa u a e a opological bound in ol ing a opo- logical index. These equa ions a e ui±iijuj=0 (5.7) o he CP(1) model and iijui¯ uj (1+|u|2)2±λ0U=0 (5.8) o he BPS baby Sky me model. Hence, we ecognize a simila pa e n o he one ound in he (3 +1)-dimensional Sky me model. Le us ema k ha he e exis u he possibili ies o find BPS e sions o he Sky me model. One pa icula example, based on he same field con en s bu a di e en Lag angian was cons uc ed in [19]. I he opology o he base space mani old is changed and allows o define addi ional opological indices, hen u he BPS sec o s o he Sky me model ela ed o hese new opological indices may be ound [20,21]. Ano he op ion which equi es, how- e e , o change he field con en s by adding an infini e numbe o ec o mesons was p oposed and de eloped in [22,23]. Finally, he hidden BPS s uc u e o he s anda d Sky me model e ealed he e migh be ela ed o possible supe symme ic e - sions o he heo y [24,25], because he e is a close ela ion be- ween BPS sec o s and supe symme y, in gene al (see [26] o he baby Sky me model). 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