Deformation method for generalized Abelian Higgs-Chern-Simons models

a Xi :1305.4251 1 [hep- h] 18 May 2013 epl d a De o ma ion me hod o gene alized Abelian Higgs-Che n-Simons models L. Losano1,2,3, J. M. C. Malbouisson1,4, D. Rubie a-Ga cia5and C. dos San os1 1Cen o de F´ısica e Depa amen o de F´ısica e As onomia, Faculdade de Ciˆencias da Uni e sidade do Po o, 4169- 007 Po o, Po ugal 2Depa amen o de F´ısica, Uni e sidade Fede al da Pa a´ıba, 58051-900 Jo˜ao Pessoa, PB, B azil 3Depa amen o de F´ısica, Uni e sidade Fede al de Campina G ande, 58109-970 Campina G ande, PB, B azil 4Ins i u o de F´ısica, Uni e sidade Fede al da Bahia, 40210-340 Sal ado , BA, B azil 5Depa amen o de F´ısica, Uni e sidad de O iedo, 33007 O iedo, As u ias, Spain PACS 11.27.+d – Ex ended classical solu ions PACS 11.15.Yc – Che n-Simons gauge heo y Abs ac – We p esen an ex ension o he de o ma ion me hod applied o sel -dual solu ions o gene alized Abelian Higgs-Che n-Simons models. S a ing om a model de ined by a po en ial V(|φ|) and a non-canonical kine ic e m ω(|φ|)|Dµφ|2whose analy ical domain wall solu ions a e known, we show ha his me hod allows o ob ain an uncoun able numbe o new analy ical solu ions o new models de ined by o he unc ions e Vand eω. We p esen some examples o de o ma ion unc ions leading o new amilies o models and hei associa ed analy ic solu ions. In oduc ion. – Topological de ec s play impo - an ole in se e al impo an a eas such as high en- e gy physics [1], cosmology [2] and condensed ma e physics [3]. Such de ec s eme ge as classical solu ions o nonlinea ield heo ies which possess degene a ed acua. Typical examples a e domain walls desc ibed by kink so- lu ions o he φ4model, Ginzbu g-Landau o ices and monopoles. Usually, domain walls a e solu ions connec ing wo dis- inc acua o scala ield heo ies in one-space dimension, o in hei inse ions in highe dimensions, while o ices eme ge as solu ions o models ha couple cha ged ma e ields wi h gauge ields li ing in a (a leas ) 3-dimensional space- ime, and monopoles lie in a 4-D space- ime. In a (2+1)-dimensional space- ime, minimal coupling be ween cha ged-ma e and gauge ields can be imple- men ed by he Che n-Simons (CS) ac ion. Al hough he CS ield can no be concei ed as a ee ield, i s coupling wi h ma e ields imposes cons ain s in he dynamics which ha e e y ele an consequences, in bo h, classi- cal and quan um heo ies, wi h ei he ela i is ic o non- ela i is ic kine ics. In he non- ela i is ic (NR) ame- wo k, pa icles coupled h ough he CS ield ca y bo h elec ic cha ge and magne ic lux, and possess ac ional s a is ics [4]. Addi ionally, he NR scala CS model con- s i u es a seminal example o a Galilean-in a ian gauge- ield heo y [5]. Also, o a c i ical s eng h o a qua ic sel -in e ac ion o he scala ield, which es o es he scale in a iance [6], his model p o ides a ield- heo e ical de- sc ip ion o he Aha ono -Bohm (AB) sca e ing [7]; con- side ing he Lo en z co a ian ield heo y, ela i is ic co - ec ions o he AB sca e ing a e ob ained [8]. Sel -dual soli on solu ions ha e been ound in he el- a i is ic, U(1)-in a ian , Abelian, Higgs-Che n-Simons (HCS) gauge- heo y whe e he symme y-b eaking po en- ial o he Higgs ield is U(ϕ)∼ |ϕ|2(|ϕ|2− 2)2[9]; o - ex and domain-wall solu ions ha e been ob ained o his model [10]. This model was gene alized by conside ing a non-canonical kine ic e m o he complex scala ield, W(|ϕ|)|Dµϕ|2, p o iding sel -dual o ex [11] and domain- wall [12] solu ions. Models wi h noncanonical kine ic e ms (k- ields) ind also applica ions in s ong-in e ac ion physics [13] and in cosmology [14]. Due o he nonlinea i y, he e is no gene al in eg a- ion me hod o sol e analy ically he equa ions o mo- ion o non-linea ield heo ies; only o a small se o models, solu ions o he equa ions o mo ion can be di- ec ly de e mined. Howe e , o scala ields in (1+1)- dimensions, s a ing om a nonlinea model wi h known solu ions, in ini ely many new models and hei co e- p-1 L. Losano e al. sponding s a ic solu ions can be ound using he de o ma- ion me hod [15]. This me hod wo ks as ollows. Choos- ing a de o ma ion unc ion (φ), he model de ined by he de o med po en ial e V(φ) = V[( (φ))]/[ ′(φ)]2, whe e ′means he de i a i e o , possesses s a ic solu ions gi en by e φ(x) = −1(φ(x)), whe e φ(x) is a solu ion o he s a ic equa ion o mo ion o he o iginal model wi h po- en ial V(φ). This p ocedu e has been applied o gene a e de ec solu ions o many models ha ing polynomial in e - ac ions [16] and new amilies o sine-Go don and mul i- sine-Go don models [17]. Also, an o bi -based ex ension o his me hod has been applied o models in ol ing wo in e ac ing scala ields [18]. The pu pose o his Le e is o ex end he de o ma ion me hod o gauge- ield models conside ing speci ically he Abelian HCS heo y, ocusing pa icula ly on he Jackiw- Lee-Weinbe g (JLK) domain-wall solu ion [10]. In Sec- ion II, we p esen he gene alized Abelian HCS models and w i e down he i s -o de equa ions obeyed by he Bogomol’nyi-P asad-Somme eld (BPS) [19] domain-wall solu ions. In Sec ion III, he de o ma ion me hod is ex- ended o domain-wall solu ions o gene alized Abelian HCS models and some examples a e gi en, illus a ing he powe o he p ocedu e in gene a ing new models wi h hei s a ic solu ions. Finally, some ema ks a e made. BPS domain walls in he gene alized Abelian HCS model. – We conside he gene alized (2 + 1)- dimensional Abelian HCS model de ined by he La- g angian densi y [11] LS=W(|ϕ|)|Dµϕ|2−U(|ϕ|) + κ 4ǫαβγAαFβγ,(1) whe e ϕis he complex Higgs ield, Dµ=∂µ+ieAµis he co a ian de i a i e and Fµν =∂µAν−∂νAµis he ield s eng h enso o he gauge po en ial Aµ. The sel - in e ac ion po en ial, U(|ϕ|), is assumed o implemen a symme y-b eaking mechanism and he non-canonici y o he kine ic e m is engende ed by he unc ion W(|ϕ|); ak- ing W ≡ 1, one eco e s he s anda d Abelian HCS model. No e ha , in he CS e m, ǫαβγ is he ully an isymme - ic enso and he elec ic and he magne ic CS ields a e Ei=Fi0=−˙ Ai−∇iA0and B=~ ∇× ~ A=∂2A1−∂1A2, espec i ely. I is con enien o wo k wi h dimensionless quan i ies. In (2 + 1) dimensions, he scala ield ϕhas mass dimen- sion equal o 1/2, he same one we ake o he gauge ield; his choice ensu es ha he mass dimension o Aα ag ees wi h he one ob ained i a Maxwell e m we e added o LS. I ollows ha he elec ic cha ge eand he CS pa ame e κhas mass dimensions equal o 1/2 and 1, espec i ely, so ha e2/κ is dimensionless. We can ge an addi ional simpli ica ion i we abso b he pa ame e s eand κby ede ining space- ime coo dina es and ields. Thus, wi h Mbeing a mass scale o he model, we de- ine ¯xµ=Me2xµ/κ,φ=√κϕ/√Me,Aµ=κAµ/Me, V=κ2U/M3e4and ω=e2W/κ; he dimensionless La- g angian densi y is hen gi en by L=κ2LS/M3e4and he ac ion becomes S=κ e2Rd3¯xL. Fo a simple no a- ion, we supp ess he ba o e he space- ime coo dina es and use, om now on, only dimensionless quan i ies. Va ia ion o he ac ion leads o he equa ions o mo ion ωDµDµφ+∂µωDµφ−|Dµφ|2∂ω ∂φ∗+∂V ∂φ∗= 0 ,(2) 1 2ǫαβγFβγ =−Jα,(3) whe e he cu en densi y, Jα= (ρ,~ j), is gi en by Jα=iω [φ(Dαφ)∗−φ∗Dαφ].(4) The ime componen o eq. (3) s a es ha he magne ic ield is equal o he plana elec ic-cha ge densi y, B=ρ, which is he CS Gauss law. Also, o s a ic ield con igu- a ions, we ind B=ρ= 2A0|φ|2ω(|φ|), Ea=ǫabjb,(5) which shows ha he elec ic-cu en densi y is pe pen- dicula o he elec ic ield. The ene gy-momen um enso is gi en by Tµν =ω[Dµφ(Dνφ)∗+Dνφ(Dµφ)∗] −gµν ω|Dαφ|2−V(|φ|)(6) om which we ob ain he ene gy densi y, ε=T00, and he p essu e componen s, P1=T11 and P2=T22. We a e in e es ed in s a ic domain-wall solu ions. Fi s ly, no e ha he complex phase o he scala ield φcan be supp essed by a sui able gauge ans o ma ion. Then, ixing he Coulomb gauge, we can sea ch o solu- ions o he o m [10,12] φ=h(x), Aµ=A0(x), A1= 0, A2=A(x),(7) whe e h(x) and A(x) a e eal unc ions and xdeno es he x1-coo dina e. This ansa z co esponds o domain-walls (ac ually lines in he plane) pa allel o he x2-axis. In his case, he s a ic equa ions o mo ion educes o [2ωh′]′= 2hω A2−A2 0+dV dh ,(8) A′ 0=−2ωh2A , (9) and he Gauss law A′=−2ωh2A0,(10) whe e he p ime deno es de i a ion wi h espec o x. F om eqs. (9) and (10) we in e ha A0A′ 0=AA′, so ha ime and space componen s o he gauge iled a e cons ained by A2 0=A2−C , (11) whe e Cis a eal cons an . Also, consis ency wi h eq. (8) imposes a ela ion be ween he unc ion ω(h) and he po- en ial V(h) exp essed as d dh"pV/ω h#=−2ωh . (12) p-2 De o ma ion me hod o Abelian HCS models Now, he s abili y condi ion P1=P2= 0 leads o he i s -o de equa ions [20] h′=±hA (13) A′=−2ωh2A0(14) wi h V=h2ωA2 0.(15) Fo h≥0 and A≥0, he signal + (−) in eq. (13) co e- sponds o he kink (an i-kink) like solu ion o he Higgs ield, h(+) (h(−)). No e ha , he i s -o de equa ions (13) and (14) sol e he equa ions o mo ions (2) and (3). The s a ic solu ions a e physically cha ac e ized by hei cha ge and ene gy. Now, e u ning o eq. (6), o non- nega i e V(h) and ω(h), he ene gy o s a ic solu ions can be ew i en in he o m E=Z∞ −∞ dx T00 =Z∞ −∞ dx (V+ωh′+ 2ωh2A2 0+Cωh2) =Z∞ −∞ dx h√V±√ωhA2 02+√ωh′±√ωhA2 +p−A0A′±√2ωhA02i +Z∞ −∞ dx 2√ωV hA0±2ωhh′A ±2p−2ωA0A′hA0+A0A′−2ωh2A2 0,(16) which is minimized i eqs. (13), (14), and (15) a e obeyed, esul ing in E=Z∞ −∞ dx (4V) = A2(−∞)−A2(+∞),(17) o C= 0. In his case A2 0=A2, so he sys em o i s - o de equa ions decouples and is sol ed simply by (13) wi h A(h) = −2Zωh dh +c , (18) whe e cis an in eg a ion cons an sui able o he bounda y condi ions equi ed o he gauge ield. And, om (5) and (6), he elec ic cha ge, Q, and Noe he cha ge, P, a e gi en by Q=Z∞ −∞ dxρ(x) = A(−∞)−A(+∞),(19) P=Z∞ −∞ dx T02 =1 2A2(−∞)−A2(+∞),(20) which a e bo h conse ed due o he U(1) symme y and he ansla ional in a iance along x2-di ec ion, espec- i ely. This shows ha , o hin a ange such ha ω(h)≥0 and V(h)≥0, he BPS solu ions o he i s -o de eqs. (13) and (14), wi h (15), indeed co espond o solu ions o min- imum ene gy and hei ene gy and cha ge can be calcu- la ed knowing only he asymp o ic beha io o he gauge ield. Co espondingly, he Higgs ield, o bo h kink and an i-kink solu ions, connec s wo consecu i e acua o he po en ial, while a lump-like solu ion s a s and e mina es on he same acuum when x→ ±∞. S anda d sel -dual domain walls. The simples Abelian HCS model ha suppo s sel -dual domain wall solu ions is he JLW model [10], which is de ined by he Lag angian 1 wi h canonical kine ic e m (ω= 1) and he (dimensionless) po en ial V(h) = h2(1 −h2)2,(21) plo ed in ig. 1. In his case, he use o eq. (18) (wi h Fig. 1: The po en ial (21) as unc ion o he Higgs ield. c= 1) p o ides he esul A= 1 −h2,(22) which, subs i u ing in (13), gi es he solu ions h(+)(x) = 1/p1 + e−2x, A(−)(x) = 1/(1 + e2x),(23) and h(−)(x) = 1/p1 + e2x, A(+)(x) = 1/(1 + e−2x),(24) which a e displayed in ig. 2. We see ha he scala ield, in bo h cases, in e pola es be ween he symme ic and he asymme ic acua. ig. 3shows he ene gy and Fig. 2: The Higgs ield (solid line) and he gauge ield (dashed line), (h(+)(x), A(−)(x)) om eq. (23) on he le , and (h(−)(x), A(+)(x)) om eq. (24) on he igh . elec ic-cha ge densi ies o bo h wall solu ions. We ind ha he spa ial dis ibu ion o he elec ic cha ge is sym- me ic a ound he o igin, while o he ene gy he axis o symme y a e displaced om he o igin. And, om eqs. (17), (19) and (20), o he solu ions (h(+), A(−)) and (h(−), A(+)), we ha e he cha ges Q= 1, P= 1/2, and Q=−1, P=−1/2, espec i ely, and he same ene gy, E= 1. p-3 L. Losano e al. Fig. 3: Ene gy densi y o he solu ions (h(+)(x), A(−)(x)) (dashed line) and (h(−)(x), A(+)(x)) (dashed-do ed line), and module o elec ic-cha ge densi y o bo h solu ions (solid line). The de o ma ion me hod. – Le us now de elop he de o ma ion me hod o gene alized Abelian HCS models ollowing he spi i o he p ocedu e in oduced o scala ields [15]. As we shall show, by de o ming si- mul aneously he Higgs and he CS ields, we a e able o cons uc many new gene alized HCS models and hei s a ic domain-wall solu ions. The o iginal and he de- o med models a e mapped in o each o he h ough he de o ma ion unc ion. Deno e by ˜ φ(x) and e A(x) new ields whose dynamics is go e ned by he (dimensionless) Lag angian densi y e L=eω(|˜ φ|)|Dµ˜ φ|2−e V(|˜ φ|) + 1 4ǫαβγ e Aαe Fβγ ,(25) whe e e V(|˜ φ|) and eω(|˜ φ|) a e new unc ions speci ying his model. As in sec. II, we assume ha he sel -dual BPS domain-wall solu ions o his model ake he o m ˜ φ=˜ h(x),e Aµ=e A0(x),e A1= 0,e A2=e A(x),(26) and sa is y he i s -o de equa ions o mo ion ˜ h′=±˜ he A , (27) e A′=−2eω˜ h2e A0,(28) whe e ˜ h′≡d˜ h/dx and e A′≡de A/dx, wi h he cons ain s e V=h2eωe A2 0and e A2 0=e A2. Now, in oduce he de o ma ion unc ion such ha he Higgs ields o he wo models a e mapped in o each o he , h= (˜ h), which is assumed o be in e ible (in a p esc ibed domain o de ini ion) and di e en iable. Also, conside ha he de o med CS-gauge ield is ob ained om Aby he p esc ip ion e A(˜ h) = (˜ h)A[h→ (˜ h)] ˜ h e h ,(29) whe e e h=d /deh. Then, i ollows om eqs. (27) and (28), using eq. (29), ha he model de ined by Lag angian densi y (25), wi h he de o med unc ion eωand he de- o med po en ial e Vgi en by eω(˜ h) = 1 2e A˜ h ˜ h,e V(˜ h) = ˜ h2e A2eω(˜ h),(30) whe e e A˜ h=de A/d˜ h, possesses s a ic BPS solu ions ˜ h(x) = −1[h(x)] ,e A(x) = e A −1[h(x)],(31) whe e h(x) is a s a ic solu ion o he o iginal model (1). I should be no ed ha all he conside a ions and e- la ions p esen ed in sec. II, ela i e o ene gy and con- se ed cha ges, a e held unchanged o he de o med sys- em. In he ollowing, aking as he s a ing poin he JLW domain-wall solu ions desc ibed in sec. II.A, we conside some illus a i e examples o he me hod. Example I. Fi s ly, we conside he couple o de o - ma ion unc ion (˜ h)(±)= (±)1−˜ h2 1 + ˜ h2,(32) which, using eqs. (22) and (29), gi es e A(±)(˜ h) = (˜ h)(±); and, om eq. (30), i ollows ha ˜ω=2 (1 + ˜ h2)2,e V=2˜ h2(1 −˜ h2)2 (1 + ˜ h2)4.(33) These unc ions, which a e plo ed in ig. 4, de ine he gen- e alized Abelian HCS model employed in Re . [12]. No e ha , he h ee acua a ˜ h= 0,1,+∞es ablish wo walls, one be ween ˜ h= 0 and ˜ h= 1, and o he be ween ˜ h= 1 and ˜ h= +∞. F om he in e se o he de o ma ion unc- Fig. 4: The po en ial (33) ( op panel) and he co esponding unc ion w(bo om panel), as unc ion o he Higgs ield. Fig. 5: The Higgs ield (solid line) and he gauge ield (dashed line), (˜ h(+)(x), e A(−)(x)) om eq. (34) on he le , and (˜ h(−)(x), e A(+)(x)) om eq. (35) on he igh , o 0 ≤˜ h≤1 ion (32) and eqs. (23) and (24), o he ange 0 ≤˜ h≤1, we ob ain he solu ions ˜ h(+)(x) = p1 + e−2x−e−x,e A(−)(x) = 1/p1 + e2x, (34) ˜ h(−)(x) = 1/p1 + 2e2x,e A(+)(x) = 1/(1 + e−2x),(35) p-4 De o ma ion me hod o Abelian HCS models while o ˜ h≥1 we ha e ˜ h(+)(x) = p1 + 2e2x,e A(+)(x) = 1/(1 + e−2x),(36) ˜ h(−)(x) = p1 + e−2x+e−x,e A(−)(x) = 1/(p1 + e2x). (37) In igs. 5 and 6, we display hese domain wall solu ions. The walls o 0 ≤˜ h≤1 and ˜ h≥1 ha e he same gauge ields, bu wi h he asymp o ic alue o x=±∞ changed. Then, o bo h anges he walls ha e he same ene gy, E= 1, and cha ges Q= 1 and P= 1/2, o e A(−), and Q=−1 and P=−1/2, o e A(+). This makes possible o ha e a ac i e o epulsi e o ce be ween he wo walls. In ig. 7, we display he ene gy and cha ge densi ies Fig. 6: The ields (˜ h(+)(x), e A(+)(x)), eq. (36) (solid line), and (˜ h(−)(x), e A(−)(x)), eq. (37) (dashed line), o ˜ h≥1. Fig. 7: Module o elec ic cha ge o solu ions A(−)(x) (dashed line) and A(+)(x) (dashed-do ed line), and ene gy densi y (solid line) o bo h walls. o he wo walls. The compa ison wi h he walls o he JLW model shows ha , no wi hs anding he walls ha e he same cha ges and ene gy, he JLW walls ha e sym- me ic spa ial dis ibu ions o ene gy and cha ge, while he e only he dis ibu ion o ene gy is symme ic and all he co esponding dis ibu ions a e mo e sp ead ou . The model de ined by eqs. (33), which was ob ained by de o m- ing he JLW model, was s udied in Re . [12] bu only he solu ion sa is ying 0 ≤˜ h≤1 was conside ed. The de o ma ion unc ion (32) is a pa icula case o he de o ma ion unc ion (˜ h) = cos[αa c an(˜ h)], co e- sponding o α= 2; om ha new amily o models can be gene a ed o αin ege . Example II. As a second example, conside he se o de o ma ion unc ions [16] α(˜ h)] = cos[αa ccos(˜ h)] = Tα(˜ h),(38) whe e he in ege α > 2 and Tαis he Chebyshew poly- nomials o i s kind. Using his de o ma ion in eq. (29), wi h eq. (22), we ha e he gauge ield e Aα(˜ h) = (1 −˜ h2)1/2sin[2αa ccos(˜ h)] /2α˜ h , = (1 −˜ h2)U2α−1(˜ h)/2α˜ h , (39) whe e Uis he Chebyshew polynomials o second kind; which explici esul s, o α= 2,3, a e e A2(˜ h) = (1 −˜ h2)(2˜ h2−1) ,(40) e A3(˜ h) = 1 3(1 −˜ h2)(1 −2˜ h2)(3 −4˜ h2).(41) In his case, om eqs. (30) and (38), we ha e a amily o models de ined by he unc ion eωα(˜ h) and he po en ial e Vα(˜ h) w i en in polynomial o m as eωα(˜ h) = |[2α˜ h T2α(˜ h) + U2α−1(˜ h)]/4α˜ h3|,(42) e Vα(˜ h) = (1 −˜ h2)2U2 2α−1(˜ h)eωα(˜ h)/4α2.(43) Then, each alue o he pa ame e αspeci ies a model o his amily. The explici esul s o α= 2,3 a e eω2(˜ h) = 3 −4˜ h2,(44) e V2(˜ h) = ˜ h2(1 −˜ h2)2(1 −2˜ h2)2eω2(˜ h),(45) eω3(˜ h) = 1 3(19 −64˜ h2+ 48˜ h4),(46) e V3(˜ h) = 1 9˜ h2(1 −˜ h2)2(1 −2˜ h2)2(3 −4˜ h2)2eω3(˜ h). (47) Fo hese models, om he in e se o de o ma ion unc ion Fig. 8: The po en ial (45) ( op panel) and he unc ion w(44) (bo om panel), as unc ion o ˜ h. (38), we ob ain he s a ic Higgs ield solu ions in he o m ˜ h(±)(x) = cos h(a ccos(h(±)(x)) + (m−1)π)/αi,(48) whe e h(±)(x) is gi en by eqs. (23) and (24), and mis an in ege , which gene a es dis inc solu ions only o m= 0, ..., α −1 . Fi s ly, we examine he model o α= 2, de ined by eqs. (44) and (45) displayed in ig. 8. We see ha , he p-5 L. Losano e al. po en ial is posi i e only o ˜ h≤p3/2. Then, he e a e wo kind o s a ic solu ions o he Higgs ield, one pai kink/an i-kink like solu ion be ween 0 ≤˜ h≤1/√2, and a lump-like solu ion be ween 1/√2≤˜ h≤p3/2. In Re . [21] is conside ed a model ha p esen s a cha ged lump-like solu ion. He e, he lump-like solu ion p esen s anishing cha ges and ene gy, hence we examine only he wall o 0 ≤˜ h≤1/√2. In ig. 9, we display he Higgs ield (48) and he gauge ield (39) solu ions. These walls ha e he same o al ene gy and cha ges o he walls o he s an- da d JLW model, bu wi h di e en spacial dis ibu ion o he ene gy and cha ge densi ies, as shown in ig. 10. Fig. 9: The Higgs ield (48) o m= 0 (solid line) and he ma ching gauge ield (40) (dashed line), o ˜ h(+)(x) and e A(−)(x), on he le , and o ˜ h(−)(x) and e A(+)(x), on he igh . Fig. 10: The ene gy densi y (on he le ) and he module o cha ge densi y (on he igh ), o solu ion A(−)(x) (solid line), and o A(+)(x) (dashed-line). Ending Commen s. – The examples p esen ed abo e illus a e how he de o ma ion me hod may be used o gene a e many new gene alized Abelian HCS models and hei de ec solu ions. This is achie ed wi hou e- qui ing o di ec ly sol e he nonlinea equa ions o mo ion o he new models. The me hod also allows he cons uc- ion o new de ec solu ions con olling impo an ea u es such as hei heigh , wid h o he opological cha ac e . 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