Deformation method for generalized Abelian Higgs-Chern-Simons models

arXiv:1305.4251v1 [hep-th] 18 May 2013 epl draft Deformation method for generalized Abelian Higgs-Chern-Simons models L. Losano1,2,3, J. M. C. Malbouisson1,4, D. Rubiera-Garcia5and C. dos Santos1 1Centro de F´ısica e Departamento de F´ısica e Astronomia, Faculdade de Ciˆencias da Universidade do Porto, 4169007 Porto, Portugal 2Departamento de F´ısica, Universidade Federal da Para´ıba, 58051-900 Jo˜ao Pessoa, PB, Brazil 3Departamento de F´ısica, Universidade Federal de Campina Grande, 58109-970 Campina Grande, PB, Brazil 4Instituto de F´ısica, Universidade Federal da Bahia, 40210-340 Salvador, BA, Brazil 5Departamento de F´ısica, Universidad de Oviedo, 33007 Oviedo, Asturias, Spain PACS 11.27.+d – Extended classical solutions PACS 11.15.Yc – Chern-Simons gauge theory Abstract – We present an extension of the deformation method applied to self-dual solutions of generalized Abelian Higgs-Chern-Simons models. Starting from a model defined by a potential V(|φ|) and a non-canonical kinetic term ω(|φ|)|Dµφ|2whose analytical domain wall solutions are known, we show that this method allows to obtain an uncountable number of new analytical solutions of new models defined by other functions e Vand eω. We present some examples of deformation functions leading to new families of models and their associated analytic solutions. Introduction. – Topological defects play important role in several important areas such as high energy physics [1], cosmology [2] and condensed matter physics [3]. Such defects emerge as classical solutions of nonlinear field theories which possess degenerated vacua. Typical examples are domain walls described by kink solutions of the φ4model, Ginzburg-Landau vortices and monopoles. Usually, domain walls are solutions connecting two distinct vacua of scalar field theories in one-space dimension, or in their insertions in higher dimensions, while vortices emerge as solutions of models that couple charged matter fields with gauge fields living in a (at least) 3-dimensional space-time, and monopoles lie in a 4-D space-time. In a (2+1)-dimensional space-time, minimal coupling between charged-matter and gauge fields can be implemented by the Chern-Simons (CS) action. Although the CS field can not be conceived as a free field, its coupling with matter fields imposes constraints in the dynamics which have very relevant consequences, in both, classical and quantum theories, with either relativistic or nonrelativistic kinetics. In the non-relativistic (NR) framework, particles coupled through the CS field carry both electric charge and magnetic flux, and possess fractional statistics [4]. Additionally, the NR scalar CS model constitutes a seminal example of a Galilean-invariant gaugefield theory [5]. Also, for a critical strength of a quartic self-interaction of the scalar field, which restores the scale invariance [6], this model provides a field-theoretical description of the Aharonov-Bohm (AB) scattering [7]; considering the Lorentz covariant field theory, relativistic corrections to the AB scattering are obtained [8]. Self-dual soliton solutions have been found in the relativistic, U(1)-invariant, Abelian, Higgs-Chern-Simons (HCS) gauge-theory where the symmetry-breaking potential of the Higgs field is U(ϕ)∼ |ϕ|2(|ϕ|2−v2)2[9]; vortex and domain-wall solutions have been obtained for this model [10]. This model was generalized by considering a non-canonical kinetic term for the complex scalar field, W(|ϕ|)|Dµϕ|2, providing self-dual vortex [11] and domainwall [12] solutions. Models with noncanonical kinetic terms (k-fields) find also applications in strong-interaction physics [13] and in cosmology [14]. Due to the nonlinearity, there is no general integration method to solve analytically the equations of motion of non-linear field theories; only for a small set of models, solutions of the equations of motion can be directly determined. However, for scalar fields in (1+1)- dimensions, starting from a nonlinear model with known solutions, infinitely many new models and their correp-1 L. Losano et al. sponding static solutions can be found using the deformation method [15]. This method works as follows. Choosing a deformation function f(φ), the model defined by the deformed potential e V(φ) = V[(f(φ))]/[f′(φ)]2, where f′means the derivative of f, possesses static solutions given by e φ(x) = f−1(φ(x)), where φ(x) is a solution of the static equation of motion of the original model with potential V(φ). This procedure has been applied to generate defect solutions of many models having polynomial interactions [16] and new families of sine-Gordon and multisine-Gordon models [17]. Also, an orbit-based extension of this method has been applied to models involving two interacting scalar fields [18]. The purpose of this Letter is to extend the deformation method to gauge-field models considering specifically the Abelian HCS theory, focusing particularly on the JackiwLee-Weinberg (JLK) domain-wall solution [10]. In Section II, we present the generalized Abelian HCS models and write down the first-order equations obeyed by the Bogomol’nyi-Prasad-Sommerfeld (BPS) [19] domain-wall solutions. In Section III, the deformation method is extended to domain-wall solutions of generalized Abelian HCS models and some examples are given, illustrating the power of the procedure in generating new models with their static solutions. Finally, some remarks are made. BPS domain walls in the generalized Abelian HCS model. – We consider the generalized (2 + 1)- dimensional Abelian HCS model defined by the Lagrangian density [11] LS=W(|ϕ|)|Dµϕ|2−U(|ϕ|) + κ 4ǫαβγAαFβγ,(1) where ϕis the complex Higgs field, Dµ=∂µ+ieAµis the covariant derivative and Fµν =∂µAν−∂νAµis the field strength tensor of the gauge potential Aµ. The selfinteraction potential, U(|ϕ|), is assumed to implement a symmetry-breaking mechanism and the non-canonicity of the kinetic term is engendered by the function W(|ϕ|); taking W ≡ 1, one recovers the standard Abelian HCS model. Note that, in the CS term, ǫαβγ is the fully antisymmetric tensor and the electric and the magnetic CS fields are Ei=Fi0=−˙ Ai−∇iA0and B=~ ∇× ~ A=∂2A1−∂1A2, respectively. It is convenient to work with dimensionless quantities. In (2 + 1) dimensions, the scalar field ϕhas mass dimension equal to 1/2, the same one we take for the gauge field; this choice ensures that the mass dimension of Aα agrees with the one obtained if a Maxwell term were added to LS. It follows that the electric charge eand the CS parameter κhas mass dimensions equal to 1/2 and 1, respectively, so that e2/κ is dimensionless. We can get an additional simplification if we absorb the parameters eand κby redefining space-time coordinates and fields. Thus, with Mbeing a mass scale of the model, we define ¯xµ=Me2xµ/κ,φ=√κϕ/√Me,Aµ=κAµ/Me, V=κ2U/M3e4and ω=e2W/κ; the dimensionless Lagrangian density is then given by L=κ2LS/M3e4and the action becomes S=κ e2Rd3¯xL. For a simpler notation, we suppress the bar over the space-time coordinates and use, from now on, only dimensionless quantities. Variation of the action leads to the equations of motion ωDµDµφ+∂µωDµφ−|Dµφ|2∂ω ∂φ∗+∂V ∂φ∗= 0 ,(2) 1 2ǫαβγFβγ =−Jα,(3) where the current density, Jα= (ρ,~ j), is given by Jα=iω [φ(Dαφ)∗−φ∗Dαφ].(4) The time component of eq. (3) states that the magnetic field is equal to the planar electric-charge density, B=ρ, which is the CS Gauss law. Also, for static field configurations, we find B=ρ= 2A0|φ|2ω(|φ|), Ea=ǫabjb,(5) which shows that the electric-current density is perpendicular to the electric field. The energy-momentum tensor is given by Tµν =ω[Dµφ(Dνφ)∗+Dνφ(Dµφ)∗] −gµν ω|Dαφ|2−V(|φ|)(6) from which we obtain the energy density, ε=T00, and the pressure components, P1=T11 and P2=T22. We are interested in static domain-wall solutions. Firstly, note that the complex phase of the scalar field φcan be suppressed by a suitable gauge transformation. Then, fixing the Coulomb gauge, we can search for solutions of the form [10,12] φ=h(x), Aµ=A0(x), A1= 0, A2=A(x),(7) where h(x) and A(x) are real functions and xdenotes the x1-coordinate. This ansatz corresponds to domain-walls (actually lines in the plane) parallel to the x2-axis. In this case, the static equations of motion reduces to [2ωh′]′= 2hω A2−A2 0+dV dh ,(8) A′ 0=−2ωh2A , (9) and the Gauss law A′=−2ωh2A0,(10) where the prime denotes derivation with respect to x. From eqs. (9) and (10) we infer that A0A′ 0=AA′, so that time and space components of the gauge filed are constrained by A2 0=A2−C , (11) where Cis a real constant. Also, consistency with eq. (8) imposes a relation between the function ω(h) and the potential V(h) expressed as d dh"pV/ω h#=−2ωh . (12) p-2 Deformation method for Abelian HCS models Now, the stability condition P1=P2= 0 leads to the first-order equations [20] h′=±hA (13) A′=−2ωh2A0(14) with V=h2ωA2 0.(15) For h≥0 and A≥0, the signal + (−) in eq. (13) corresponds to the kink (anti-kink) like solution for the Higgs field, h(+) (h(−)). Note that, the first-order equations (13) and (14) solve the equations of motions (2) and (3). The static solutions are physically characterized by their charge and energy. Now, returning to eq. (6), for nonnegative V(h) and ω(h), the energy of static solutions can be rewritten in the form 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 if eqs. (13), (14), and (15) are obeyed, resulting in E=Z∞ −∞ dx (4V) = A2(−∞)−A2(+∞),(17) for C= 0. In this case A2 0=A2, so the system of firstorder equations decouples and is solved simply by (13) with A(h) = −2Zωh dh +c , (18) where cis an integration constant suitable to the boundary conditions required for the gauge field. And, from (5) and (6), the electric charge, Q, and Noether charge, P, are given by Q=Z∞ −∞ dxρ(x) = A(−∞)−A(+∞),(19) P=Z∞ −∞ dx T02 =1 2A2(−∞)−A2(+∞),(20) which are both conserved due to the U(1) symmetry and the translational invariance along x2-direction, respectively. This shows that, for hin a range such that ω(h)≥0 and V(h)≥0, the BPS solutions of the first-order eqs. (13) and (14), with (15), indeed correspond to solutions of minimum energy and their energy and charge can be calculated knowing only the asymptotic behavior of the gauge field. Correspondingly, the Higgs field, for both kink and anti-kink solutions, connects two consecutive vacua of the potential, while a lump-like solution starts and terminates on the same vacuum when x→ ±∞. Standard self-dual domain walls. The simplest Abelian HCS model that supports self-dual domain wall solutions is the JLW model [10], which is defined by the Lagrangian 1 with canonical kinetic term (ω= 1) and the (dimensionless) potential V(h) = h2(1 −h2)2,(21) plotted in fig. 1. In this case, the use of eq. (18) (with Fig. 1: The potential (21) as function of the Higgs field. c= 1) provides the result A= 1 −h2,(22) which, substituting in (13), gives the solutions 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 are displayed in fig. 2. We see that the scalar field, in both cases, interpolates between the symmetric and the asymmetric vacua. fig. 3shows the energy and Fig. 2: The Higgs field (solid line) and the gauge field (dashed line), (h(+)(x), A(−)(x)) from eq. (23) on the left, and (h(−)(x), A(+)(x)) from eq. (24) on the right. electric-charge densities for both wall solutions. We find that the spatial distribution of the electric charge is symmetric around the origin, while for the energy the axis of symmetry are displaced from the origin. And, from eqs. (17), (19) and (20), for the solutions (h(+), A(−)) and (h(−), A(+)), we have the charges Q= 1, P= 1/2, and Q=−1, P=−1/2, respectively, and the same energy, E= 1. p-3 L. Losano et al. Fig. 3: Energy density of the solutions (h(+)(x), A(−)(x)) (dashed line) and (h(−)(x), A(+)(x)) (dashed-dotted line), and module of electric-charge density for both solutions (solid line). The deformation method. – Let us now develop the deformation method for generalized Abelian HCS models following the spirit of the procedure introduced for scalar fields [15]. As we shall show, by deforming simultaneously the Higgs and the CS fields, we are able to construct many new generalized HCS models and their static domain-wall solutions. The original and the deformed models are mapped into each other through the deformation function. Denote by ˜ φ(x) and e A(x) new fields whose dynamics is governed by the (dimensionless) Lagrangian density e L=eω(|˜ φ|)|Dµ˜ φ|2−e V(|˜ φ|) + 1 4ǫαβγ e Aαe Fβγ ,(25) where e V(|˜ φ|) and eω(|˜ φ|) are new functions specifying this model. As in sec. II, we assume that the self-dual BPS domain-wall solutions of this model take the form ˜ φ=˜ h(x),e Aµ=e A0(x),e A1= 0,e A2=e A(x),(26) and satisfy the first-order equations of motion ˜ h′=±˜ he A , (27) e A′=−2eω˜ h2e A0,(28) where ˜ h′≡d˜ h/dx and e A′≡de A/dx, with the constraints e V=h2eωe A2 0and e A2 0=e A2. Now, introduce the deformation function fsuch that the Higgs fields of the two models are mapped into each other, h=f(˜ h), which is assumed to be invertible (in a prescribed domain of definition) and differentiable. Also, consider that the deformed CS-gauge field is obtained from Aby the prescription e A(˜ h) = f(˜ h)A[h→f(˜ h)] ˜ hfe h ,(29) where fe h=df/deh. Then, it follows from eqs. (27) and (28), using eq. (29), that the model defined by Lagrangian density (25), with the deformed function eωand the deformed potential e Vgiven by eω(˜ h) = 1 2e A˜ h ˜ h,e V(˜ h) = ˜ h2e A2eω(˜ h),(30) where e A˜ h=de A/d˜ h, possesses static BPS solutions ˜ h(x) = f−1[h(x)] ,e A(x) = e Af−1[h(x)],(31) where h(x) is a static solution of the original model (1). It should be noted that all the considerations and relations presented in sec. II, relative to energy and conserved charges, are held unchanged for the deformed system. In the following, taking as the starting point the JLW domain-wall solutions described in sec. II.A, we consider some illustrative examples of the method. Example I. Firstly, we consider the couple of deformation function f(˜ h)(±)= (±)1−˜ h2 1 + ˜ h2,(32) which, using eqs. (22) and (29), gives e A(±)(˜ h) = f(˜ h)(±); and, from eq. (30), it follows that ˜ω=2 (1 + ˜ h2)2,e V=2˜ h2(1 −˜ h2)2 (1 + ˜ h2)4.(33) These functions, which are plotted in fig. 4, define the generalized Abelian HCS model employed in Ref. [12]. Note that, the three vacua at ˜ h= 0,1,+∞establish two walls, one between ˜ h= 0 and ˜ h= 1, and other between ˜ h= 1 and ˜ h= +∞. From the inverse of the deformation funcFig. 4: The potential (33) (top panel) and the corresponding function w(bottom panel), as function of the Higgs field. Fig. 5: The Higgs field (solid line) and the gauge field (dashed line), (˜ h(+)(x), e A(−)(x)) from eq. (34) on the left, and (˜ h(−)(x), e A(+)(x)) from eq. (35) on the right, for 0 ≤˜ h≤1 tion (32) and eqs. (23) and (24), for the range 0 ≤˜ h≤1, we obtain the solutions ˜ 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 Deformation method for Abelian HCS models while for ˜ h≥1 we have ˜ 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 figs. 5 and 6, we display these domain wall solutions. The walls for 0 ≤˜ h≤1 and ˜ h≥1 have the same gauge fields, but with the asymptotic value for x=±∞ changed. Then, for both ranges the walls have the same energy, E= 1, and charges Q= 1 and P= 1/2, for e A(−), and Q=−1 and P=−1/2, for e A(+). This makes possible to have attractive or repulsive force between the two walls. In fig. 7, we display the energy and charge densities Fig. 6: The fields (˜ h(+)(x), e A(+)(x)), eq. (36) (solid line), and (˜ h(−)(x), e A(−)(x)), eq. (37) (dashed line), for ˜ h≥1. Fig. 7: Module of electric charge for solutions A(−)(x) (dashed line) and A(+)(x) (dashed-dotted line), and energy density (solid line) for both walls. for the two walls. The comparison with the walls of the JLW model shows that, notwithstanding the walls have the same charges and energy, the JLW walls have symmetric spatial distributions of energy and charge, while here only the distribution of energy is symmetric and all the corresponding distributions are more spread out. The model defined by eqs. (33), which was obtained by deforming the JLW model, was studied in Ref. [12] but only the solution satisfying 0 ≤˜ h≤1 was considered. The deformation function (32) is a particular case of the deformation function f(˜ h) = cos[αarctan(˜ h)], corresponding to α= 2; from that new family of models can be generated for αinteger. Example II. As a second example, consider the set of deformation functions [16] fα(˜ h)] = cos[αarccos(˜ h)] = Tα(˜ h),(38) where the integer α > 2 and Tαis the Chebyshew polynomials of first kind. Using this deformation in eq. (29), with eq. (22), we have the gauge field e Aα(˜ h) = (1 −˜ h2)1/2sin[2αarccos(˜ h)] /2α˜ h , = (1 −˜ h2)U2α−1(˜ h)/2α˜ h , (39) where Uis the Chebyshew polynomials of second kind; which explicit results, for α= 2,3, are e A2(˜ h) = (1 −˜ h2)(2˜ h2−1) ,(40) e A3(˜ h) = 1 3(1 −˜ h2)(1 −2˜ h2)(3 −4˜ h2).(41) In this case, from eqs. (30) and (38), we have a family of models defined by the function eωα(˜ h) and the potential e Vα(˜ h) written in polynomial form 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 value of the parameter αspecifies a model of this family. The explicit results for α= 2,3 are 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) For these models, from the inverse of deformation function Fig. 8: The potential (45) (top panel) and the function w(44) (bottom panel), as function of ˜ h. (38), we obtain the static Higgs field solutions in the form ˜ h(±)(x) = cos h(arccos(h(±)(x)) + (m−1)π)/αi,(48) where h(±)(x) is given by eqs. (23) and (24), and mis an integer, which generates distinct solutions only for m= 0, ..., α −1 . Firstly, we examine the model for α= 2, defined by eqs. (44) and (45) displayed in fig. 8. We see that, the p-5 L. Losano et al. potential is positive only for ˜ h≤p3/2. Then, there are two kind of static solutions for the Higgs field, one pair kink/anti-kink like solution between 0 ≤˜ h≤1/√2, and a lump-like solution between 1/√2≤˜ h≤p3/2. In Ref. [21] is considered a model that presents a charged lump-like solution. Here, the lump-like solution presents vanishing charges and energy, hence we examine only the wall for 0 ≤˜ h≤1/√2. In fig. 9, we display the Higgs field (48) and the gauge field (39) solutions. These walls have the same total energy and charges of the walls of the standard JLW model, but with different spacial distribution of the energy and charge densities, as shown in fig. 10. Fig. 9: The Higgs field (48) for m= 0 (solid line) and the matching gauge field (40) (dashed line), for ˜ h(+)(x) and e A(−)(x), on the left, and for ˜ h(−)(x) and e A(+)(x), on the right. Fig. 10: The energy density (on the left) and the module of charge density (on the right), for solution A(−)(x) (solid line), and for A(+)(x) (dashed-line). Ending Comments. – The examples presented above illustrate how the deformation method may be used to generate many new generalized Abelian HCS models and their defect solutions. 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