aa r X i v : . [ h e p - t h ] A p r Prepared for submission to JHEP
Exact Self-Dual Skyrmions
L.A. Ferreira † and Ya. Shnir ‡ ⋆ § † Instituto de F´ısica de S˜ao Carlos; IFSC/USP; Universidade de S˜ao Paulo, USP; Caixa Postal369, CEP 13560-970, S˜ao Carlos-SP, Brazil ‡ BLTP, JINR, Dubna 141980, Moscow Region, Russia ⋆ Department of Theoretical Physics and Astrophysics, BSU, Minsk 220004, Belarus § Department of Theoretical Physics, Tomsk State Pedagogical University, Russia
Abstract:
We introduce a Skyrme type model with the target space being the sphere S and with an action possessing, as usual, quadratic and quartic terms in field derivatives.The novel character of the model is that the strength of the couplings of those two termsare allowed to depend upon the space-time coordinates. The model should therefore beinterpreted as an effective theory, such that those couplings correspond in fact to low ener-gy expectation values of fields belonging to a more fundamental theory at high energies.The theory possesses a self-dual sector that saturates the Bogomolny bound leading toan energy depending linearly on the topological charge. The self-duality equations areconformally invariant in three space dimensions leading to a toroidal ansatz and exactself-dual Skyrmion solutions. Those solutions are labelled by two integers and, despitetheir toroidal character, the energy density is spherically symmetric when those integersare equal and oblate or prolate otherwise. ontents R
65 Conclusions 11A Toroidal Coordinates 12
Self-dual field configurations possess very nice physical and mathematical properties, andthey are important in the study of non-linear aspects of field theories possessing topologicalsolitons. The best known examples are the instanton solutions of the Yang-Mills theoryin four dimensional Euclidean space [1] and the self-dual Bogomol’nyi-Prasad-Sommerfield(BPS) monopoles in the 3+1 dimensional Yang-Mills-Higgs theory [2, 3]. The self-dualsolitons satisfy first order differential equations which yields the absolute minimum of theenergy, and by construction they are also solutions of the full dynamical system of the fieldequations. Another feature of the self-dual field configurations is that the correspondingtopological solitons always saturate the topological bound, their static energy (or the Eu-clidean action in the case of the Yang-Mills instantons) depends linearly on the topologicalcharge. Moreover, there are very elegant mathematical methods of construction of vari-ous multi-soliton configurations in these models, the Nahm equation [4] and the algebraicAtiyah-Hitchin-Drinfeld-Manin scheme [5].However the usual Skyrme model [6],[7], which can be suggested as an effective low-energy theory of pions, do not support self-dual equations [8], the mass of the solitonsolutions for this model, the Skyrmions, is always above the topological lower bound ina given topological sector [9], even though in compact spaces it is possible to saturate abound [10, 11]. As a consequence, there is no exact mathematical scheme of constructionof multi-soliton solutions of the Skyrme model, the only way to obtain these solutions inany topological sector, is to implement various numerical methods, some of them are rathersophisticated, they usually need a large amount of computational power.– 1 –ecently some modification of the Skyrme model was proposed to construct the solitonsolutions which satisfy the first-order Bogomol’nyi-type equation [12–14]. In the first casethe conventional Skyrme model was drastically changed via replacement of the usual sigmamodel term and the quartic Skyrme term with a term sextic in first derivatives and apotential [12, 13]. In the second case the usual Skyrme model is coupled to the infinitetower of vector mesons [14]. These self-dual models are directly related to the usual Skyrmemodel since they can be considered as submodels of a general model of that type. Further, itwas shown very recently that the standard Skyrme model without the potential term can beexpressed as a sum of two BPS submodels with different solutions [15]. The correspondingsubmodels, however, are not directly related to the generalized Skyrme model of any type.Another modification of the Skyrme model, which supports self-dual solutions and hasan exact BPS bound, was suggested in [11]. Similar to the usual Skyrme model, or itsgeneralizations, the field of the new model is a map from compactified coordinate space S to the SU (2) group space. The corresponding first order equations are equivalent tothe so-called force free equation well known in solar and plasma physics, see e.g. [16].The drawback of this construction is that, due to an argument by Chandrasekhar [17],these equations does not possess finite energy solutions on R , although it supports regularsolutions on three sphere S [11].In this paper we propose generalization of the self-dual Skyrme-type model discussedin [11], which possess the regular solution on R . Similar to the usual Skyrme model, weconsider a nonlinear scalar sigma-model, parameterized by two complex scalar fields Z a , a = 1 ,
2, satisfying the constraint Z ∗ a Z a = 1. The action of the model is given by S = Z d x (cid:18) m ( x ρ )2 A µ − e ( x ρ ) H µν (cid:19) (1.1)with two couplings m ( x ρ ) and e ( x ρ ), dependent upon the space-time coordinates, whichare of dimension of mass and dimensionless, respectively. In addition, µ , ν = 0 , , ,
3, and A µ = i Z ∗ a ∂ µ Z a − Z a ∂ µ Z ∗ a ) and H µν = ∂ µ A ν − ∂ ν A µ . (1.2)The model (1.1) is similar to the one considered in [11], with the main difference beingthe fact that the coupling constants now are allowed to depend upon the space-time co-ordinates. That plays a crucial role in the properties of the model. In the first place, itcircumvents the famous Chandrasekhar’s argument [17] that prevents the existence of finiteenergy solutions extending over the whole R space. In addition, as we explain below, itrenders the self-duality equations conformally invariant in the three dimensional space R .As it is usual in many effective field theories, coupling constants that depend upon thespace-time coordinates correspond in fact to low energy expectation values of fields be-longing to a more fundamental theory at higher energies. At the end of the paper we shalldiscuss some possibilities for the introduction of a dilation type field that could account– 2 –or the space-time dependent coupling constants appearing in (1.1). We now discuss theproperties of the self-dual sector of the theory (1.1). It will be convenient for our purposes to represent the corresponding static energy functionalvia the dual of H ij , defined as B i = 12 ε ijk H jk , i, j, k = 1 , , E = 12 Z d x (cid:18) m ( ~r ) A i + 1 e ( ~r ) B i (cid:19) (2.2)In order to have finite energy solutions the fields Z a , a = 1 ,
2, have to approach fixedconstant values at spatial infinity, and so as long as topological arguments are concerned,we can compactify the physical space R to S . Thus the field of the model (1.1) becomesa map Z a : S → S . The mapping is labeled by the topological invariant Q = π ( S ),which is the winding number of the field configuration, and it can be calculated by thefollowing integral Q = 112 π Z d x ε abcd ε ijk Φ a ∂ i Φ b ∂ j Φ c ∂ k Φ d = 14 π Z d x A i B i , (2.3)where we have written Z ≡ Φ + i Φ , Z ≡ Φ + i Φ , and a, b, c, d = 1 , , , Q in terms of the vectors A i and B i defined in (1.2) and (2.1), respectively. Evidently, this structure reminds the Hopfinvariant used in the theories with the target space being S , like in the Skyrme-Faddeevmodel [18]. However, our target space is still S and we are not projecting the map downto S as is the case of the first Hopf map.Next we follow the arguments presented in [19]. Let us denote by χ α , α = 1 , , S . The topological charge Q given in (2.3) isinvariant under infinitesimal smooth (homotopic) deformations of the fields δχ α , and so,without the use of the equations of motion, one finds that δQ = 0. Since the variations arearbitrary one gets from (2.3) that the vectors A i and B i have to satisfy B i δA i δχ α − ∂ j (cid:18) B i δA i δ∂ j χ α (cid:19) + A i δB i δχ α − ∂ j (cid:18) A i δB i δ∂ j χ α (cid:19) = 0 . (2.4)On the other hand, the static Euler-Lagrange equations associated to (1.1) are given by m ( ~r ) A i δA i δχ α − ∂ j (cid:18) m ( ~r ) A i δA i δ∂ j χ α (cid:19) + 1 e ( ~r ) B i δB i δχ α − ∂ j (cid:18) e ( ~r ) B i δB i δ∂ j χ α (cid:19) = 0 . (2.5)– 3 –f one now imposes the self-duality equation m ( ~r ) A i = ± e ( ~r ) B i (2.6)one gets that (2.4) becomes ± m ( ~r ) e ( ~r ) A i δA i δχ α − ∂ j (cid:18) ± m ( ~r ) e ( ~r ) A i δA i δ∂ j χ α (cid:19) ± m ( ~r ) e ( ~r ) B i δB i δχ α − ∂ j (cid:18) ± m ( ~r ) e ( ~r ) B i δB i δ∂ j χ α (cid:19) = 0 . (2.7)If, in addition, we impose that m ( ~r ) = m f ( ~r ) e ( ~r ) = e f ( ~r ) (2.8)with m and e being constants, then (2.7) becomes the same as (2.5). The conclusion isthat, for the choice (2.8), the self-duality equation (2.6) implies (2.5), when the identity(2.4), coming from the topological charge, is used. Therefore, using (2.8), the static energy(2.2) becomes E = 12 Z d x (cid:20)(cid:18) m f A i + 1 e f B i (cid:19)(cid:21) (2.9)which can be written as E = 12 Z d x (cid:18) m f A i ∓ e f B i (cid:19) ± m e Z d x A i B i (2.10)Therefore the lower energy bound is E ≥ π m e | Q | (2.11)which is saturated for the solutions of the self-duality equation m e f A i = ± B i (2.12)For such self-dual field configurations we have E = m Z d x f A i = 1 e Z d x B i f (2.13)Note that if we treat f as an independent field then the static Euler-Lagrange equationcoming from (2.9) is given by m f A i = 1 e f B i (2.14)which certainly follows from the self-duality equation (2.12). Therefore, the first orderself-duality equations (2.12) imply all the static second order Euler-Lagrange equationsassociated to the theory (1.1), when the coupling constants have the form given in (2.8).As we shall see, the dilaton function f ( r ) can regularize solutions of the self-duality equationproviding a way to evade the usual arguments [11, 17] that there can be no finite energysolutions of the force free equation. – 4 – Conformal symmetry of the model
Remarkably, the the self-dual sector of the model (1.1) is invariant under conformal trans-formations in three space dimensions. In order to see it, we will follow the approach of [20]and consider a general infinitesimal space transformations of the form δx i = ζ i , such that δZ a = 0; δ∂ i Z a = − ∂ i ζ j ∂ j Z a . (3.1)Therefore δA i = − ∂ i ζ j A j δH ij = − ∂ i ζ k H kj − ∂ j ζ k H ik , (3.2)and δB i = − ε ijk ∂ j ζ l H lk = − ∂ j ζ l ε ijk ε lkm B m = ∂ j ζ i B j − ∂ j ζ j B i (3.3)Let us consider how the self-duality equations (2.12) change under such transformations.It is convenient to write them in the formΛ i ≡ λ f A i − B i = 0 λ = η m e η = ± , (3.4)and so δ Λ i =2 δff λ f A i − ∂ i ζ j λ f A j − ∂ j ζ i B j + ∂ j ζ j B i = (cid:20) δff δ ij − ( ∂ i ζ j + ∂ j ζ i ) + ∂ l ζ l δ ij (cid:21) λ f A j . (3.5)Hence, in order to remain invariant with respect to the transformations (3.1), the variationsof the space coordinates ζ i must satisfy ∂ i ζ j + ∂ j ζ i = 2 D δ ij (3.6)for some function D . Therefore, δ Λ i = (cid:20) δff + D (cid:21) λ f A i (3.7)and the self-duality equation (2.12) remains invariant if δf = − D f (3.8)As is was shown in [20], the transformations satisfying (3.6) are actually the conformaltransformations. Indeed, we have that the possibilities are ζ ( P j ) i = ε ( P j ) δ ij D ( P j ) = 0 (translations) ζ R jk i = ε ( R jk ) ( δ ki x j − δ ji x k ) j = k D ( R ij ) = 0 (rotations) ζ ( d ) i = ε ( d ) x i D ( d ) = ε ( d ) (dilatations) ζ ( c j ) i = ε ( c j ) (cid:0) x i x j − x l δ ij (cid:1) D ( c j ) = ε ( c j ) x j (special conf.) (3.9)– 5 –herefore the self-duality equations (2.12) are invariant under conformal transformationsin three dimensional space. Note that f is a scalar field under translations and rotationsbut not under dilatations and special conformal transformations. Further, one can checkthat δA i = − D A i ; δB i = − D B i ; δ ( A i B i ) = − D A i B i (3.10)and the volume element transforms as δ ( d x ) = 3 D d x (3.11)Hence both the static energy functional (2.9) and the topological charge (2.3) are confor-mally invariant. R Here we again follow the reasonings of [20] to construct an ansatz for our self-dualityequations, which is invariant under the diagonal subgroup of two commuting U (1)’s in theconformal group and other two commuting U (1)’s in the internal symmetry group of themodel (1.1). Note that the model (1.1) is invariant under the U (2) global transformations Z Z ! → U Z Z ! U ∈ U (2) (4.1)The Cartan subgroup of the U (2) includes two commuting U (1) elements, namely Z → e iα Z Z → Z (4.2)and Z → Z Z → e iβ Z (4.3)In addition we also have, in the conformal group in three dimensions, two commuting U (1)elements, which correspond to the vector fields V ζ = ζ i ∂ i with (see [20]) ∂ ϕ ≡ V ϕ = x ∂ − x ∂ (4.4) ∂ ξ ≡ V ξ = x a ( x ∂ + x ∂ ) + 12 a (cid:0) a + x − x − x (cid:1) ∂ (4.5)with a being a length scale factor, and where we have introduced two angles, ϕ and ξ ,such that the vectors fields, V ϕ and V ξ , generate rotations along those angular directions.Note, that V ϕ is the generator of rotations on the plane x x . On the other hand, V ξ isa linear combination of the special conformal generator V ( c ) = x x i ∂ i − x l ∂ , and thetranslation generator V ( P ) = ∂ (see (3.9)). One can easily check that indeed [ ∂ ϕ , ∂ ξ ] = 0.The third curvilinear coordinate in R which is orthogonal to ϕ and ξ is z = 4 a ( x + x )( x + x + x + a ) (4.6)– 6 –ne can check that indeed ∂ ϕ z = ∂ ξ z = 0. It turns out that ( z , ξ , ϕ ) constitute thetoroidal coordinates in R defined as x = ap √ z cos ϕ ; x = ap √ z sin ϕ ; x = ap √ − z sin ξ (4.7)where p = 1 − √ − z cos ξ ≤ z ≤ ≤ ϕ , ξ ≤ π (4.8)We now want field configurations that are invariant under the diagonal subgroup of thetensor product of the internal U (1) defined in (4.2) and the external U (1) generated by ∂ ϕ given in (4.4). In addition we want those same field configurations to be invariant underthe diagonal subgroup of the tensor product of the internal U (1) defined in (4.3) and theexternal U (1) generated by ∂ ξ given in (4.5). That brings us to the toroidal ansatz definedby Z = p F ( z ) e i n ϕ Z = p − F ( z ) e i m ξ (4.9)where m and n are two integers, to keep the configuration single valued in R .In order to proceed it is convenient to write the self-duality equation in terms of vectorcalculus notation, and so we have that (2.12) can be written as ~ ∇ ∧ ~A = η m e f ( ~r ) ~A ; η = ± . (4.10)Writing ~A in terms of the unit vectors of the toroidal coordinates as ~A = V z h z ~e z + V ξ h ξ ~e ξ + V ϕ h ϕ ~e ϕ (see (A.5)), we have that V ζ = i ( Z ∗ a ∂ ζ Z a − Z a ∂ ζ Z ∗ a ), with ζ ≡ z, ξ, ϕ , and where the scalingfactors h ζ are defined in (A.2). Therefore, (4.10) can be written in components as κ f V z = p z (1 − z ) ∂ ξ V ϕ κ f V ξ = − − z ) p ∂ z V ϕ κ f V ϕ = 2 z p [ ∂ z V ξ − ∂ ξ V z ] (4.11)where we have introduced the dimensionless quantity κ ≡ η m e a , with η = ± ∂ ξ F = 0 κ m f p (1 − F ) = − − z ) n ∂ z Fκ n f p F = − z m ∂ z F (4.12)Now we can eliminate the derivative ∂ z F from this system, it yields a simple algebraicsolution of the self-duality equations (4.12) for any values of the integers m and nF = m zm z + n (1 − z ) f = 2 pm e a | m n | [ m z + n (1 − z )] (4.13) We have replaced the usual toroidal coordinate η by z , these coordinates are related as z = tanh η ,with η > – 7 –here, to keep f real, we had to choose the sign of ( m n ) to be related to the sign η of theself-duality as η = − sign ( m n ). Thus, the explicit form of the solution for the self-dualmodel (1.1) on R is Z = s m zm z + n (1 − z ) e i n ϕ Z = s n (1 − z ) m z + n (1 − z ) e i m ξ f = s | m n | (cid:0) − √ − z cos ξ (cid:1) m e a [ m z + n (1 − z )] (4.14)The vector field ~A takes the following form when evaluated on the solutions (4.14) ~A = − m n p/am z + n (1 − z ) (cid:2) ~e ξ n √ − z + ~e ϕ m √ z (cid:3) (4.15)and so A i = m n p /a m z + n (1 − z ) (4.16)Note that ~A is tangent to the toroidal surfaces defined by z = constant. On the circle onthe x x plane defined by z = 1 (see the appendix A), one has that ~A circle = − n ~e ϕ /a . Atspatial infinity, where z = 0 and ξ = 0, one has that ~A infinity = 0. On the x -axis, where z = 0, one has ~A x − axis = − ( m/a )(1 − cos ξ ) ~e ξ .Evaluating the static energy (2.13) on the solutions (4.14), we get E = m Z d x f ~A = 4 π m e | m n | m n Z dz m z + n (1 − z )] (4.17)Using the fact that Z dz m z + n (1 − z )] = 1 m n one gets E = 4 π m e | m n | (4.18)Further, using (2.12) into the definition of the topological charge (2.3), we get that thesolutions (4.14) have topological charges given by Q = 14 π Z d x A i B i = η m e π Z d x f A i = − m n (4.19)where we have used that the sign η = ±
1, in the self-duality equation (2.12) is related to m n as η = − sign ( m n ) (see below (4.13)). Thus, these field configurations exactly saturatethe topological bound (2.11) for any values of n, m .Note that the solutions (4.14) are very similar to those exact solutions constructedin [21], and possessing in fact the same topological charges. The model in [21] however,– 8 –s defined on target space S and it does not possess a self-dual sector, even though itpresents conformal symmetry in three space dimensions.If one of the integers, m or n vanishes, the solutions (4.14) become trivial with f = 0everywhere in space. Note that in the particular case where n = ± m the general solution(4.14) is reduced to Z = √ z e i n ϕ , Z = √ − z e ± i n ξ , f = r m e a (1 − p (1 − z ) cos ξ ) (4.20)Clearly, the field Z a in (4.20) resembles the form of the solution of the model (1.1) on S ,constructed in [11], however the angular variables in the latter case are related with theangular coordinates on the three-sphere and the function f , which is a regulator on R ,does not appear there.Remind that in the toroidal coordinates (4.7) the spacial infinity corresponds to z = 0, ξ = 0 and the origin corresponds to z = 0, ξ = π . Evidently, for the solutions (4.14) wehave the asymptotic behavior Z ( r → ∞ ) = 0 , Z ( r → ∞ ) = 1 , f ( r → ∞ ) = 0 (4.21)and Z ( r →
0) = 0 , Z ( r →
0) = − f ( r →
0) = 2 √ m e a r(cid:12)(cid:12)(cid:12) mn (cid:12)(cid:12)(cid:12) (4.22)which agrees with the topological boundary conditions imposed on the field Z a . The generalsolution is axially symmetric, and on the x -axis, corresponding to z = 0, we have Z (0 , , x ) = 0 , Z (0 , , x ) = e i n ξ , f (0 , , x ) = s | m || n | − cos ξ ) m e a (4.23)thus, the solutions are regular everywhere in space.In Figs. 1-2 we show the function f in terms of spherical coordinates r, θ . For n = m the solutions possess spherical symmetry. For n = m , the configuration becomes axiallysymmetric, it oblate for n > m and it is prolate for n < m .The solutions (4.14) can be written in the spherical coordinates in a more transparentform. Indeed, using the expressions (A.1), we can write the energy density of the generalsolution (4.14) as E = 16 m e a | mn | (( r/a ) + 1)[4( ρ/a ) ( m − n ) + n (( r/a ) + 1) ] . (4.24)If m = n , the configuration becomes spherically symmetric, then E = 16 m e a n (( r/a ) + 1) (4.25)– 9 – igure 1 : The function f ( r, θ )) for the solutions (4.14) of the model (1.1) at a = 1, m = 1, e = 1, for n = 1 , m = 1 (left plot), n = 1 , m = 4 (middle plot), and n = 4 , m = 1 (right plot) f r n=1,m=1n=1,m=2n=1,m=4n=2,m=1n=4,m=1 Figure 2 : The function f ( r, θ = π/ n, m , at a = 1, m = 1, e = 1. Note that in both cases the energy density decays as 1 /r as r → ∞ . In addition, itscales as 1 /a , and so, the total energy is scale invariant and that is a consequence of theconformal invariance of the model.In Fig. 3 we display the energy density iso-surfaces (see (4.24)) for the cases ( n =1 , m = 4), ( n = 2 , m = 2), and ( n = 4 , m = 1). Note that all these configurations have thesame total energy.Finally, let us note that the solutions for the Z a fields given in (4.14) do not depend– 10 – igure 3 : The isosurfaces of the energy density of the n = 1 , m = 4 (left plot), n = 2 , m = 2(middle plot), and n = 4 , m = 1 (right plot) solutions of the model (1.1) at a = 1, m = 1, e = 1. on the arbitrary scale parameter a , as one should expect since they scalar under conformaltransformations (see (3.1)). The function f however scales as 1 / √ a , and that is a conse-quence of the fact it is not a scalar under dilatations and special conformal transformations,see (3.8). Thus, similar to the self-dual soliton solution of the non-linear O (3) sigma modelin 2+1 dimensions [22] the instanton solution of the Yang-Mills theory in Euclidian four-dimensional space [1], and the exact Hopfions constructed in [21] those field configurations(except for f ) are scale invariant. The main purpose of this work was to construct exact analytical and regular self-dualsolutions of a Skyrme theory with target space S . The crucial ingredient that madethat possible was the conformal symmetry of the self-duality equations in three spacedimensions. On its turn, such symmetry was possible due to the fact that the strengthsof the couplings of the quadratic and quartic terms in the action have a space dependenceencoded in a quantity f . The physical nature of such quantity is still to be understood,but it is quite natural to relate it to low energy expectation values of fields of a morefundamental theory in higher energies that would contain our Skyrme model as a lowenergy effective theory. Note from (3.8) and (3.10) that the quantity f transforms underthe conformal group, in the same way as ( A i B i ) / , i.e. a fractional power of the topologicalcharge density. In fact, f and ( A i B i ) / differ by a multiplicative constant, when evaluatedon the solutions (4.14) for the case m = n , i.e the solutions with spherically symmetricenergy densities. Such a fact could perhaps be a hint on how one could try to extend ourmodel by a scalar dilation type field or even vector fields.Certainly our results open the way for further investigations on the properties of theproposed Skyrme model, and perhaps on its possible physical applications. Of course, it– 11 –ould be interesting to study how the conformal symmetry could be broken leading toscale dependent solutions and bringing a physical scale to the theory. The introduction ofa potential or even of the dilation field mentioned above are some of the possibilities. Itwould also be important to investigate the rotational modes of the solutions and their semi-classical quantization. Rotating solutions not only would break the conformal symmetrybut also would split the energy degeneracies of our self-dual spectrum. We hope to reporton those issues elsewhere. A Toroidal Coordinates
Here we give some useful formulas related to the toroidal coordinates (4.7), and that areneeded for the explicit calculations leading to the exact solutions (4.14). Inverting therelations (4.7) one gets that z = 4 a (cid:0) x + x (cid:1)(cid:0) x + x + x + a (cid:1) ; ξ = ArcTan " a x (cid:0) x + x + x − a (cid:1) ; ϕ = ArcTan (cid:18) x x (cid:19) (A.1)The metric in toroidal coordinates is ds = h z dz + h ξ dξ + h ϕ dϕ , with scaling factorsbeing h z = ap p z (1 − z ) h ξ = ap √ − z h ϕ = ap √ z (A.2)The volume element is then dx dx dx = 12 a p dz dξ dϕ (A.3)Note that r = x + x + x = a (cid:0) √ − z cos ξ (cid:1)(cid:0) − √ − z cos ξ (cid:1) (A.4)Therefore the spatial infinity corresponds to z = 0, and ξ = 0 (or 2 π ). The x -axiscorresponds to z = 0, for 0 < ξ < π . The origin corresponds to z = 0, and ξ = π . Inaddition, z = 1 corresponds to the circle x + x = a , and x = 0.The unit vectors are defined as ~e ζ = h ζ d ~rd ζ , for ζ ≡ z , ξ , ϕ , and so we have that ~e z = 1 p (cid:2) ( √ − z − cos ξ ) cos ϕ ~e + ( √ − z − cos ξ ) sin ϕ ~e − √ z sin ξ ~e (cid:3) ~e ξ = − p (cid:2) √ z cos ϕ sin ξ ~e + √ z sin ϕ sin ξ ~e + ( √ − z − cos ξ ) ~e (cid:3) (A.5) ~e ϕ = − sin ϕ ~e + cos ϕ ~e where ~e i , i = 1 , ,
3, are the unit vectors in Cartesian coordinates.– 12 – cknowledgements:
The authors are grateful to Profs. Wojtek Zakrzewski and NobuyukiSawado for valuable discussions. LAF is partially supported by CNPq-Brazil. YS thanksIlya Perapechka for relevant discussion. YS is grateful to Funda¸c˜ao de Apoio `a Pesquisado Estado de S˜ao Paulo, FAPESP, for the financial support under the grant 2015/25779-6,he also gratefully acknowledges support from the Russian Foundation for Basic Research(Grant No. 16-52-12012), the Ministry of Education and Science of Russian Federation,project No 3.1386.2017, and DFG (Grant LE 838/12-2). YS would like to thank the Insti-tuto de F´ısica de S˜ao Carlos for its kind hospitality.
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