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It is widely known that quantum aspects of the soliton solutions fourth power time derivative terms. A form of the Lagrangian ad- exhibit a special property (“fractional” spin-statistics) when the Hopf equate for quantization is obtained imposing a condition term (theta term) is included in the action of the model [24]. Since 1 (3) Π3(CP ) = Z, then such a term became the Hopf invariant and β + 2γ = 0 therefore it can be represented as a total derivative which has no in- which eliminates some unwanted terms. We shall analyze in this pa- fluence on classical equations of motion [25]. On the other hand, per some solutions of the 2+1 dimensional model (a planar case). since Π (CP 1) is trivial, the coupling constant (prefactor) Θ is not 4 In such a case the coupling constants have different physical dimen- quantized. As shown in [24], when the Hopf Lagrangian is included sions to those in the 3+1 dimensional model, i.e., M has dimension in the model, the solitons with unit topological charge acquire frac- of mass1/2 and three other coupling constants e−2,β,γ have dimen- tional spin Θ . For a fermionic model coupled with CP N field, Θ 2π sion of mass. can be determined at least perturbatively [26, 27]. N According to the previous paper [5], one can parametrize the Π3(CP ) is trivial for N > 1 and then the Hopf term is perturba- model in terms of N complex fields ui, where i = 1, ..., N. We tive, i.e., it is not a homotopy invariant. It means that the contribution assume an (N + 1)-dimensional defining representation where the from this term can be fractional even for an integer n in the anyon SU(N + 1) valued element g is of the form angle Θ = nπ. It was pointed out in [28] that an analogue of the Wess-Zumino-Witten term appears for the CP N field and it plays a 1 ∆ iu g ϑ 1+ u† u (4) similar role as the Hopf term for N = 1 [29]. Consequently, the ≡ ϑ iu† 1 ≡ · soliton can be quantized as an anyon with statistics angle Θ and also   p such Hopf-like term. and where ∆ is the Hermitian N N-matrix, × The paper [10] contains discussion of the influence of this term on ∗ uiuj † † quantum spectra for N > 1. The author has taken into account the ∆ij = ϑδij which satisfies ∆ u = u and u ∆= u . − 1+ ϑ · · field being a trivial extension of the case N = 1 (i.e., including only a single winding number). In this paper we shall give more thorough The principal variable takes the form and complete discussions of quantum spectra for N > 1 implement- † 2 IN×N 0 2 u u iu ing a set of winding numbers n1, n2, . We shall present the quan- Ψ(g)= g = + − ⊗ . (5) · · · 0 1 ϑ2 iu† 1 tum spectra within a standard semiclassical zero mode quantization  −    scheme. It has been shown recently that the model (1) possesses vortex so- The paper is organized as follows. In Sec.II we give a brief review lutions. There exists a family of exact solutions in the model with- C N of the extended Skyrme-Faddeev model on the P target space, out potentials where in addition the coupling constants satisfy the its classical solutions and their topological charges. The Hopf La- condition βe2 + γe2 = 2. The solutions satisfy the zero curvature grangian is presented in the final part of this section. In Sec.III we µ condition ∂µui∂ uj = 0 for all i, j = 1, ..., N and therefore one briefly discuss the quantization scheme and fractional spin of solitons can construct the infinite set of conserved currents. Furthermore, ac- in the model with N = 1 (baby skyrmion). Section IV contains gen- cording to numerical study there exist vortex solutions which do not eralization of the collective coordinate quantization scheme for the belong to the integrable sector. Such solutions have been found for case N = 2. In Sec.V we present the analysis of the spectrum. Fi- the potential nally in Sec.VI we generalize our formula for N ≧ 3 and we present −1 a −1 b the energy plot of the quantized system. Section VII contains sum- V = Tr(1 Ψ0 Ψ) Tr(1 Ψ∞ Ψ) (6) mary of the paper. − − with a 0,b > 0 where Ψ0 and Ψ∞ are a vacuum value of the field Ψ at≥ origin and spatial infinity respectively. The potential (6) is an analog of potentials for the baby Skyrme model. The numerical II. THE CP N EXTENDED SKYRME-FADDEEV MODEL solutions and holomorphic exact solutions corresponding with the same set of winding numbers have common boundary behavior.

N The Lagrangian (1) is invariant under the global symmetry Ψ The extended Skyrme-Faddeev model on the CP target space † Ψ , , SU(N + 1). Since we restrict the analysis to 2+1→ has been proposed in [5]. The coset space C N P = SU(N + AdimensionsB A B then ∈ it is natural to consider a diagonal subgroup. For 1)/SU(N) U(1) is an example of a symmetric space and it can this reason we transform the variable Ψ into the Hermitian one be naturally⊗ parametrized in terms of so called principal variable Ψ(g) := gσ(g)−1, with g SU(N + 1), σ being the order two au- 2 u u† iu ∈ X := IN+1×N+1 + − ⊗† (7) tomorphism under which the subgroup SU(N) U(1) is invariant ϑ2 iu 1 i.e. σ(h) = h for h SU(N) U(1). The principal⊗ coordinate  − −  Ψ(g) defined above satisfies∈ Ψ(gh⊗)=Ψ(g). Therefore we have just which in addition satisfies X−1 = X. Now the Lagrangian (1) be- one matrix Ψ(g) for each coset in SU(N + 1)/SU(N) U(1). comes ⊗ The first term of the Lagrangian (1) is quadratic in Ψ and corre- 2 N M µ 1 2 sponds with the Lagrangian of the CP model. The quartic term = Tr (∂µX∂ X)+ Tr ([ ∂µX,∂ν X ]) L 2 e2 proportional to e−2 is the Skyrme term whereas other quartic terms β µ 2 2 2 constitute the extension of standard Skyrme-Faddeev model. The + [Tr (∂µX∂ X)] + γ [Tr (∂µX ∂ν X)] µ V (X) . (8) non-Skyrme type quartic terms introduce to the Lagrangian some 2 − An analysis of zero modes of the classical solutions is much easier in approach involving a variable X and in practice enables to apply the quantization scheme. In order to explain this statement let us note and the 2+1 baby Skyrme model. The former possesses the symmetry U → that for the variable Ψ in the Lagrangian (1) the boundary conditions † AUB† while the latter only has the diagonal ones. It originates in the fact which result in Ψ∞ = Ψ∞ break partially the symmetry asso- that the chiral symmetry can only be defined for odd space dimensions. ciated with the transformationA B Ψ Ψ †. Unlike for the standard → A B 3

2 2 r 2 skyrmion, where the chiral field U goes to U∞ = I and the symme- 0 for each k = 1,...,N, where µ˜ := M2 µ and symbols Cj take try is simply broken down to = , the Ψ∞ has a nontrivial value the form which depends on winding numbers.A B Moreover, one still has to deter- 2 ω mine the pair of ( , ) for the zero-modes. On the contrary, for (8) C1 = 1+(βe 1) , − − ρ2 the symmetry transformationA B becomes diagonal, i.e., X X †, 2 2 2 (16) and then an explicit form of can be easily determined as→ expansion A A C2 = ρ + ρ (βe 1)θ, A − − in basis of the standard Gell-Mann matrices. C3 = 3iζ. It is worth it to stress that for the planar case the classical equa- tions of motion and their classical solutions have exactly the same The energy of the static solution is given by the integral form for both parametrizations. Furthermore, since the quantization 2 2 ω 3ζ 2 θω 2 procedure is based on properties of classical solutions then the re- Mcl = 2πM ρdρ θ + + (βe 1) µ˜ V . − ρ2 ρ2 − − ρ2 − sulting quantum spectra must correspond. Z   The variable X has close relation with a well-known Hermitian (17) projector P that satisfies According to discussion in [30] and also in [6] one can introduce P † = P, TrP = 1, P 2 = P . (9) two-dimensional topological charges associated with vortex config- urations. Such charges are closely related with a topological current The projector P is defined as that has the following form in terms of the principal variable

† P (V )= (10) µ i µνλ Z ⊗ Z j (X)= ǫ Tr(X∂ν X∂λX). (18) 16π where the symbol stands for the N-component complex vector Z T † = (u1,...,uN , i) /√1+ u u which depends on two vari- Since the solutions behave as holomorphic functions near the bound- ablesZ z,z∗. The form of the projector· allows to express X in the aries then the topological charges are equal to the number of poles of form ui, including those at infinity, i.e.,

0 2 X = IN+1×N+1 2P . (11) Qtop = j (X)d x = nmax + nmin (19) − | | We introduce dimensionless coordinates (t,ρ,ϕ) Z where nmax is the highest positive integer in the set ni, i = 0 1 2 x = r0t, x = r0ρ cos ϕ, x = r0ρ sin ϕ (12) 1, 2, , N and nmin is the lowest negative integer in the same set. The· · ·conserved current (18) defines a gauge potential where the length scale r0 is defined in terms of coupling constants 2 2 2 4 µ i µνλ M > 0 and e < 0 i.e. r := and the light speed is c = 1 j = ǫ ∂ν aλ (20) 0 − M 2e2 − 2π in the natural units. The linear element ds2 reads where aµ is determined up to the gauge freedom aµ aµ ∂µΛ. → − 2 2 2 2 2 2 As it was pointed out in [24], aµ is a nonlocal function of X. The ds = r0(dt dρ ρ dϕ ). − − straightforward calculation shows that aµ can be written as We shall consider the axial symmetric planar solutions −2 ν λ µ aµ = 2πi∂ [ǫµνλ∂ j ], in the gauge ∂ aµ = 0. (21) inj ϕ − uj = fj (ρ)e (13) In the alternative approach the gauge potential aµ is given in terms where the constants ni form the set of integer numbers and fi(ρ) are of the complex vector real-valued functions. Equivalently, the ansatz (13) in matrix form Z † reads iλϕ where . In order to sim- aµ = i ∂µ (22) u = f(ρ)e λ = diag(n1,...,nN ) − Z Z plify the form of some formulas below, we introduce the functions where the U(1) rotation acting on induces the gauge transforma- defined as follows Z tion on aµ. 4 2 ′T ′ ′T T ′ The “Hopf Lagrangian” is defined in terms of and it reads θ = ϑ f .f (f .f)(f .f ) , aµ − ϑ4 − h i Θ µνλ 4 2 T 2 T 2 (23) ω = ϑ f .λ .f (f .λ.f) , (14) Θ Hopf = 2 ǫ aµ∂ν aλ. − ϑ4 − L − 4π 4 h i ζ = ϑ2 f ′T .λ.f (f T .λ.f)(f ′T .f) This Lagrangian is invariant under U(1) gauge transformation and − ϑ4 − the value of the prefactor Θ is essentially undetermined. Since h i Π (CP 1) = Z then (23) is exactly the Hopf invariant for N = 1 where derivative with respect to ρ is denoted by d =′ and T stands 3 dρ and consequently it can be expressed as a total derivative. For this for matrix transposition. The classical equations of motion written in reason it does not contribute to the classical equations of motion. On dimensionless coordinates take the form N the contrary, Π3(CP ) is trivial for N > 1 and therefore the Hopf ′ T 1 ′ ′ i C3 1 2 term is not a homotopic invariant in this case. It means that the con- (1 + f .f) ρC1f + (λ.f)k C2(λ .f)k ρ k ρ ρ − ρ4 tribution from the Hopf term is always fractional even for an integer    
m in the anyon angle Θ= mπ. Note that even though the Hopf term T ′ ′ 1 T 2 C1(f .f )f C2(f .λ.f)(λ.f)k is not a total derivative anymore, it still does not affect the classical − k − ρ4   soliton solutions because it is linear in time derivative. N In the following part we quantize the model containing the La- fk δV δV +µ ˜2 (1 + f T .f)2 + f 2 = 0 (15) grangian (8) extended by the Hopf term (23) and examine the spin 4 δf 2 i δf 2 " k i=1 i # statistics of the C N solitons. X P 4

III. COLLECTIVE COORDINATE QUANTIZATION OF The expression (29) parametrized by a complex coordinate Z reads THE BABY SKYRMIONS † † X(r,A(x0)) = A(x0)(1 2Z(r)Z (r))A (x0) − † It became quite instructive to present a scheme of quantization = 1 2 A(x0)Z(r) A(x0)Z(r) (30) for the model with the CP 1 target space before going to the main − question which is a quantization of the model with the CP N target which allows us to conclude that

space. The model with N > 1 is technically more complex because it contains many fields. For this reason we shall begin presenting Z(r,A(x0)) = A(x0)Z(r) . (31) analysis of the CP 1 baby skyrmions. The full canonical quantization of the model has already been studied [15], however, in absence of Plugging (29) into the Lagrangian (27) and also (31) into the Hopf the Hopf term. We consider the collective coordinate quantization term (23) we obtain an effective Lagrangian taking into account the Hopf term and discuss the spin of the baby skyrmions. 1 Θ Leff = IabΩaΩb + ΛaΩa Mcl (32) 2 4π −

where the collective angular velocities Ωa appear in expansion of A. The model and the quantized energy † τa the operator iA ∂x0 A = 2 Ωa and where τa are Pauli matrices a = 1, 2, 3. The inertia tensor Iab is given in terms of X(r) The baby Skyrme model [31, 32] is a mimic of a hadronic Skyrme model. Its solutions (baby skyrmions) are considered as possible 4 τa τb Iab = ρdρdϕ Tr , X , X candidates for vortices or spin textures. − e2 − 2 2 The model is given in terms of a vectorial triplet ~n =(n1, n2, n3) Z    1 τa  τb   with the constraint ~n ~n = 1. Performing stereographic projection + Tr , X ,∂kX , X ,∂kX . (33) S2 on a complex plane· one can parametrize the model by a complex 8 2 2 h ih i scalar field u related to the triplet ~n by formula     We consider the well known “hedgehog” ansatz 1 ~n = (u + u∗, i(u u∗), u 2 1). (24) 1+ u 2 − − | | − ~n = (sin g(ρ) cos nϕ, sin g(ρ) sin nϕ, cos g(ρ)) | | g(0) = π, g( ) = 0.0. (34) Instead, we shall make use of another alternative parametriza- ∞ tion that is convenient for any CP N space, in particular, also for SU(2)/U(1) = CP 1 coset space. In such a case the Hermitian obtained from (24) by the following parametrization of the complex principal variable (7) X is a function of just one complex field u field u g(ρ) u = cot einϕ. (35) 1 1 u 2 2iu 2 X = − | ∗| 2 . (25) 1+ u 2 2iu u 1 The topological charge of solutions (34) takes integer values accord-  − | | −  | | ing to It can be also expressed in terms of components of the unit vector ~n ∞ 3 2 1 n ′ X = n τ3 n τ1 n τ2. (26) qtop = dρ sin gg = n. (36) − − − − 2 0 The Lagrangian of the baby Skyrme model parametrized by the Z variable X takes the form The components of the moment of inertia read 2 M µ 1 2 2 (27) I11 = I22 bS = Tr(∂µX∂ X) 2 Tr([∂µX,∂ν X] ) µ V 2 8e ∞ L − − 4π n2 sin2 g = ρdρ 2 + 2 cos2 g + + cos2 gg′2 where M 2,e2 are coupling constant of the model and V is a potential − e2 ρ2 Z0   which we shall not specify for a moment because its explicit form is (37) irrelevant for current discussion. We shall consider a model ∞ := 16π L 2 ′2 (38) bS +Θ Hopf which constitutes extension of the model (27) due I33 = 2 ρdρ sin g(1+2g ) L L 1 − e 0 to the Hopf term (23). Since Π3(CP ) = Z, the Hopf term can Z be represented as a total derivative so it does not contribute to the and Iab = 0 for a = b. Note that rotation is allowed only around the classical equations of motion. The complex coordinate Z and the 6 2 third axis because I11 = I22 = . Hermitian principal variable X in the case of the CP 1 target space ∞ Taking into account that Π (CP 1) = Z we obtain integer values are related as X := 1 2Z Z†. The topological charge is given 3 for expressions Λ that appear in the Hopf term by − ⊗ a

i 2 † 3 † † † 3 q = d xǫ Tr(X∂ X∂ X), i, j = 1, 2 . (28) Λ3 = i dρ (Z τ Z)∂ρ(Z ∂ϕZ) (Z ∂ϕZ)∂ρ(Z τ Z) top 16π ij i j − − Z Z   = n (39) Note that expressions (27) and (28) are invariant under rotation − realized by a unitary matrix A according to transformation X AXA†. The standard procedure proceeds by promoting the parame-→ ter A to the status of dynamical variable A(x0). Then the dynamical ansatz adopted in collective coordinate quantization reads 2 Our results are essentially equivalent with (19) of Ref.[15] up to constant. However, their formulation (20) (corresponding to our (37)) was incorrect † X(r; A(x0)) = A(x0)X(r)A (x0). (29) about sign of the coefficient. 5

and Λ1 = Λ2 = 0. The (body-fixed) isospin operator J3 can be in- IV. COLLECTIVE COORDINATE QUANTIZATION OF troduced as a symmetry transformation generator via Noether’s the- THE CP 2 MODEL orem

τ3 i ϑ3 It has been already mentioned that the Lagrangian density (1) is A Ae 2 , → invariant under transformation Ψ Ψ † where , are some nΘ → A B A B J3 = I33Ω3 + . (40) unitary constant matrices. This symmetry could remain also for the − 4π Lagrangian density written in new variable X = CΨ where C is a The Legendre transform of the Lagrangian leads to the following diagonal constant matrix. However, for X = X† only the diagonal expression for the Hamiltonian symmetry = is allowed. Moreover, the topological charge (19) is invariantA underB such transformation only for . It leads 2 = g33 nΘ to the conclusion that for a model which supportsA topologicaB l soliton Heff = Mcl + J3 . (41) 2 − 4π solutions the only allowed symmetry is a diagonal one X X †.   In fact there is another restriction on , namely for the→ asymptotic A A where g33 is inverse of the moments of inertia I33 i.e. g33 := 1/I33. A ∂ field X∞ it must hold If one represents the isospin operator J3 := i ∂α as acting on the ba- −ikα † sis ℓ e 0 , with k being an integer or a half-integer numbers, X∞ = X∞, (48) then| i≡ the energy| eigenvaluei is given by the expression A A otherwise the moments of inertia corresponding to the modes di- 2 g33 nΘ verge. Note that expression satisfying (48) depends on numbers E = Mcl + k . (42) A 2 − 4π (n1, n2) because these numbers determine the form of X∞. The vor-   tex solutions are symmetric under exchange of n1, n2. It is enough to One can substitute the quantum number k by an integer-valued index study configurations with n1 > n2 where the cases n1 > 0, n1 < 0 i.e., ℓ 2k what gives are treated separately. ≡ 2 In analogy to baby skyrmions we shall adopt following dynamical g˜33 nΘ E = Mcl + ℓ , ℓ Z (43) ansatz in collective coordinate quantization 2 − 2π ∈   † X(r; (x0)) = (x0)X(r) (x0). (49) where g˜33 = g33/4. This is a familiar result: for nΘ = 0 or in A A A general (even number) π, the angular momentum is integer then one Substituting (49) into the Lagrangian (8) one obtains a Lagrangian × † gets boson, while for nΘ = π or (odd number) π, the angular which depends on the collective angular velocity operator i ∂x0 . momentum is half integer then one gets fermion. × Such an operator possesses expansion on the set of collectivAe coordi-A nates which appear in the resulting effective Hamiltonian.

B. The fermionic effective model and the anyon angle A. The case n1 > 0 It is well known that for a fermionic effective model coupled to a baby skyrmion with a constant gap m the integrating out the Dirac In this case the asymptotic value of the principal variable X is field leads to effective Lagrangian containing a kind of baby Skyrme X∞ = diag( 1, 1, 1). The generators Fa ,a = 1, 2, 3, 8 of the model and some topological terms including the Hopf term [26, 27]. symmetry (48)− have the form { } The Euclidean path integral of the partition function, which enables λ6 λ7 1 us to examine the topological term after integrating out the Dirac F1 := , F2 := , F3 := (λ3 √3λ8) field, is of the form 2 2 − 4 − 1 1 F8 := λ3 + λ8 . 3 − 2 √ Γ(X,Aµ)= ψ ψ¯ exp d xψiDψ¯ (44) 3 D D   Z Z  They satisfy the commutation relations where the U(1) gauged Dirac operator reads [Fa, Fb]= iǫabcFc, [Fa, F8] = 0 a,b,c = 1, 2, 3 (50) µ iD := iγ (∂µ iAµ) mX. (45) − − what shows that the symmetry (48) is in fact a residual symmetry A number of articles extensively describe the derivative expansion of SU(2) U(1). The rotation matrix is parametrized by four Euler × A the effective action Seff that appears in Γ := exp(Seff ). It contains angles ϑi, (i = 1, 2, 3, 8) in the following way both the action of the model (in the real part) and the topological terms (in the imaginary part). After a bit lengthy calculation (see = e−iF3ϑ1 e−iF2ϑ2 e−iF3ϑ3 e−iF8ϑ8 . (51) Appendix A) one gets A The angular velocities Ωa of the collective coordinates became the † m 3 µ 3 expansion coefficients of the operator i ∂x in a basis of gener- ReSeff = | | d xTr(∂µX∂ X)+ O(∂X ), (46) 0 2π ators F of the residual symmetry. TheA expansionA takes the form Z a 3 µ † ImSeff = d x j (X)Aµ π sgn(m) Hopf (X) . (47) i ∂x = FaΩa. (52) − L A 0 A Z   The explicit form of the current jµ coincides with (18). Conse- The effective Lagrangian contains a term quadratic in Ωa which quently, as pointed out in [26, 27, 29], the anyon angle Θ is deter- comes from the Skyrme-Faddeev part of the total Lagrangian and a minable in this fermionic context. It means that the soliton became a term linear in Ωa having origin in the Hopf Lagrangian fermion for odd topological charges and a boson for even topological 1 Θ charges. Leff = IabΩaΩb + ΛaΩa Mcl. (53) 2 4π − 6

Figure 1: The finite components of inertia tensor (54) of the holomorphic solutions for the topological charge Qtop = 3, i.e., (n1, n2) =(3, 1)) in unit of ( 4/e2). −

Figure 2: The finite components of inertia vector (56) of the holomorphic solutions for the topological charge Qtop = 3, i.e., (n1, n2) =(3, 1)) in unit of ( 4/e2). −

where the symmetric inertia tensor Iab is given as the integral of the that expressions containing the Hermitian principal variable X(r) and (r, (x0)) = (x0) (r) . (55) they read Z A A Z 4 The inertial vector Λa has the following form Iab = 2 ρdρdϕ Tr [Fa, X] [Fb, X] e † † Z    Λa = 2i dρ Fa ∂ρ ∂ϕ − Z Z Z Z + Tr [[Fa, X] ,∂kX] [[Fb, X] ,∂kX] Z n  †   † 2  ∂ϕ ∂ρ Fa . (56) βe − Z Z Z Z + Tr [Fa, X][Fb, X] Tr(∂kX∂kX)    o 2 Explicit form of components of (54) and (56) obtained after imposing    (13) is presented in Appendix B. Tr [Fa, X]∂kX Tr [Fb, X]∂kX . (54) − In virtue of axial symmetry imposed by (13) the effective La-     grangian (53) contains the following relevant terms The symmetry of components I38 = I83 and equality I11 = I22 originate in the axial symmetry imposed in the ansatz (13). 1 2 2 2 2 Leff = I11(Ω1 + Ω2)+ I33Ω3 + 2I38Ω3Ω8 + I88Ω8 In generality the Hopf term in the Lagrangian (53) is nonlocal in 2 fields X. However, if we translate the field into using transfor-  Θ  + Λ3Ω3 + Λ8Ω8 Mcl. (57) mation (11) it has a local form. From dynamical ansatzZ (49) we find 4π { }− 7

One can construct the explicit form of operators that satisfy (63)

∂ ∂ cos ϑ1 ∂ 1 = i cos ϑ1 cot ϑ2 + sin ϑ1 , I ∂ϑ1 ∂ϑ2 − sin ϑ2 ∂ϑ3   ∂ ∂ sin ϑ1 ∂ 2 = i sin ϑ1 cot ϑ2 cos ϑ1 , I ∂ϑ1 − ∂ϑ2 − sin ϑ2 ∂ϑ3   ∂ ∂ 3 = i , 8 = i , I − ∂ϑ1 I − ∂ϑ8

cos ϑ3 ∂ ∂ ∂ 1 = i sin ϑ3 cot ϑ2 cos ϑ3 , K − sin ϑ2 ∂ϑ1 − ∂ϑ2 − ∂ϑ3   sin ϑ3 ∂ ∂ ∂ 2 = i + cos ϑ3 cot ϑ2 sin ϑ3 , K sin ϑ2 ∂ϑ1 ∂ϑ2 − ∂ϑ3   ∂ ∂ Figure 3: The quantum correction of the energy eigenvalue (69) of 3 = i , 8 = i , K ∂ϑ3 K ∂ϑ8 the holomorphic solutions for the topological charge Qtop = 3, 2 ∂ 1 ∂ i.e.,(n1, n2) = (2, 1)) in unit of ( e /4). The quantum numbers = in2 i (2n1 n2) . (64) are (l,k,Y )=(1, 1−, 1) and the anyon− angle Θ= π. J ∂ϑ3 − 2 − ∂ϑ8 The SU(2) Casimir operator

3 3 I2 := 2 = 2 =: K2, (65) Ii Ki The Lagrangian (53) possesses several global continuous sym- i=1 i=1 X X metries that lead to corresponding conserved Noether currents as well as the generators 3, 8, 3, 8 and are diagonalizable. a, a, , namely I I K K J I K J The Lagrangian (53) is a function of the four Euler angles ϑi and their time derivatives ϑ˙i ∂x ϑ, i.e. eff = eff (ϑi, ϑ˙ i). The ≡ 0 L L Legendre transform of the Lagrangian eff leads to the Hamiltonian L (ϑi,πi) := πiϑ˙i eff which depends on the Euler angles and L H −L −iFaξa ˙ (i) left SUL(2) : e , the canonical momenta πi := ∂ eff /∂ϑi. The Hamiltonian takes A→ A L Θ the form a = IbcΩbRac + Λa, (58) I 4π g11 2 2 R Heff = Mcl + 1 + 2 ii iFaξa 2 K K ( ) right SUR(2) : e , 2 2 A → A g33 Θ
g88 Θ Θ + 3 + Λ3 + 8 + Λ8 a = IabΩb Λa, (59) 2 K 4π 2 K 4π K − − 4π     (iii) the circular symmetry X(ρ, ϕ) X(ρ, ϕ + ϕ0), Θ Θ → + g38 3 + Λ3 8 + Λ8 (66) ¯ K 4π K 4π iF ϕ0 1 or e , F¯ := n2F3 (2n1 n2)F4, (60)    A → A − 2 − where we have introduced the components of inverse of inertia tensor 1 = n2 3 (2n1 n2) 4, (61) g whose explicit form in the current case is given by J K − 2 − K ab

1 I88 g11 := , g33 := 2 , I11 I33I88 I − 38 I38 I33 where the symbol Rab is defined by g38 := − 2 , g88 := 2 . I33I88 I I33I88 I − 38 − 38 The diagonalization problem can be solved using the standard Wigner function (for example, [33]) † λa = λbRba. (62) l −iY ϑ8 lmk; Y = (ϑ1,ϑ2,ϑ3)e 0 (67) A A | i Dm,k | i 1 2 where l,m,k are integer/half-integer and Y has 3 , 3 , . Then the Hamiltonian eigenvalues read · · · where stands for the Gell-Mann matrices. Here, and are λa a a g11 2 I K E = Mcl + l(l + 1) k called the coordinate-fixed isospin and the body-fixed isospin respec- 2 − tively. is a generator of the spatial rotation around the third axis. 2 2 J g33  Θ g 88 Θ They act on as + k + Λ + Y + Λ A 2 4π 3 2 4π 8     Θ Θ + g k + Λ Y + Λ . (68) 38 4π 3 4π 8    [ a, ]= Fa , [ a, ]= Fa I A − A K A A (63) It is convenient to express all the quantum numbers in terms of in- [ , ]= F¯ . teger numbers, i.e., l′ 2l,k′ 2k and Y ′ 3Y . It leads to the J A A ≡ ≡ ≡ 8

V. THE ANALYSIS

1.0

g We shall examine quantum spectra of the model (8) in the inte-

1 grable sector βe2 + γe2 = 2 where do exist a class of holomorphic

0.8 g 2 vortex solutions. We choose

2 2 0.6 βe = 4,γe = 2. (74)

4.9 4.5 −

4.3

4.7 4.1 in order to satisfy the condition (3) simultaneously with the previous

4.02 0.4

4.5 4.9 one. In the integrable sector the model possesses exact solutions

4.3

4.7

4.1 nj inj ϕ 0.2 2 uj = cj ρ e (75) e = 4.02

where cj are arbitrary scale parameters. The lowest nontrivial vortex

0.0 configurations with topological charge (19) taking the value Qtop =

0.0 0.2 0.4 0.6 0.8 1.0 2 are given by (n1, n2)=(2, 1), (1, 1). Note that the first term of y the inertia tensor (54) −

Nlσ 4 Figure 4: The classical solutions with the topological charge Qtop = I = ρdρdϕTr [F , X] [F , X] (76) ab e2 a b 3, i.e.,(n1, n2) =(3, 1)). Z   is logarithmically divergent. It means that it has no proper quantum numbers unless one employ a suitable regularization scheme. A sim- following expression for the energy ilar situation has been identified for baby-skyrmions in an antiferro- g˜11 ′ ′ ′2 magnetism [13]. In such a case the moment of inertia corresponding E = Mcl + l (l + 2) k 2 − to the solution with the winding number n = 1 diverges and be- 2 2 cause of it no quantized states emerge. The authors have introduced g˜33 ′  Θ g˜88 ′ 3Θ + k + Λ3 + Y + Λ8 a regularization term in order to get the finite value of the integral. 2 2π 2 4π     Here we shall study some configurations of vortices with finite ′ Θ ′ 3Θ moments of inertia characterized by Qtop = 3. The values of +g ˜38 k + Λ3 Y + Λ8 (69) 2π 4π the moments of inertia (54) for holomorphic solutions are shown in    Fig.1. The moments are shown in dependence on dilatation param- where g˜11 g11/4, g˜33 g33/4, g˜38 g38/6, g˜88 g88/9 (in 2 ≡ ≡ ′ ′ ′ ≡′ ≡ eters (c1, c2) defined by (75). The unit scale is given by 4/e . further analysis we shall omit of k ,l ,Y for simplicity. ) − The derivative expansion of partition function obtained for baby- Similarly, the finite components of inertia vector (56) of the holo- morphic solutions have been shown in Fig.2. Note that for c2 0 skyrmions can be straightforwardly applied to the model with N = → 2. In such a case the Dirac operator is of the form the component becomes trivial then the value of the vector becomes topological, i.e., Λ3 Qtop, what is consistent with the analyt- µ → − i := iγ (∂µ iAµ) mX (70) ical calculation for the baby-skyrmions (39). In Fig.3 we plot the D − − and real and imaginary parts of the effective action read dimensionless quantum energy correction ∆E corresponding to the moments of inertia shown in Fig.1. 2 2 m 3 µ 3 For coupling parameters such that the condition βe + γe = 2 ReSeff = | | d xTr(∂µX∂ X)+ O(∂X ), (71) 2π does not hold the solution is no longer holomorphic. In such a case Z 3 µ the numerical analysis is required in order to compute the quantum ImSeff = d x j (X)Aµ π sgn(m) Hopf (X) . (72) − L corrections. The numerical analysis for the classical solutions of (15) Z   has been extensively studied in [7]. Here we shall employ the poten- In analogy to the previous section, one can fix the anyon angle Θ as tial Θ π sgn(m) provided that the vortices are coupled with fermionic ≡ N 2 2 field. However, since Π3(CP ) is trivial, the Hopf term itself Hopf (1 + f ) L V = 2 . (77) is perturbative and the value of the integral depends on the back- (1 + f 2 + f 2)2 ground classical solutions. Consequently, one could not expect that 1 2 this value became an integer number. As a result, the solitons are The corresponding classical solutions for several values of βe2 are always anyons even if Θ= nπ, n Z. shown in Fig.4. In Fig.5 we present the corresponding energies and ∈ values of the isoscalar root mean square radius

1/2 B. Case: n1 < 0 4 ρ2 ρ2j0(ρ)dρ . (78) h i≡ − M 2e2   Z  In this case the asymptotic value of the Hermitian principal vari- p The components of the moment of inertia tensors Iab and also the able X at spatial infinity reads X∞ = diag(1, 1, 1). It follows that − inertia vectors Λa are shown in Fig.6. When N > 1 there are several four generators of the symmetry Fa ,a = 1, 2, 3, 8 can be chosen as { } excitation modes which are labeled by the quantum numbers l,k,Y . In Fig.7 we plot the excitation modes in dependence on l,k with λ1 λ2 λ3 λ8 F1 := , F2 := , F3 := , F8 := . (73) fixed Y = 0 (“l-mode”), and also in dependence on Y with fixed 2 2 2 √3 (l,k)=(0, 0) (“Y -mode”). 2 The quantum Hamiltonian can be diagonalized in the same way as The dimensionless classical energy M˜ cl Mcl/8πM is topo- ≡ for n1 > 0 what leads to the energy spectrum (69). logical for the holomorphic solutions, i.e., it equals to the topological 9

The root mean square radius The classical energy

40

3.6

3.4

35 3.2

3.0

2.8

30 1/2

>

2.6 2 <

2.4

25 2.2

2.0

1.8

20

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

2 2

e e

Figure 5: The energies (in unit of 4M 2) and the root mean square radius ρ2 (in unit of 4/M 2e2) of the classical solutions with the topological charge Q = 3, i.e., (n , n )=(3, 1). h i − top 1 2 p

charge Qtop. Clearly, for nonholomorphic solutions the energy devi- estimated as Mcl 10.4 eV and ≃ ates from Qtop, however, it still can be useful to introduce a coupling 2 2 strength for the quantum correction e 1 e . 2 α := 4 8πM2 = 32πM2 e ′′ 2 2 − × − ∆E 20.5 keV “Y mode (Y = 1, Θ=0), (81) The coupling constants M ,e are some free model parameters, − 4 ≃ however, their values must be determined by underlying physics. 38.1 keV “l mode′′ (l = 1,k = 0, Θ=0).(82) In order to get some rough idea about properties of quantum ex- ≃ citations, it might be instructive to estimate value of the quantum The coupling constant α takes the approximate value α 2.8 106. excitations for a given energy scale. For hadronic scale analysis ≃ × 2 Note that presented above estimations were performed in order to one usually fixes the coefficient of the second order terms fπ in the 4 get some rough idea about the order of magnitude of excitations. For standard Skyrme model as being equal to the pion decay constant more definite analysis one needs further inputs derived from under- f 64.5MeV. Similarly, we put the coupling constant M 2 of the π lying physics. extended≃ Skyrme-Faddeev model as being equal to the pion decay 2 2 constant M fπ 10 MeV and we also employ the isoscalar charge density∼ of the≃ nucleon, i.e., ρ2 0.7 fm in (78). For instance, in the case of our numerical solutionh i ≃ with βe2 = 4.1 one can easily estimate e2/4 3.6 p103 MeV. For such a choice of VI. GENERALIZATION TO HIGHER N − ≃ × parameters the classical energy is about Mcl 10.4 GeV and the ≃ quantum corrections read Though it seems to be certainly involved, a generalization of our scheme to higher N ≧ 3 is straightforward. In order to simplify the 2 formulation we restrict consideration the case of anyon angle Θ = 0. e ′′ ∆E 10.4 MeV “Y mode (Y = 1, Θ=0), (79) We also assume that is the highest positive integer in the sequence − 4 ≃ n1 ′′ ni, i = 1, 2, , N. We present below results for the case N = 3. 19.3 MeV “l mode (l = 1,k = 0, Θ=0)(80). · · · (3) ≃ The generators Fa read { } The coupling constant takes the value α 1.4. Unfortunately, no ≃ (3) 1 0 0 (3) 1 0 0 (3) 1 0 0 physical candidate for such small energy excitation for the nucleon F1 = , F2 = , F3 = , 2 0 λ1 2 0 λ2 2 0 λ3 are known. One has to stress that the result was obtained as a crude       estimation. Moreover, the present model is only a two-dimensional (3) 1 0 0 (3) 1 0 0 F4 = , F5 = , mimic of the realistic 3+1 Skyrme model. 2 0 λ4 2 0 λ5     Another example of such estimation can be done for an antifer- (3) 1 0 0 (3) 1 0 0 F6 = , F7 = , romagnetic material. A simple estimation of a different type of 2 0 λ6 2 0 λ7 quantum correction in the case of a continuum limit of a Heisen-     (3) 1 0 0 (3) 1 3 0 berg model was demonstrated in [11]. In the continuum Heisenberg F8 = , F15 = − . (83) 2 √ 0 λ8 4 0 I3×3 model, the parameter M can be assigned as the exchange coupling 3     constant or a a spin stiffness. In antiferromagnetic La2CuO4 the spin wave velocity c is of the order ~c 0.04 eV nm and the lattice Following the procedure presented for the C 2 case, we substitute ≥ P constant is of the order a 0.5 nm. The exchange coupling con- the dynamical ansatz (49) into Lagrangian (8) and expand the opera- 2 ≃ stant is roughly M ~c/a 0.1 eV. The soliton size ρ2 is † (3) tor i ∂x in basis of generators Fa responsible for the size∼ of the excitation≃ then it may be estimatesh i as A 0 A { } ρ2 a 0.5 nm. In the case of our numerical solutionp with h2 i ∼ ≃ 2 6 † (3) βe = 4.1 we have e /4 7.1 10 eV. The classical energy is i ∂x = F Ωa. (84) p − ≃ × A 0 A a 10

600 650

I 550 I

600

33 11

500

550

450

500

400

450

350

400

300

250 350

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

2 2

e e

750

340

700

I I

38 650 88

320 600

550

500

300

450

400

280

350

300

260 250

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

2

2

e e

0.4 3.4

0.0

8 3 3.2

-0.4

3.0

-0.8

2.8

-1.2

2.6

-1.6

-2.0 2.4

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

2

2 e

e

Figure 6: Components of the tensor and the vector of inertia corresponding to the solutions of Fig.4.

The effective Lagrangian takes the form where k1 = k3,k4 = (2k3 + 3k8)/4 and k6 = (2k3 3k8)/4. The second order Casimir operator of the SU(3) group can− be expressed Iaa 2 2 Leff = Ω + Ω as 2 a a+1 a=1,4,6 2 2 X
4k3 + 3k8 Iab C2(SU(3)) = l1(l1+1)+l4(l4+1)+l6(l6+1) . (88) + ΩaΩb Mcl. (85) − 8 2 − a=3,8,15 X b=3X,8,15 The generalization to an arbitrary N is almost straightforward. We The Legendre transformation of the Lagrangian leads to the can define the SU(N) U(1) generators similarly to (83) using a × N Hamiltonian higher N generalization of Gell-Mann matrices λa which are the gaa 2 2 standard SU(N) generators. We take the diagonal components of Heff = a + a+1 2 K K SU(N) part (counterparts of F3 and F8) a=1,4,6 X
gab 1 0 0 + a b + Mcl (86) (N) (89) 2 K K Fa2−1 = N a = 2, 3, , N , a=3,8,15 a 0 λ 2 · · · X b=3X,8,15  a −1  where a are coordinate-fixed isospin operators and gab stand for and the off diagonal ones, like F and F , in the form K 1 4 components of inverse of the inertia tensor Iab. The energy spectrum is obtained after diagonalization of the Hamiltonian (86) in base of (N) 1 0 0 Fb = N , states belonging to a SU(3) U(1) irrep and it reads 2 0 λb ×   gaa 2 2 2 Erot = la(la + 1) k where b are integer numbers from 1 to N 1 excluding a 1. The 2 − a − − a=1,4,6 last U(1) generator is defined as X  g + ab k k , (87) 2 a b (N) 1 N 0 a=3,8,15 b=3,8,15 F 2 = − . (90) N −2N N + 1 0 IN×N X X   11

` ` l = 1 mode'' excitations ` ` l = 0 mode'' excitations

0.016 0.035

0.014 = k = -1 0.030

=

k = 0

0.012 =

0.025 k = +1

0.010

0.020

k = -1, +1 E 0.008 E =

k = 0

0.015

0.006

0.004 0.010

0.002

0.005

0.000

0.000

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

2 2

e e

` ` l = 2 mode'' exciations ``Y mode" excitations

0.070 0.070

k = -2

Y = 2

0.060 0.060 k = -1 =

Y = 1

k = 0 =

0.050 0.050

k = +1

Y = 2

k = +2 =

0.040 0.040

Y = 1

k = -2, +2 E E =

k = -1, +1

0.030 0.030

k = 0

0.020 0.020

0.010 0.010

0.000

0.000

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

2

2

e e

Figure 7: The excitation energies corresponding to the solutions of Fig.4. We plot the l = 0, 1, 2 modes with Y = 0 and for the Y mode we fix (l,k)=(0, 0).

After lengthy calculation involving generators one gets the quantum The energy formula (91) contains the first N(N 1)/2 terms of ξ − { } energy formula and the N terms of ηi , e.g. { } N i−1 for N = 2 : ξ = 1, ηi = 3, 8 gξξ 2 Erot = lξ(lξ + 1) k for N = 3 : ξ = 1, 4, 6 ηi = 3, 8, 15. 2 − ξ i=1 j=1 X X  One can check that for this values expression (91) certainly reduces N N gη η to (68) for N = 2 (and Θ = 0) and to (87) for N = 3. + i j k k (91) 2 ηi ηj i=1 j=1 X X 2 VII. SUMMARY where ξ = (i 1) + 2(j 1) and ηi = i(i + 2). Symbols lξ and − − kηi represent independent quantum numbers whereas kξ are given in The present paper aims at the problem of quantum spectra of terms of kηi according to solutions in the extended CP N Skyrme-Faddeev model. In order

i to obtain the quantum energy spectra of excitations we applied the 1 1 method of collective coordinate quantization based on a rigid body kξ = kη + kη + kη (92) 2 − (j−1) h h i  h=j approximation. Further, within this approximation, we discussed X spin statistics of the CP N soliton taking into account the Hopf   Lagrangian. According to discussion presented in previous papers with k0 = 0. Note that ξ and ηi describe the numerical sequence [24, 25, 28, 29], for N = 1 the Hopf Lagrangian is topological so the solitons are quantized as anyons with the angle . On the other ξ : 1, 4, 6, 9, 11, 13, 16, 18, 20, 22, (93) Θ { } · · · hand, for N > 1, the Hopf term is perturbative thus the solutions ηi : 3, 8, 15, 24 . (94) { } · · · became always anyons. 12

A fermionic effective model coupled with the skyrmion of a con- [34] with the CP 1 principal variable (25). The partition function is stant gap mX has been examined [26, 27]. After integrating out the given by the integral Dirac field the resulting effective Lagrangian contains the Skyrme- Faddeev model plus some topological terms. For N = 1, the anyon angle Θ became fixed, then in contrary to the previous papers, the solitons cannot be anyon at all, whereas for N > 1, the Θ became fixed again but since in this case the Hopf term is perturbative then the solution became an anyon. The further part contains the study of excitation energy of the so- lutions. The excited energy for N = 1 is described by (a third com- Θ ponent of) angular momentum S := ℓ . According to [10], 3 ¯ − 2π Γ= ψ ψe¯ R d xψiDψ = det iD (A1) for Θ = π the baby skyrmions are fermions and the ground state is D D twice degenerated. For N > 1 the situation is quite different. The Z solutions are always anyon type because of the fact that the Hopf term is no longer topological. It follows that there are no degener- acy. The excitations are parametrized by three numbers l,k,Y , so ΘΛ3 3ΘΛ8 S3 := k + 2π ,S8 := Y + 4π are components of the anyonic angular momenta. The paper contain plots of some energy levels in dependence on the model parameters βe2. The presented values of the energy are dimensionless. We gave some rough estimations of typical energy scales being where ψ and ψ¯ are Dirac fields, Aµ is a U(1) gauge field and iD = of order of few dozens MeV for a hadronic scale and of order of few i∂/ + A/ mX. The gamma matrices are defined as γµ = iσµ. dozens keV for a condensed matter scale. The energy excitation com- − − The effective action Seff = ln det iD can be split in its real and pared with the classical solution energy is subtle for a hadronic scale imaginary part whereas it is virtually too big in the case of the condensed matter ex- ample. Such discrepancy can be understand to some extent. For in- 2 stance, if we choose a solution parameters Qtop = 3 and βe = 4.1 2 2 then estimation of the classical energy gives Mcl 1.1 10 M and for the quantum excitation energy of the lowest≃ Y mode× it has the value e2/4 ∆E 5.5 102/(M 2 ρ2 ). The ratio of this two energies− is given× by ≃ × h i

2 e ∆E 1 4 (95) 1 † 1 iD − 5 4 2 ReSeff = ln det D D, ImSeff = ln det . (A2) Mcl ≃ × M ρ 2 2i iD† h i − where dimensions of M 2, ρ2 are [eV, nm] or [MeV, fm]. One can easily see that a huge discrepancyh i of the classical/quantum en- ergy for the antiferromagneticp material is fixed almost by a value of the spin wave velocity. However, some systems may support dif- ferent values of the parameters M 2 and ρ2 . The systems with higher values of coupling constant M 2 (whichh determinesi the energy scale) and with a large characteristic excitationp size ρ2 can sup- h i port the existence of excitations whose energy is comparable with the For Aµ 0 one can easily see that classical energy. We shall leave this problem for thep future. → One has to bear in mind that the collective coordinate quantization is an approximated method. An alternative method which can be ap- plied to quantization of the vortex systems is a canonical quantization method. Such approach would be more suitable in full understanding of the quantum aspects of the model. The work is in progress and its results will be reported in a subsequent paper [35].

Acknowledgement D†D = ∂2 + m2 + im∂X,/ − The authors would like to thank L. A. Ferreira for discussions and DD† = ∂2 + m2 im∂X/ . (A3) comments. We thank to A. Nakamula and K. Toda for their useful − − comments. Also we are grateful to Y. Fukumoto for giving us many valuable information.

Appendix A: The derivative expansion of the fermionic model

The method of perturbation for the partition function (44) is quite common and widely examined for the analysis of the spin-statistics For the variation D D + δD,D† D† + δD†, the real part of → → of solitons coupled with fermions. Here we employ notation used in the effective action in Aµ 0 is → 13

1 1 † 1 † the calculation is almost similar and the first nonzero component con- δReSeff = Sp D δD + DδD 2 D†D DD† tains product of the three derivatives   1 d3k = d3x e−ik·x 3 sgn(m) 3 µνλ 2 (2π) δImSeff = d xǫ Tr(∂µX∂ν X∂λXXδX) . (A7) Z Z − 32π 2 2 −1 2 Z Tr ( ∂ + m + im∂X/ ) (im∂δX/ + m XδX) × − †  In terms of new variable aµ := iZ ∂µZ the last formula can be − +( ∂2 + m2 im∂X/ )−1( im∂δX/ + m2XδX) eik·x written as − − −  sgn(m) 1 d3k δImS(3) = d3xǫµνλδa ∂ a . (A8) = d3x eff 2π µ ν λ 2 (2π)3 Z Z Z 2 2 2 −1 2 As it was argued in [26], the term itself should be zero, i.e., Tr (k + m 2ik∂ ∂ + im∂X/ ) (im∂δX/ + m XδX) × − − (3) in the pertubative calculation because the homotopy  ImSeff = 0 2 2 2 −1 2 2 Z +(k + m 2ik∂ ∂ im∂X/ ) ( im∂δX/ + m XδX) group π3(S ) = is nontrivial. In order to find the form for − − − − N = 1, we generalize the model into N ≧ 2 such as Z = (A4) T N (z1,z2, 0, , 0) . In the case when the homotopy π3(CP ) is · · · where Sp stands for a full trace containing a functional and also a trivial one gets matrix trace involving the flavor and the spinor indices and Tr stands (3) sgn(m) 3 µνλ for usual matrix trace. Expanding the above expression in powers of ImS = d xǫ aµ∂ν aλ . (A9) 2 eff 4π 2ik∂ + ∂ and∂X / one gets the lowest nonzero term Z

(2) m 3 For N > 1, the above manipulation is not a trick and we directly δS = | | d xTr(∂X/ ∂δX/ ) . (A5) † Re 8π obtain the form (A9) with aµ := i ∂µ . Z − Z Z For Aµ = 0 we also have the two derivative component After taking the spinor trace and switching to the Minkowski metric 6 one gets the action (46). (2) 1 3 µνλ For the variation of the imaginary part δImS Aµ=06 = δAµ d xǫ Tr(∂ν X∂λXX) .(A10) eff | − 16πi Z 1 1 † 1 † δImSeff = Sp D δD DδD (A6) Both contribute to the final expression (47). 2i D†D − DD†   14

2 Appendix B: Inertia tensor Iab and inertia vector Λa for CP

In this appendix we give the explicit form of the relevant components of inertia tensor Iab (54) and inertia vector Λa (56) written explicitly ′ df1 ′ df2 by the radial profile functions f1(ρ), f2(ρ) and their derivatives f1 := dρ , f2 := dρ . They read

2 2 2 4 8π 1+ f1 + f1 f2 + f2 I11 = 2 ρdρ 2 2 2 e − (1 + f1 + f2 ) Z "
2 2 4 2 4 6 ′ 2 2 2 2 4 ′ ′ 4 1+ f1 1 + 8f2 + f2 + f2 + f2 + f2 f 8f1f2 4+ f1 f2 + 4 3f2 + f2 f f 1 − 1 2 + 2 2 4 2 2 4  (1 + f1 +
f2 ) −  (1 + f1 +
f2 ) 2 4 4 2 2 2 2 ′ 2 4 2 2 2 2 2 2 2 2 4 f2 + 2 f1 + f2 3f2 + 4 f1 + 2 f2 1 f 4 f2 (n1 n2) f1 + 2n + 2n 1+ f1 f2 + n f1 − − 2 − 2 2 1 + 2 2 4 + 2 2 2 3 
(1 + f1 + f2
)
 r (1 + f
1 + f2 )
2 2 2 4 2 ′ 2 ′ ′ 2 ′ 2 2 1+ f1 + f1 f2 + f2 1+ f2 f 2f1f2f f + 1+ f1 f 2 1 − 1 2 2 + 2βe 2 2 4 (−
(1 + f1 +
f2 )
2 2 2 4 2 2 2 2 2 2 2 2 2 2 2 2 2 1+ f1 + f1 f2 + f2 n f1 +(n1 n2) f1 f2 + n f2 f2 2n1f1 n2 1+ f1 f2 1 − 2 − − (B1) 2 2 2 4 + 2 2 2 4 , − r (1
 + f1 + f2 )  r (1 + f1 + f2 )
)# 2 2 2 2 2 4 2 4 ′ 2 8π 1+ f2 f1 + 4f2 16 f1 1 f2 f2 + f2 + f2 f1 − − − I33 = 2 ρdρ 2 2 2 + 2 2 4 e − (1 + f1 + f2 ) (1 + f1 + f2 ) Z " 

2 2 2 ′ ′ 2 4 2 4 2 ′ 2 2 2 2 16f1f2 f1 2f1 + 5 f2 + 3 f1f2 4 f1 + f1 + 16 + 17f1 + 4f1 f2 f2 4 (2n n ) f f − 1 2 1 2 + 2 2 4 + 2 2 4 + 2 − 2 2 3  (1 + f1 + f2
)  (1 + f1 + f2 )
r (1 + f1 + f2 ) 2 2 2 2 ′ 2 ′ ′ 2 ′ 2 1+ f2 f1 + 4f2 1+ f2 f 2f1f2f f + 1+ f1 f 2 1 − 1 2 2 + 4βe 2 2 4 (− 
(1 + f1
+ f2 )
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1+ f2 f1 + 4f2 n 1+ f2 f1 2n1n2f1 f2 + n 1+ f1 f2 n1f1 1 f2 + n2 2+ f1 f2 1 − 2 − 2 2 2 4 + 4 , − r (1 + f1 + f2 ) 2 2 2 


 r (1 +
f1 +f2 )
)# (B2) 2 2 2 4 ′ 2 2 2 2 ′ ′ 16π f1 1 f2 16f1 1 f2 f1 8f1f2 f1 4f1 + 3 f2 3 f1f2 − − − − I38 = 2 ρdρ 2 2 2 2 2 4 + 2 2 4 e (1 + f1 + f2 ) − (1 + f1 + f2 ) (1 + f1 + f2 ) Z "


2 2 2 2 ′ 2 2 2 4f1 1+ 1 4f2 f1 7f2 f2 4 (2n n ) n f f − − 1 2 2 1 2 2 2 4 + 2 − 2 2 3 −  (1 + f1 +
f2 ) r (1 + f1 + f2 ) 2 2 2 ′ 2 ′ ′ 2 ′ 2 1 f2 f1 1+ f2 f 2f1f2f f + 1+ f1 f 2 − 1 − 1 2 2 + 4βe 2 2 4 (
 (1
+ f1 + f2 )
2 2 2 2 2 2 2 2 2 2 f1 1 f2 n 1+ f2 f1 2n1n2f1 f2 + n 1+ f1 f2 − 1 − 2 + 2 2 2 4
 r
(1 + f1 + f2 )
2 2 2 2 2 2 2 f1 n1f1 1 f2 + n2 f1 + 2 f2 n1 1+ f2 n2f2 − − (B3) 2 2 2 4 , −
r (1 + f1 +
f2
)

)# 2 2 2 3 ′ ′ 2 2 2 ′ 2 32π f1 1+ f2 32 1+ f2 f1 f2f1f2 16 1+ f2 f1 f1 I88 = 2 ρdρ 2 2 2 2 2 4 + 2 2 4 e − (1 + f1 + f2 ) − (1 + f1 + f2 ) (1 + f1 + f2 ) Z "


2 2 2 2 2 ′ 2 2 2 2 4 1+ f1 1+ f2 + 4f1 f2 f1 f2 4n2f1 f2 + 2 2 4 + 2 2 2 3 (1 + f1 + f2 )
r (1 + f1 + f2 ) 2 2 2 ′ 2 ′ ′ 2 ′ 2 f1 1+ f2 1+ f2 f 2f1f2f f + 1+ f1 f 2 1 − 1 2 2 + 4βe 2 2 4 (−
 (1
+ f1 + f2 )
2 2 2 2 2 2 2 2 2 2 4 2 2 2 f1 1+ f2 n 1+ f2 f1 2n1n2f1 f2 + n f1 + 1 f2 f1 n1 1+ f2 n2f2 1 − 2 − (B4) 2 2 2 4 + 2 2 2 4 , −
 r
(1 + f1 + f2 )
r (1 + f1 +
f2 ) )# and n f f ′ + n f f ′ + n f f (f f ′ f f ′ ) 1 1 1 2 2 2 1 1 2 1 2 2 1 (B5) Λ3 = 2 dρ 2 2 2 − , − (1 + f1 + f2 ) Z 2 ′ 2 ′ ′ ′ n1 1+ f2 f1f + n2 1 2f1 f2f f1f2 (n1f1f 2n2f2f ) Λ = 4 dρ 1 − 2 − 2 − 1 . (B6) 8 3(1+ f 2 + f 2) 2 Z
1
2 15

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