- Email: [email protected]
Infinite symmetries in the Skyrme model
5 April 2001
Physics Letters B 504 (2001) 195–200 www.elsevier.nl/locate/npe
Infinite symmetries in the Skyrme model L.A. Ferreira a , J. Sánchez Guillén b a Instituto de Física Teórica — IFT/UNESP, Rua Pamplona 145, 01405-900, São Paulo, SP, Brazil b Departamento de Física de Partículas, Facultad de Física, Universidad de Santiago, E-15706 Santiago de Compostela, Spain
Received 30 November 2000; accepted 14 December 2000 Editor: L. Alvarez-Gaumé
Abstract We show that the Skyrme theory possesses a submodel with an infinite number of local conserved currents. The constraints leading to the submodel explore a decomposition of SU(2) with a complex field parametrizing the symmetric space SU(2)/U (1) and a real field in the direction of U (1). We demonstrate that the Skyrmions of topological charges ±1 belong to such integrable sector of the theory. Our results open ways to the development of exact methods, compensating for the non-existence of a BPS type sector in the Skyrme theory. 2001 Published by Elsevier Science B.V. PACS: 12.39.Dc; 11.10.Lm; 05.45.Yv; 11.27.+d
Skyrme introduced in the sixties a very original idea which nowadays is widely used in physics and mathematics, that the classical soliton solutions of a given field theory may correspond to particles in its quantum spectrum. He proposed a theory, now called Skyrme model [1], to describe the low energy interactions of hadrons, where the nucleons are interpreted as topological solitons. The model found applications not only in nuclear physics, but also in particle and condensed matter physics [2–4]. The theory received a boost in the eighties when it was shown to correspond indeed to the low energy limit of QCD for large Nc [5,6]. The soliton solutions have been calculated by numerical methods [7–9] and there is now a vast amount of evidence that Skyrme’s first spherical ansatz (the hedgehog) does lead to the solutions of unit topological charge. The generalizations of that ansatz based on rational maps lead to good approxi-
E-mail address: [email protected] (J. Sánchez Guillén).
mations of the minimal energy solutions [8,9]. Such ansatzes are inspired in the BPS monopoles by sharing its symmetries and instanton methods. But unlike the well understood instantons, one easily shows that in three space dimensions skyrmions cannot saturate self-duality bounds. The non-existence of a BPS type sector is perhaps one of the factors preventing the development of exact methods for the Skyrme model. In this Letter we propose a way of addressing such problems by showing that there exist a submodel in the Skyrme theory possessing an infinite number of local conserved currents. They form in fact a continuous set of currents since they are parametrized by special functionals of the fields. The submodel is constructed by imposing Lorentz invariant constraints following the methods of [10] for studying integrable theories in any dimension. The constraints explore a suitable decomposition of the SU(2) group elements where the Skyrme’s fields are a complex scalar field u parametrizing the symmetric space SU(2)/U (1) and a real scalar field ζ parametrizing the U (1) subgroup. It
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turns out that, in the static case, particular solutions to the constraints are obtained by taking ζ to depend only upon the radial variable r, and u to be a meromorphic function of a complex variable z, built out of the two angles of the spherical polar coordinates. The original Skyrme hedgehog ansatzes for topological charges ±1 fall into such class of configurations and so solve our constraints. We proof that they solve indeed our submodel equations of motion. The candidates for higher charges solutions were constructed numerically in [8], and it would be very interesting to verify if they satisfy our constraint. We point out that the existence of such submodel containing the Skyrme’s hedgehog and possessing an infinite number of conservation laws is highly non-trivial. It became possible now, about forty years after Skyrme’s original paper, due to the recent developments on the study of integrable theories in higher dimensions [10]. Another interesting point is that the generalizations to higher charges of the original Skyrme’s hedgehog obtained through the rational map ansatzes are particular solutions of our constraints. We show that such configurations can not be solutions of Skyrme’s model. However, since they provide approximations to the numerical candidates to true solutions [9], our constraints may give hints on how to correct them to obtain of a new ansatzes for higher charges solutions. In any case, our results establish an interesting connection between soliton solutions and infinite number of conservation laws in 3+1 field theory of great interest in physics. Similar results have already been found for the Skyrme–Faddeev model [11] and other related (3 + 1)-dimensional theories defined on S 2 [12], as well as in theories defined on homogenous spaces like CPN and chiral models [13–16]. The Lagrangian density for the Skyrme model, with the pion mass set to zero, can be written as [1–3]: 1 f2 L = π tr U † ∂µ U U † ∂ µ U 4 2 1 tr U † ∂µ U, U † ∂ν U , − (1) 2 32e where fπ and e are phenomenological constants, and U is a SU(2) unitary matrix. The equations of motion 1 We are using the space–time metric ds 2 = (dt)2 − (d x)2 , and
so the relative signs in the Lagrangian are set by the positivity of the energy.
derived from this Lagrangian are: ∂ µ Jµ = 0 with Jµ ≡ U † ∂µ U − λ U † ∂ ν U, U † ∂µ U, U † ∂ν U , (2) where λ = 4e21f 2 . π We shall use a special parametrization of the fields of Skyrme model which leads quite naturally to a submodel possessing an infinite number of conservation laws and containing the unit charge soliton solutions. Denote U ≡ eiζj τj , where τj , j = 1, 2, 3, are the Pauli matrices. Then one gets that U = eiζ T = 1 cos ζ + iT sin ζ, where ζ ≡ ζ12 + ζ22 + ζ32 , and 2 1 −2iu |u| − 1 T≡ . 1 − |u|2 2iu∗ 1 + |u|2
(3)
(4)
The complex field u is obtained by making the stereographic projection of the unit vector ζi /ζ , as 1 ζ ≡ −i(u − u∗ ), u + u∗ , |u|2 − 1 . 2 ζ 1 + |u|
(5)
The matrix T is conjugated to −τ3 = diag(−1, 1) as T = −W † τ3 W, 1 1 W≡ ∗ 1 + |u|2 iu
iu 1
(6)
and consequently we get the decomposition of the SU(2) group elements U = W † e−iζ τ3 W.
(7)
Therefore,
U † ∂µ U = W † −i∂µ ζ τ3 2 sin ζ iζ −iζ ∗ e ∂µ uτ+ − e ∂µ u τ− W, + 1 + |u|2 (8) with τ± = (τ1 ± iτ2 )/2. Denoting Jµ ≡ W † Bµ W , it follows that Skyrme’s equations of motion are then written as D µ Bµ = ∂ µ Bµ + Aµ , Bµ = 0 (9)
L.A. Ferreira, J. Sánchez Guillén / Physics Letters B 504 (2001) 195–200
with Aµ ≡ −∂µ W W † 1 − i∂µ uτ+ − i∂µ u∗ τ− = 1 + |u|2 1 + (u∂µ u∗ − u∗ ∂µ u)τ3 , (10) 2 2 sin ζ iζ e Sµ τ+ − e−iζ Sµ∗ τ− , Bµ ≡ −iRµ τ3 + 1 + |u|2 (11) where sin2 ζ ∗ N , + N µ µ (1 + |u|2 )2 2 sin2 ζ Sµ ≡ ∂µ u + 4λ Mµ − K µ (1 + |u|2 )2 Rµ ≡ ∂µ ζ − 8λ
(12)
Rµ ∂ µ u = 0,
Kµ ∂ µ ζ = Nµ∗ ∂ µ u,
Mµ ∂ µ ζ = 0,
Nµ ∂ µ u = 0. Consequently, the second constraint in (14) can also be written as Sµ ∂ µ ζ = 0.
(15)
By substituting the constraints into (9), or equivalently (2), one finds the equations of motion for the submodel to be ∂ µ Sµ = 0, 4 sin ζ cos ζ ∂ µ Rµ = Sµ ∂ µ u∗ , 2 2 (1 + |u| )
(16)
Jµ ≡ G1 (ζ, u, u∗ )Sµ + G2 (ζ, u, u∗ )Sµ∗ + G3 (u, u∗ )Rµ ,
(13)
Notice that Bµ has a vanishing covariant divergence w.r.t. a flat connection Aµ . It has been shown in [10] that this corresponds to a local zero curvature condition generalizing to higher dimensions the well known two dimensional Lax–Zakharov–Shabat equation. An important point of the formalism in [10] is that Bµ should transform under some representation of the algebra defined by Aµ . In fact, the number of conserved currents is equal to the dimension of the representation. The Skyrme model has three conserved currents and the representation relevant is the triplet. Indeed, (11) lies in the adjoint of SU(2). There is a systematic algebraic procedure [13,14] to find additional conditions for the construction to hold true in an infinite number of representations. However, one can directly obtain a submodel of the Skyrme theory with an infinite number of local conserved currents by imposing the following constraints 2 Sµ ∂ µ u = 0,
and their complex conjugates. Using (13) one notices that 3
where we have used the fact that Sµ ∂ µ u∗ = Sµ∗ ∂ µ u. The construction of an infinite number of conserved currents is now quite simple. Consider the quantity
and Kµ ≡ ∂ ν u∂ν u∗ ∂µ u − (∂ν u)2 ∂µ u∗ , Mµ ≡ ∂ ν u∂ν ζ ∂µ ζ − (∂ν ζ )2 ∂µ u, Nµ ≡ ∂ ν u∂ν u∗ ∂µ ζ − ∂ν ζ ∂ ν u ∂µ u∗ .
197
(14)
2 These constraints will suppress the higher spin terms in the
commutator in (9) when extending the representation where Bµ lies, beyond the triplet.
(17)
with Gi being arbitrary functionals of the fields as shown in (17), but not of their derivatives. Using (14), (15) and (16) one gets δG1 δG2 4 sin ζ cos ζ µ + + G3 Sµ ∂ µ u∗ . ∂ Jµ = ∗ 2 2 δu δu (1 + |u| ) (18) Therefore, by choosing the functionals such that δG1 δG2 4 sin ζ cos ζ + + G3 = 0, δu∗ δu (1 + |u|2 )2
(19)
one gets that Jµ is conserved. There are basically two classes of conserved currents. The first one corresponds to the choice G1 = δG/δu, G2 = −δG/δu∗ and G3 = 0, leading to the currents δG δG (20) Sµ − ∗ Sµ∗ , δu δu with G being an arbitrary functional of ζ , u and u∗ , but not of their derivatives. The second class corresponds to the choice Ga = 4 sin ζ cos ζ Ha (u, u∗ ), with a = 1, 2, and G3 = −(1 + |u|2 )2 (δH1 /δu∗ + δH2 /δu), JµG =
3 Other useful relations: K ∂ µ u = 0, M ∂ µ u∗ + N ∂ µ ζ = 0. µ µ µ
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leading to the currents Jµ(H1 ,H2 ) = 4 sin ζ cos ζ H1 Sµ + H2 Sµ∗ δH2 2 2 δH1 + − 1 + |u| Rµ , δu∗ δu
(21)
with Ha being arbitrary functionals of u and u∗ , but not of their derivatives. The three Noether currents Jµ in (2) are linear combinations of currents belonging to the two classes (20) and (21). In terms of the fields, the first constraint in (14), namely Sµ ∂ µ u = 0, reads (∂µ u∂ µ ζ )2 , (∂µ u)2 = −4λ 1 − 4λ(∂ν ζ )2
(22)
and the second constraint in (14), after (∂µ u)2 is eliminated using (22), reads sin2 ζ µ ∂µ u∂ ζ 1 − 8λ ∂ν u∂ ν u∗ (1 + |u|2 )2 (∂ν u∂ ν ζ )(∂ρ u∗ ∂ ρ ζ ) = 0. (23) + 4λ 1 − 4λ(∂σ ζ )2 Therefore, the conditions µ 2 ∂ u = 0, ∂ µ ζ ∂µ u = 0,
(24)
are sufficient for the constraints (14) to be satisfied. In the static case however, the conditions (24) are equivalent to the constraints (22) and (23). Indeed, one can easily show, using the real part of (22), that for fields not depending upon time sin2 ζ ∂ν u∂ ν u∗ 1 − 8λ (1 + |u|2 )2 (∂ν u∂ ν ζ )(∂ρ u∗ ∂ ρ ζ ) + 4λ 1 − 4λ(∂σ ζ )2 1 )2 sin2 ζ (∇u = 1 + 16λ (1 + |u|2 )2 )2 4λ(∇ζ )2 1 · ∇ζ (∇u 1, × 1− 1 )2 (∇ζ )2 1 + 4λ(∇ζ )2 (∇u (25) denotes where we have denoted u = u1 + iu2 , and ∇f the spatial gradient of f . Consequently, the constraints (22) and (23) can only be satisfied if · ∇u = 0, ∇u
· ∇ζ = 0. ∇u
(26)
Notice that the conditions (26) can be easily solved by an ansatz which contains as a particular case the rational map ansatz used to approximate the soliton solutions [8,9]. We denote the three Cartesian space coordinates as xi , i = 1, 2, 3, and the radial variable
as r ≡ x12 + x22 + x32 . We shall use the stereographic projection to parametrize the points on a sphere of radius r by a complex coordinate z. So we define 1 x ≡ −i(z − z∗ ), z + z∗ , |z|2 − 1 , 2 r 1 + |z|
(27)
The metric is then given by ds 2 = (dr)2 +
4r 2 dz dz∗ . (1 + |z|2 )2
(28)
We consider time independent solutions such that ζ = ζ(r),
u = u(z),
u∗ = u∗ (z∗ ).
(29)
Then due to the metric (28), it follows that (26), and consequently the constraints (14), are satisfied. Let us now analyze the configurations (29) that satisfy the equations of motion (16) of the submodel. Substituting (29) into (12) one finds that S0 = Sr = Sz∗ = 0, sin2 ζ 2 ∂z u, Sz = 1 + 4λ (∂r ζ ) + 2 β r
(30)
where β≡
(1 + |z|2 )2 ∂z u∂z∗ u∗ . (1 + |u|2 )2
(31)
Then the first equation in (16) and its complex conjugate imply that ∂z β = ∂z∗ β = 0.
(32)
Since by definition β is not a function of r, it follows that β = constant. The second equation in (16) gives the equation for the profile function ζ as sin 2ζ 2 sin2 ζ ζ + 4λβ 2 (ζ )2 + ζ 1 + 8λβ 2 r r r 2 sin 2ζ 2 sin 2ζ sin ζ − 4λβ = 0. r2 r4 Notice that due to (29) and (31) we can write
−β
2i du∗ ∧ du 2i dz∗ ∧ dz β= 2 2 (1 + |z| ) (1 + |u|2 )2
(33)
(34)
L.A. Ferreira, J. Sánchez Guillén / Physics Letters B 504 (2001) 195–200
and so it follows that β is the pull-back of the area form on the target sphere of the map z → u. Therefore, the topological charge becomes 2i du∗ ∧ du ∧ dζ 1 Q= 2 4π (1 + |u|2 )2 β 2i dz∗ ∧ dz ∧ dr = ζ 2 4π (1 + |z|2 )2 ζ (0) − ζ (∞) . =β (35) π For the hedgehog solution u = z, and so β = 1, Eq. (33) becomes the well known ODE for the original Skyrme profile function. With the standard boundary conditions ζ (∞) = 0 and ζ (0) = π , one then gets Q = 1. In order to get the solution with Q = −1 one has to interchange z by z∗ in (29) and take u = z∗ . Consequently, the Skyrmions with charges Q = ±1 are solutions of our submodel. The question now is if there are other holomorphic solutions within the ansatz (29). The answer is that only those configurations given by u=
az + b , cz + d
ad − bc = 1,
(36)
lead to solutions, and therefore one obtains through (29) solutions of charge Q = ±1 only. Notice that (36) correspond to rotations of the solution u = z. The simplest way to prove it is to observe that the map z → u provided by the configuration implies that the metric on the two spheres are related by dz dz∗ du du∗ =β . 2 2 (1 + |u| ) (1 + |z|2 )2
(37)
But the only maps satisfying that are the modular transformations (36) which imply that β = 1. Notice that the rational map ansatz u = p(z)/q(z), with p and q being polinomials in z with degree higher than one, are particular cases of (29). Consequently, they can not lead to exact solutions of the Skyrme model, even though they constitute good approximations to the minimal energy numerical configurations [8,9]. It would be very interesting to check if the numerical higher charge solutions in [8] satisfy the static constraints (26), which has a much simpler form than the full constraints (14) since they do not involve the scale determined by the coupling constant λ. A positive outcome of such check would encourage the use of those
199
constraints in the construction of new ansatzes for the higher charge solutions. In conclusion, we have identified an integrable submodel containing the Skyrmions of charges Q = ±1, showing that they are the only solutions within the meromorphic ansatz (29). The submodel, which is defined by the constraints (14) and equations of motion (16), possesses the infinite set of local conserved currents given by Eqs. (20), (21). That is our main result since it unravels an infinite symmetry inside the Skyrme model which opens ways to the construction of exact methods.
Acknowledgements J.S.G. gratefully acknowledges Fapesp (Proj. Temático 98/16315-9) for financial support and IFT/UNESP for hospitality. L.A.F. is very grateful to the Department of Physics of University Santiago de Compostela for the hospitality and acknowledges the financial support through the Spanish CICYT AEN99-0589 grant. We are grateful to O. Alvarez and P.M. Sutcliffe for discussions.
References [1] T.H.R. Skyrme, Proc. R. Soc. London 260 (1961) 127. [2] V.G. Makhankov, Y.P. Rybakov, V.I. Sanyuk, The Skyrme Model: Fundamentals, Methods and Applications, SpringerVerlag, 1993. [3] I. Zahed, G.E. Brown, Phys. Rep. 142 (1986) 1. [4] T. Gisiger, M.B. Paranjape, Phys. Rep. 306 (1998) 109–211, hep-th/9812148. [5] E. Witten, Nucl. Phys. B 223 (1983) 422; E. Witten, Nucl. Phys. B 223 (1983) 433; G.S. Adkins, C.R. Nappi, E. Witten, Nucl. Phys. B 228 (1983) 552; G.S. Adkins, C.R. Nappi, Nucl. Phys. B 233 (1984) 109. [6] A.P. Balachandran, V.P. Nair, S.G. Rajeev, A. Stern, Phys. Rev. D 27 (1983) 1153. [7] E. Braaten, S. Townsend, L. Carson, Phys. Lett. B 235 (1990) 147. [8] R.A. Battye, P.M. Sutcliffe, Phys. Rev. Lett. 79 (1997) 363, hep-th/9702089. [9] N.S. Manton, B.M.A.G. Piette, hep-th/0008110; C.J. Houghton, N.S. Manton, P. Sutcliffe, Nucl. Phys. B 510 (1998) 507–537, hep-th/9705151. [10] O. Alvarez, L.A. Ferreira, J. Sánchez-Guillén, Nucl. Phys. B 529 (1998) 689, hep-th/9710147.
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[11] L. Faddeev, in: 40 Years in Mathematical Physics, World Scientific, 1995; L. Faddeev, A. Niemi, Nature 387 (1997) 58, hep-th/9610193. [12] H. Aratyn, L.A. Ferreira, A.H. Zimerman, Phys. Rev. Lett. 83 (1999) 1723, hep-th/9905079; H. Aratyn, L.A. Ferreira, A.H. Zimerman, Phys. Lett. B 456 (1999) 162, hep-th/9902141. [13] D. Gianzo, J.O. Madsen, J. Sánchez-Guillén, Nucl. Phys. B 537 (1999) 586, hep-th/9805094. [14] E.E. Leite, L.A. Ferreira, Nucl. Phys. B 547 (1999) 471, hepth/9810067. [15] O. Alvarez, L.A. Ferreira, J. Sánchez-Guillén, Integrable theories in any dimension: a perspective, in: H. Falomir, R.E.
Gamboa Saraví, F. Schaposnik (Eds.), Trends in Theoretical Physics II, AIP Conf. Proc., Vol. 484, 1999, pp. 81–90. [16] K. Fujii, Y. Homma, T. Suzuki, Phys. Lett. B 438 (1998) 290, hep-th/9806084; K. Fujii, Y. Homma, T. Suzuki, Mod. Phys. Lett. A 14 (1999) 919–928, hep-th/9809149; K. Fujii, T. Suzuki, Lett. Math. Phys. 46 (1998) 49–59, hepth/9802105; T. Suzuki, Nucl. Phys. B 578 (2000) 515–523, hepth/0002003.
Physics Letters B 504 (2001) 195–200 www.elsevier.nl/locate/npe
Infinite symmetries in the Skyrme model L.A. Ferreira a , J. Sánchez Guillén b a Instituto de Física Teórica — IFT/UNESP, Rua Pamplona 145, 01405-900, São Paulo, SP, Brazil b Departamento de Física de Partículas, Facultad de Física, Universidad de Santiago, E-15706 Santiago de Compostela, Spain
Received 30 November 2000; accepted 14 December 2000 Editor: L. Alvarez-Gaumé
Abstract We show that the Skyrme theory possesses a submodel with an infinite number of local conserved currents. The constraints leading to the submodel explore a decomposition of SU(2) with a complex field parametrizing the symmetric space SU(2)/U (1) and a real field in the direction of U (1). We demonstrate that the Skyrmions of topological charges ±1 belong to such integrable sector of the theory. Our results open ways to the development of exact methods, compensating for the non-existence of a BPS type sector in the Skyrme theory. 2001 Published by Elsevier Science B.V. PACS: 12.39.Dc; 11.10.Lm; 05.45.Yv; 11.27.+d
Skyrme introduced in the sixties a very original idea which nowadays is widely used in physics and mathematics, that the classical soliton solutions of a given field theory may correspond to particles in its quantum spectrum. He proposed a theory, now called Skyrme model [1], to describe the low energy interactions of hadrons, where the nucleons are interpreted as topological solitons. The model found applications not only in nuclear physics, but also in particle and condensed matter physics [2–4]. The theory received a boost in the eighties when it was shown to correspond indeed to the low energy limit of QCD for large Nc [5,6]. The soliton solutions have been calculated by numerical methods [7–9] and there is now a vast amount of evidence that Skyrme’s first spherical ansatz (the hedgehog) does lead to the solutions of unit topological charge. The generalizations of that ansatz based on rational maps lead to good approxi-
E-mail address: [email protected] (J. Sánchez Guillén).
mations of the minimal energy solutions [8,9]. Such ansatzes are inspired in the BPS monopoles by sharing its symmetries and instanton methods. But unlike the well understood instantons, one easily shows that in three space dimensions skyrmions cannot saturate self-duality bounds. The non-existence of a BPS type sector is perhaps one of the factors preventing the development of exact methods for the Skyrme model. In this Letter we propose a way of addressing such problems by showing that there exist a submodel in the Skyrme theory possessing an infinite number of local conserved currents. They form in fact a continuous set of currents since they are parametrized by special functionals of the fields. The submodel is constructed by imposing Lorentz invariant constraints following the methods of [10] for studying integrable theories in any dimension. The constraints explore a suitable decomposition of the SU(2) group elements where the Skyrme’s fields are a complex scalar field u parametrizing the symmetric space SU(2)/U (1) and a real scalar field ζ parametrizing the U (1) subgroup. It
0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 2 8 0 - 5
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turns out that, in the static case, particular solutions to the constraints are obtained by taking ζ to depend only upon the radial variable r, and u to be a meromorphic function of a complex variable z, built out of the two angles of the spherical polar coordinates. The original Skyrme hedgehog ansatzes for topological charges ±1 fall into such class of configurations and so solve our constraints. We proof that they solve indeed our submodel equations of motion. The candidates for higher charges solutions were constructed numerically in [8], and it would be very interesting to verify if they satisfy our constraint. We point out that the existence of such submodel containing the Skyrme’s hedgehog and possessing an infinite number of conservation laws is highly non-trivial. It became possible now, about forty years after Skyrme’s original paper, due to the recent developments on the study of integrable theories in higher dimensions [10]. Another interesting point is that the generalizations to higher charges of the original Skyrme’s hedgehog obtained through the rational map ansatzes are particular solutions of our constraints. We show that such configurations can not be solutions of Skyrme’s model. However, since they provide approximations to the numerical candidates to true solutions [9], our constraints may give hints on how to correct them to obtain of a new ansatzes for higher charges solutions. In any case, our results establish an interesting connection between soliton solutions and infinite number of conservation laws in 3+1 field theory of great interest in physics. Similar results have already been found for the Skyrme–Faddeev model [11] and other related (3 + 1)-dimensional theories defined on S 2 [12], as well as in theories defined on homogenous spaces like CPN and chiral models [13–16]. The Lagrangian density for the Skyrme model, with the pion mass set to zero, can be written as [1–3]: 1 f2 L = π tr U † ∂µ U U † ∂ µ U 4 2 1 tr U † ∂µ U, U † ∂ν U , − (1) 2 32e where fπ and e are phenomenological constants, and U is a SU(2) unitary matrix. The equations of motion 1 We are using the space–time metric ds 2 = (dt)2 − (d x)2 , and
so the relative signs in the Lagrangian are set by the positivity of the energy.
derived from this Lagrangian are: ∂ µ Jµ = 0 with Jµ ≡ U † ∂µ U − λ U † ∂ ν U, U † ∂µ U, U † ∂ν U , (2) where λ = 4e21f 2 . π We shall use a special parametrization of the fields of Skyrme model which leads quite naturally to a submodel possessing an infinite number of conservation laws and containing the unit charge soliton solutions. Denote U ≡ eiζj τj , where τj , j = 1, 2, 3, are the Pauli matrices. Then one gets that U = eiζ T = 1 cos ζ + iT sin ζ, where ζ ≡ ζ12 + ζ22 + ζ32 , and 2 1 −2iu |u| − 1 T≡ . 1 − |u|2 2iu∗ 1 + |u|2
(3)
(4)
The complex field u is obtained by making the stereographic projection of the unit vector ζi /ζ , as 1 ζ ≡ −i(u − u∗ ), u + u∗ , |u|2 − 1 . 2 ζ 1 + |u|
(5)
The matrix T is conjugated to −τ3 = diag(−1, 1) as T = −W † τ3 W, 1 1 W≡ ∗ 1 + |u|2 iu
iu 1
(6)
and consequently we get the decomposition of the SU(2) group elements U = W † e−iζ τ3 W.
(7)
Therefore,
U † ∂µ U = W † −i∂µ ζ τ3 2 sin ζ iζ −iζ ∗ e ∂µ uτ+ − e ∂µ u τ− W, + 1 + |u|2 (8) with τ± = (τ1 ± iτ2 )/2. Denoting Jµ ≡ W † Bµ W , it follows that Skyrme’s equations of motion are then written as D µ Bµ = ∂ µ Bµ + Aµ , Bµ = 0 (9)
L.A. Ferreira, J. Sánchez Guillén / Physics Letters B 504 (2001) 195–200
with Aµ ≡ −∂µ W W † 1 − i∂µ uτ+ − i∂µ u∗ τ− = 1 + |u|2 1 + (u∂µ u∗ − u∗ ∂µ u)τ3 , (10) 2 2 sin ζ iζ e Sµ τ+ − e−iζ Sµ∗ τ− , Bµ ≡ −iRµ τ3 + 1 + |u|2 (11) where sin2 ζ ∗ N , + N µ µ (1 + |u|2 )2 2 sin2 ζ Sµ ≡ ∂µ u + 4λ Mµ − K µ (1 + |u|2 )2 Rµ ≡ ∂µ ζ − 8λ
(12)
Rµ ∂ µ u = 0,
Kµ ∂ µ ζ = Nµ∗ ∂ µ u,
Mµ ∂ µ ζ = 0,
Nµ ∂ µ u = 0. Consequently, the second constraint in (14) can also be written as Sµ ∂ µ ζ = 0.
(15)
By substituting the constraints into (9), or equivalently (2), one finds the equations of motion for the submodel to be ∂ µ Sµ = 0, 4 sin ζ cos ζ ∂ µ Rµ = Sµ ∂ µ u∗ , 2 2 (1 + |u| )
(16)
Jµ ≡ G1 (ζ, u, u∗ )Sµ + G2 (ζ, u, u∗ )Sµ∗ + G3 (u, u∗ )Rµ ,
(13)
Notice that Bµ has a vanishing covariant divergence w.r.t. a flat connection Aµ . It has been shown in [10] that this corresponds to a local zero curvature condition generalizing to higher dimensions the well known two dimensional Lax–Zakharov–Shabat equation. An important point of the formalism in [10] is that Bµ should transform under some representation of the algebra defined by Aµ . In fact, the number of conserved currents is equal to the dimension of the representation. The Skyrme model has three conserved currents and the representation relevant is the triplet. Indeed, (11) lies in the adjoint of SU(2). There is a systematic algebraic procedure [13,14] to find additional conditions for the construction to hold true in an infinite number of representations. However, one can directly obtain a submodel of the Skyrme theory with an infinite number of local conserved currents by imposing the following constraints 2 Sµ ∂ µ u = 0,
and their complex conjugates. Using (13) one notices that 3
where we have used the fact that Sµ ∂ µ u∗ = Sµ∗ ∂ µ u. The construction of an infinite number of conserved currents is now quite simple. Consider the quantity
and Kµ ≡ ∂ ν u∂ν u∗ ∂µ u − (∂ν u)2 ∂µ u∗ , Mµ ≡ ∂ ν u∂ν ζ ∂µ ζ − (∂ν ζ )2 ∂µ u, Nµ ≡ ∂ ν u∂ν u∗ ∂µ ζ − ∂ν ζ ∂ ν u ∂µ u∗ .
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(14)
2 These constraints will suppress the higher spin terms in the
commutator in (9) when extending the representation where Bµ lies, beyond the triplet.
(17)
with Gi being arbitrary functionals of the fields as shown in (17), but not of their derivatives. Using (14), (15) and (16) one gets δG1 δG2 4 sin ζ cos ζ µ + + G3 Sµ ∂ µ u∗ . ∂ Jµ = ∗ 2 2 δu δu (1 + |u| ) (18) Therefore, by choosing the functionals such that δG1 δG2 4 sin ζ cos ζ + + G3 = 0, δu∗ δu (1 + |u|2 )2
(19)
one gets that Jµ is conserved. There are basically two classes of conserved currents. The first one corresponds to the choice G1 = δG/δu, G2 = −δG/δu∗ and G3 = 0, leading to the currents δG δG (20) Sµ − ∗ Sµ∗ , δu δu with G being an arbitrary functional of ζ , u and u∗ , but not of their derivatives. The second class corresponds to the choice Ga = 4 sin ζ cos ζ Ha (u, u∗ ), with a = 1, 2, and G3 = −(1 + |u|2 )2 (δH1 /δu∗ + δH2 /δu), JµG =
3 Other useful relations: K ∂ µ u = 0, M ∂ µ u∗ + N ∂ µ ζ = 0. µ µ µ
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leading to the currents Jµ(H1 ,H2 ) = 4 sin ζ cos ζ H1 Sµ + H2 Sµ∗ δH2 2 2 δH1 + − 1 + |u| Rµ , δu∗ δu
(21)
with Ha being arbitrary functionals of u and u∗ , but not of their derivatives. The three Noether currents Jµ in (2) are linear combinations of currents belonging to the two classes (20) and (21). In terms of the fields, the first constraint in (14), namely Sµ ∂ µ u = 0, reads (∂µ u∂ µ ζ )2 , (∂µ u)2 = −4λ 1 − 4λ(∂ν ζ )2
(22)
and the second constraint in (14), after (∂µ u)2 is eliminated using (22), reads sin2 ζ µ ∂µ u∂ ζ 1 − 8λ ∂ν u∂ ν u∗ (1 + |u|2 )2 (∂ν u∂ ν ζ )(∂ρ u∗ ∂ ρ ζ ) = 0. (23) + 4λ 1 − 4λ(∂σ ζ )2 Therefore, the conditions µ 2 ∂ u = 0, ∂ µ ζ ∂µ u = 0,
(24)
are sufficient for the constraints (14) to be satisfied. In the static case however, the conditions (24) are equivalent to the constraints (22) and (23). Indeed, one can easily show, using the real part of (22), that for fields not depending upon time sin2 ζ ∂ν u∂ ν u∗ 1 − 8λ (1 + |u|2 )2 (∂ν u∂ ν ζ )(∂ρ u∗ ∂ ρ ζ ) + 4λ 1 − 4λ(∂σ ζ )2 1 )2 sin2 ζ (∇u = 1 + 16λ (1 + |u|2 )2 )2 4λ(∇ζ )2 1 · ∇ζ (∇u 1, × 1− 1 )2 (∇ζ )2 1 + 4λ(∇ζ )2 (∇u (25) denotes where we have denoted u = u1 + iu2 , and ∇f the spatial gradient of f . Consequently, the constraints (22) and (23) can only be satisfied if · ∇u = 0, ∇u
· ∇ζ = 0. ∇u
(26)
Notice that the conditions (26) can be easily solved by an ansatz which contains as a particular case the rational map ansatz used to approximate the soliton solutions [8,9]. We denote the three Cartesian space coordinates as xi , i = 1, 2, 3, and the radial variable
as r ≡ x12 + x22 + x32 . We shall use the stereographic projection to parametrize the points on a sphere of radius r by a complex coordinate z. So we define 1 x ≡ −i(z − z∗ ), z + z∗ , |z|2 − 1 , 2 r 1 + |z|
(27)
The metric is then given by ds 2 = (dr)2 +
4r 2 dz dz∗ . (1 + |z|2 )2
(28)
We consider time independent solutions such that ζ = ζ(r),
u = u(z),
u∗ = u∗ (z∗ ).
(29)
Then due to the metric (28), it follows that (26), and consequently the constraints (14), are satisfied. Let us now analyze the configurations (29) that satisfy the equations of motion (16) of the submodel. Substituting (29) into (12) one finds that S0 = Sr = Sz∗ = 0, sin2 ζ 2 ∂z u, Sz = 1 + 4λ (∂r ζ ) + 2 β r
(30)
where β≡
(1 + |z|2 )2 ∂z u∂z∗ u∗ . (1 + |u|2 )2
(31)
Then the first equation in (16) and its complex conjugate imply that ∂z β = ∂z∗ β = 0.
(32)
Since by definition β is not a function of r, it follows that β = constant. The second equation in (16) gives the equation for the profile function ζ as sin 2ζ 2 sin2 ζ ζ + 4λβ 2 (ζ )2 + ζ 1 + 8λβ 2 r r r 2 sin 2ζ 2 sin 2ζ sin ζ − 4λβ = 0. r2 r4 Notice that due to (29) and (31) we can write
−β
2i du∗ ∧ du 2i dz∗ ∧ dz β= 2 2 (1 + |z| ) (1 + |u|2 )2
(33)
(34)
L.A. Ferreira, J. Sánchez Guillén / Physics Letters B 504 (2001) 195–200
and so it follows that β is the pull-back of the area form on the target sphere of the map z → u. Therefore, the topological charge becomes 2i du∗ ∧ du ∧ dζ 1 Q= 2 4π (1 + |u|2 )2 β 2i dz∗ ∧ dz ∧ dr = ζ 2 4π (1 + |z|2 )2 ζ (0) − ζ (∞) . =β (35) π For the hedgehog solution u = z, and so β = 1, Eq. (33) becomes the well known ODE for the original Skyrme profile function. With the standard boundary conditions ζ (∞) = 0 and ζ (0) = π , one then gets Q = 1. In order to get the solution with Q = −1 one has to interchange z by z∗ in (29) and take u = z∗ . Consequently, the Skyrmions with charges Q = ±1 are solutions of our submodel. The question now is if there are other holomorphic solutions within the ansatz (29). The answer is that only those configurations given by u=
az + b , cz + d
ad − bc = 1,
(36)
lead to solutions, and therefore one obtains through (29) solutions of charge Q = ±1 only. Notice that (36) correspond to rotations of the solution u = z. The simplest way to prove it is to observe that the map z → u provided by the configuration implies that the metric on the two spheres are related by dz dz∗ du du∗ =β . 2 2 (1 + |u| ) (1 + |z|2 )2
(37)
But the only maps satisfying that are the modular transformations (36) which imply that β = 1. Notice that the rational map ansatz u = p(z)/q(z), with p and q being polinomials in z with degree higher than one, are particular cases of (29). Consequently, they can not lead to exact solutions of the Skyrme model, even though they constitute good approximations to the minimal energy numerical configurations [8,9]. It would be very interesting to check if the numerical higher charge solutions in [8] satisfy the static constraints (26), which has a much simpler form than the full constraints (14) since they do not involve the scale determined by the coupling constant λ. A positive outcome of such check would encourage the use of those
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constraints in the construction of new ansatzes for the higher charge solutions. In conclusion, we have identified an integrable submodel containing the Skyrmions of charges Q = ±1, showing that they are the only solutions within the meromorphic ansatz (29). The submodel, which is defined by the constraints (14) and equations of motion (16), possesses the infinite set of local conserved currents given by Eqs. (20), (21). That is our main result since it unravels an infinite symmetry inside the Skyrme model which opens ways to the construction of exact methods.
Acknowledgements J.S.G. gratefully acknowledges Fapesp (Proj. Temático 98/16315-9) for financial support and IFT/UNESP for hospitality. L.A.F. is very grateful to the Department of Physics of University Santiago de Compostela for the hospitality and acknowledges the financial support through the Spanish CICYT AEN99-0589 grant. We are grateful to O. Alvarez and P.M. Sutcliffe for discussions.
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