The construction of axially symmetric soliton solutions with non-zero Hopf topological charges according to a theory known as the extended Skyrme-Faddeev model, was performed in [1]. In this paper we show how masses of glueballs are predicted within this model

Ferreira, L. A.

Kato, S.

Sawado, N.

Toda, K.

Acta Polytechnica

English

L. A. Ferreira

S. Kato

N. Sawado

K. Toda

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Quantized Solitons in the Extended Skyrme-Faddeev Model

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Acta Polytechnica Vol. 51 No. 1/2011Quantized Solitons in the Extended Skyrme-Faddeev ModelL. A. Ferreira, S. Kato, N. Sawado, K. TodaAbstractThe construction of axially symmetric soliton solutions with non-zero Hopf topological charges according to a theoryknown as the extended Skyrme-Faddeev model, was performed in [1]. In this paper we show how masses of glueballs arepredicted within this model.Keywords: integrable systems, solitons, monopoles, instantons, semiclassical quantizations.We construct static soliton solutions carrying non-trivial Hopf topological charges for a ﬁeld theory thathas found interesting applications in many areas ofphysics. It is a (3 + 1)-dimensional Lorentz invari-ant ﬁeld theory for a triplet of scalar ﬁelds n, livingon the two-sphere S2, n2 = 1, and deﬁned by theLagrangian densityL = M2 ∂μn · ∂μn− 1e2(∂μn ∧ ∂νn)2 + (1)β2(∂μn · ∂μn)2 ,where the coupling constants e2 and β are dimen-sionless, and M has a mass dimension. The ﬁrst twoterms correspond to the so-called Skyrme-Faddeev(SF) model [2, 3, 4] proposed by the following ideaof Skyrme [5]. It was conjectured by Faddeev andNiemi [6] that the SF model describes the low energy(strong coupling) regime of the pure SU(2) Yang-Mills theory. This was based on the so-called Cho-Faddeev-Niemi-Shabanov decomposition [6, 7, 8] ofthe SU(2) Yang-Mills ﬁeld Aμ, where its six physi-cal degrees of freedom are encoded into a triplet ofscalars n (n2 = 1), a massless U(1) gauge ﬁeld, andtwo real scalar ﬁelds. Gies has computed the one-loopWilsonian eﬀective action for the SU(2) Yang-Millstheory and has found agreements with the conjec-ture, which is provided that the SF model is modi-ﬁed by additional quartic terms in derivatives of then ﬁeld [9].One can now stereographically project S2 on aplane and work with a complex scalar ﬁeld u relatedto the triplet n byn =11 + | u |2 (u+ u∗,−i(u− u∗), | u |2 − 1). (2)The static Hamiltonian associated to (1) isHstatic = 4M2 ∂iu ∂iu∗(1 + | u |2)2 −8e2[(∂iu)2(∂ju∗)2(1 + | u |2)4 + (β e2 − 1) (∂iu ∂iu∗)2(1 + | u |2)4]. (3)Therefore, it is positive deﬁnite for M2 > 0, e2 < 0,β < 0, βe2 ≥ 1. The Euler-Lagrange equations from(1) read(1 + | u |2) ∂μKμ − 2 u∗Kμ ∂μu = 0, (4)together with its complex conjugate, whereKμ :=M2 ∂μu+ (5)4e2[(β e2 − 1) (∂νu ∂νu∗) ∂μu+ (∂νu∂νu)∂μu∗](1 + | u |2)2 .We choose to use the toroidal coordinates deﬁned asx1 =r0p√z cosϕ,x2 =r0p√z sinϕ, (6)x3 =r0p√1− z sin ξ,where p = 1 − cos ξ√1− z, xi (i = 1, 2, 3) arethe Cartesian coordinates in R3, and (z, ξ, ϕ) arethe toroidal coordinates. We have 0 ≤ z ≤ 1,−π ≤ ξ ≤ π, 0 ≤ ϕ ≤ 2π, and r0 is a free parameterwith dimension of length. We use the ansatz for thesolutionu =√1− g(z, ξ)g(z, ξ)eiΘ(z,ξ)+i n ϕ, (7)with n being an integer. We now impose the bound-ary conditionsg(z = 0, ξ) = 0,g(z = 1, ξ) = 1, for − π ≤ ξ ≤ π (8)andΘ(z, ξ = −π) = −mπ,Θ(z, ξ = π) = mπ, for 0 ≤ z ≤ 1 (9)with m being an integer.47Acta Polytechnica Vol. 51 No. 1/2011Fig. 1: The Hopf charge density isosurfaces for solutionswith various charges (m,n) = (1, 2) (left), (2, 2) (middle)and (4, 1) (right) for βe2 = 1.1The ﬁnite energy solutions of the theory (1) de-ﬁne maps from the three dimensional space R3 ∼ S3to the target space S2. These are classiﬁed into ho-motopy classes labeled by an integer QH called theHopf index, which has values of QH = mn.Substituting (7) into (4) and (5), we get two cou-pled non-linear partial diﬀerential equations in twovariables. We have constructed numerical solutionswith several Hopf charges up to four by a successiveover-relaxation method. In Fig. 1, we present some ofthe results of the Hopf charge density for the charge(m,n) = (1, 2), (2, 2) and (4, 1).Probably the main interest in studying the classof the model is to get an insight into the mass ofthe glueballs. Since our ﬁnding solutions are classi-cal, they must be properly quantized. Making thefollowing replacement for the ﬁelds [10]n(r) · τ → n′(r, t) · τ := (10)A(t)[n(R(B(t))r − X) · τ]A†(t),one obtains the kinetic contribution to the energyT = 12aiUijaj − aiWijbj + 12biVijbj (11)with ai := −itr(τiA†A˙) and bi := itr(τiB˙B†). HereA and B are matrices in SU(2), and B works throughSO(3) form Rij(B) =12tr(τiBτjB−1). b is an angu-lar velocity, and a means an angular velocity in anisospace. Quantizing these coordinates, one ﬁnallyﬁnds the quantized mass of the Hopf solitonEquanta :=M|e|Estatic −Me2|e|[i(i+ 1)2U11+j(j + 1)2V11+ (12)k232(1U33− 1U11− n2V11)],where Estatic is the energy corresponding to thehamiltonian (3) and the inertia tensors Uab, Vab arethe function of the classical ﬁelds n. Numerically U11has a quite large value, so we omit the terms withU11. As a result, the states are essentially labeled bytwo quantum numbers (j, k3).Krusch applied the basic FR constraints to theSkyrme-Faddeev model, and ﬁnally found that thequantum states with even topological charge QHcould be bosonic[11]. They may be possible candi-dates for glueballs.Fig. 2: The quantized spectra for the topological charge(m,n) = (1, 2) and (4, 1) (bold line), compared with re-sult of the lattice simulation [12]We plot the lowest three states of (m,n) = (1, 2)and (4, 1). First, we have ﬁtted the ﬁrst two spectrato the result of the lattice simulation [12] and havecomputed the third lowest spectrum correspondingto JPC = 2−+. For (1, 2), the third state is lower en-ergy than the second state because the second termon the right hand side of (12) has a negative contribu-tion to the quantum energy. On the other hand, if weemploy the solution of (4, 1), the third state appearsnear to the prediction of the lattice. This is quitepromising. In [12], the authors also predicted theroot mean square radius√〈r2〉 ∼ 0.481 [fm]. In ourcalculation, the radius is√〈r2〉 = 0.466 (for (1, 2))or 0.536 (for (4, 1)) [fm]; the results are consistent.We summarize our analysis. We have foundnew Hopf soliton solutions for the extended Skyrme-Faddeev model. The model is a low energy eﬀectivemodel of QCD, and the solutions are possible candi-dates for the glueballs. We have performed the collec-tive coordinate quantization for the obtained classicalsolutions and have computed the quantum energies.Some of our results are in good agreement with thestudy of the lattice gauge simulation.AcknowledgementThe authors acknowledge ﬁnancial support fromFAPESP (Brazil). One of the authors (L.A.F.) is par-tially supported by CNPq. This work was partially48Acta Polytechnica Vol. 51 No. 1/2011supported by a Grant-in-Aid for the Global COEProgram “The Next Generation of Physics, Spunfrom Universality and Emergence” from the Ministryof Education, Culture, Sports, Science and Technol-ogy (MEXT) of Japan. One of the authors (K.T.) ispartially supported by the FY 2010 Researcher Ex-change Program between JSPS and the German Aca-demic Exchange Service. One of the authors (N.S)expresses his gratitude to the conference organizersof ISQS19 for kind accommodation and hospitality.References[1] Ferreira, L. A., Sawado, N., Toda, K.: Sta-tic Hopﬁons in the extended Skyrme-Fad-deev model, Journal of High Energy Physics,JHEP11(2009)124.[2] Faddeev, L. D.: Quantization of solitons,Princeton preprint IAS Print-75-QS70 (1975).Faddeev, L. D., Niemi, A. J.: Knots and par-ticles, Nature 387, 58 (1997).[3] Battye, R. A., Sutcliﬀe, P. M.: To be or knot tobe?, Phys. Rev. Lett. 81, 4798 (1998).[4] Hietarinta, J., Salo, P.: Faddeev-Hopf knots:Dynamics of linked un-knots, Phys. Lett. B 451,60 (1999).[5] Skyrme, T. H. R.: A Nonlinear ﬁeld theory,Proc. Roy. Soc. Lond. A 260, 127 (1961).[6] Faddeev, L. D., Niemi, A. J.: Partially dualvariables in SU(2) Yang-Mills theory, Phys. Rev.Lett. 82, 1624 (1999).[7] Cho, Y. M.: A Restricted Gauge Theory, Phys.Rev. D 21, 1080 (1980). Cho, Y. M.: ExtendedGauge Theory And Its Mass Spectrum, Phys.Rev. D 23, 2415 (1981).[8] Shabanov, S. V.: An eﬀective action formonopoles and knot solitons in Yang-Mills the-ory, Phys. Lett. B 458, 322 (1999).[9] Gies, H.: Wilsonian eﬀective action for SU(2)Yang-Mills theory with Cho-Faddeev-Niemi-Shabanov decomposition, Phys. Rev. D 63,125023 (2001).[10] Kondo, K. I., Ono, A., Shibata, A., Shino-hara, T., Murakami, T.: Glueball mass fromquantized knot solitons and gauge-invariantgluon J. Phys. A 39, 13767 (2006).[11] Krusch, S., Speight, J. M.: Fermionic quanti-zation of Hopf solitons, Commun. Math. Phys.264, 391 (2006).[12] Morningstar, C. J., Peardon, M. J.: The glue-ball spectrum from an anisotropic lattice study,Phys. Rev. D 60, 034509 (1999).About the authorNobuyuki Sawado is currently working as an Asso-ciate Professor at the Institute of Science and Tech-nology, Department of Physics at Tokyo Universityof Science. His research interest is in topological soli-tons and related areas, such as eﬀective models ofQCD, brane world scenarios and some topics in con-densed matter physics.Luiz Agostinho FerreiraInstitute of Physics of Sa˜o CarlosIFSC/USP, University of Sa˜o Paulo – USPCaixa Postal 369, CEP 13560-970, Sa˜o Carlos-SP,BrazilSatoru KatoDepartment of PhysicsInstitute of Science and TechnologyTokyo University of Science, Noda, Chiba 278-8510,JapanNobuyuki SawadoE-mail: sawado@ph.noda.tus.ac.jpDepartment of PhysicsInstitute of Science and TechnologyTokyo University of ScienceNoda, Chiba 278-8510, JapanKouichi TodaDepartment of Mathematical PhysicsToyama Prefectural UniversityKurokawa 5180, Imizu, Toyama, 939-0398, JapanandResearch and Education Center for Natural SciencesHiyoshi Campus, Keio University4-1-1 Hiyoshi, Kouhoku-ku, Yokohama, 223-8521,Japan49

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Quantized Solitons in the Extended Skyrme-Faddeev Model

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