SU(3) collective coordinate quantization to the regular solution of the B=2
axially symmetric Einstein-Skyrme system is performed. For the symmetry
breaking term, a perturbative treatment as well as the exact diagonalization
method called Yabu-Ando approach are used. The effect of the gravity on the
mass spectra of the SU(3) dibaryons and the symmetry breaking term is studied
in detail. In the strong gravity limit, the symmetry breaking term
significantly reduces and exact SU(3) flavor symmetry is recovered.Comment: 9 pages, 14 figure

Sato, H.

Sawado, N.

English

arXiv

AbstractSU(3) collective coordinate quantization to the regular solution of the B=2 axially symmetric Einstein–Skyrme system is performed. For the symmetry breaking term, exact diagonalization method called Yabu–Ando approach are used. The effect of the gravity on the mass spectra of the SU(3) dibaryons and the symmetry breaking term is studied in detail. In the strong gravity limit, the symmetry breaking term significantly reduces and exact SU(3) flavor symmetry is recovered

Sato, Hisayuki

Sawado, Nobuyuki

Elsevier - Publisher Connector 

Physics Letters B 660 (2008) 72–79www.elsevier.com/locate/physletbSU(3) dibaryons in the Einstein–Skyrme modelHisayuki Sato ∗, Nobuyuki SawadoDepartment of Physics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba 278-8510, JapanReceived 1 October 2007; received in revised form 12 October 2007; accepted 21 October 2007Available online 3 December 2007Editor: L. Alvarez-GauméAbstractSU(3) collective coordinate quantization to the regular solution of the B = 2 axially symmetric Einstein–Skyrme system is performed. For thesymmetry breaking term, exact diagonalization method called Yabu–Ando approach are used. The effect of the gravity on the mass spectra of theSU(3) dibaryons and the symmetry breaking term is studied in detail. In the strong gravity limit, the symmetry breaking term significantly reducesand exact SU(3) flavor symmetry is recovered.© 2007 Elsevier B.V.PACS: 12.39.Dc; 21.10.-k; 04.20.-q1. IntroductionThe Skyrme model [1] is considered as an unified theory of hadrons by incorporating baryons as topological solitons of pionfields, called skyrmions. The topological charge is identified as the baryon number B . Performing collective quantization for aB = 1 skyrmion, one can obtain proton and neutron states within 30% error [2]. Correspondingly multi-skyrmion (B > 1) solutionsare expected to represent nuclei [3,4].The Einstein–Skyrme (ES) model can be thought as a model of hadrons in which baryons interact with black holes. The earlystudies [5–7] have shown that the Schwarzschild black hole can support spherically symmetric (hedgehog) Skyrme hair which isa first counter example to the no-hair conjecture. Axially symmetric regular and black hole skyrmion solutions with B = 2 werefound in Ref. [8]. Also some multi-skyrmion solutions B > 2 have been obtained [9].If a skyrmion is regarded as nucleon or nuclei, it must be quantized to assign quantum numbers like spin, isospin, etc. Studyof gravitational effects on the quantum spectra of skyrmions was initiated in Ref. [10] by performing collective quantization of aB = 1 gravitating skyrmion. The study about a B = 2 axially symmetric skyrmion immediately followed to it [11]. In Ref. [11],we investigated the gravity effects to the SU(2) solutions of the ES model and found out that the effects are seen especially in theheavier dibaryon spectra. In this Letter we shall study the gravity effects to the SU(3) dibaryons because they have much largermass compared with that of SU(2) dibaryons in terms of the large flavor symmetry breaking. The effect will be more apparent onsuch dihyperons.The extensions of the Skyrme model into SU(3) have extensively been studied. The solutions are classified into two categories:embedded and nonembedded ones. For the nonembedded solutions, in Ref. [12] the classical SO(3) chiral field is simply extendedinto three flavor space. The authors constructed the solutions in the ES system and found particle-like and black hole solutions.More systematic analyses in this direction have been done in Refs. [9,13]. By using the harmonic map ansatz, the authors haveinvestigated various large flavor and multi winding number solitons.* Corresponding author.E-mail addresses: sugar@dec.rikadai.jp (H. Sato), sawado@ph.noda.tus.ac.jp (N. Sawado).Open access under CC BY license.0370-2693 © 2007 Elsevier B.V.doi:10.1016/j.physletb.2007.10.084Open access under CC BY license.H. Sato, N. Sawado / Physics Letters B 660 (2008) 72–79 73For the embedded solutions, collective coordinate method, which is essentially a natural extension of the SU(2) collectivequantization scheme, has often been used. Yabu–Ando (YA) treatment [14] is a sort of the collective quantization but the collectiveHamiltonian is exactly diagonalized by using Euler-angle parameterization of the SU(3) rotations. As a result, baryon appears to beits lowest irreducible representation (irrep) but contain significant admixture of higher representations. The application to the B = 2axially symmetric skyrmions [15] and to the multi-skyrmions (3 B  8) [16] in flat space have been studied. They observed largemixing of higher irreps especially in excited or higher winding number states.In this Letter, we investigate SU(3) axially ES system in terms of the collective quantization and YA approach. We estimatemass spectra of the dibaryons down to strangeness S = −6. Since our concern is about contributions of higher irreps to the loweststates and we shall show change of the mixing probabilities about varying coupling strength of the gravity.2. Classical gravitating B = 2 skyrmionsThe SU(3) extended Skyrme Lagrangian coupled with gravity can be written as(1)L = LG + LS + LSB + LWZwhere LG is the standard Einstein–Hilbert Lagrangian(2)LG =∫d3r√−g 116πGRand the remaining parts are about the SU(3) extension of the Skyrme Lagrangian, which are defined in terms of the chiral fieldU(x) ∈ SU(3) and using the notation lμ = U†∂μU as follows:(3)LS =∫d3r√−g[116F 2πgμν Tr(lμlν) + 132e2 gμρgνσ Tr([lμ, lν][lρ, lσ ])],(4)LSB = 116F2πm2π∫d3r√−g Tr(U + U† − 2)+ 124(F 2κ m2κ − F 2πm2π)∫d3r√−g Tr(1 − √3λ8)(U + U† − 2),(5)LWZ = − iNc240π2∫QdΣμνλρσ Tr[lμlν lλlρlσ ],where Fπ and e are basic model parameters which indicate the pion decay constant, a dimensionless parameter, respectively. TheLSB is comprised of all chiral and flavor symmetry breaking terms. The LWZ is usual Wess–Zumino term which concerns with thetopological charge of the soliton. Fκ,mκ are the kaon decay constant and the mass. Nc means a number of color.In this Letter, we employ the solution of the chiral field with axially symmetric, winding number two which is considered as apossible candidate for the B = 2 minimal energy configuration [17](6)U0(r) = exp[iF (r, θ)τ · nR],(7)nR =(sinΘ(r, θ) cosnϕ, sinΘ(r, θ) sinnϕ, cosΘ(r, θ)),and n ∈ Z is the winding number. We explore the solution with n = 2. SU(3) chiral field is constructed by trivial embedding:(8)U(r) =(U0(r) 00 1).Correspondingly, the following axially symmetric ansatz is imposed on the metric [18](9)ds2 = −f dt2 + mf(dr2 + r2 dθ2)+ lfr2 sin2 θ dϕ2where the metric functions f , m and l are the function of coordinates r and θ . This metric is symmetric with respect to the z-axis(θ = 0). Substituting these ansatz to the Lagrangian (1), one obtains the following static (classical) energy for the chiral fieldsMclass = 2πFπe∫dx dθ[√l sin θ8{x2((∂xF )2 + (∂xΘ sinF)2)+ (∂θF )2 + (∂θΘ sinF)2 + n2ml sin θsin2 F sin2 Θ}+√l sin θ2[fm(∂xF∂θΘ − ∂θF∂xΘ)2 sin2 F + n2fl sin2 θsin2 F sin2 Θ{((∂xF )2 + 1x2(∂θF )2)(10)+((∂xΘ)2 + 1x2(∂θΘ)2)sin2 F}]+ 14m√lfx2 sin θβ2π (1 − cosF)],where dimensionless variable x = eFπr and βπ = mπ are introduced.eFπ74 H. Sato, N. Sawado / Physics Letters B 660 (2008) 72–79Fig. 1. The profile functions F , Θ for α = 0.080 in the cylindrical coordinate with dimensionless variables P := efπρ, Z = efπ z.For the profile functions, the boundary conditions at the x = 0,∞ are imposed(11)F(0, θ) = π, F (∞, θ) = 0,(12)∂xΘ(0, θ) = ∂xΘ(∞, θ) = 0.At θ = 0 and π/2,(13)∂θF (x,0) = ∂θF(x,π2)= 0,(14)Θ(x,0) = 0, Θ(x,π2)= π2.For the solutions to be regular at the origin x = 0 and to be asymptotically flat at infinity, the following boundary conditions mustbe imposed(15)∂xf (0, θ) = ∂xm(0, θ) = ∂xl(0, θ) = 0,(16)f (∞, θ) = m(∞, θ) = l(∞, θ) = 1.For the configuration to be axially symmetric, the following boundary conditions are imposed at θ = 0,π/2(17)∂θf (x,0) = ∂θm(x,0) = ∂θ l(x,0) = 0,(18)∂θf(x,π2)= ∂θm(x,π2)= ∂θ l(x,π2)= 0.By taking a variation of the static energy (10) with respect to F and Θ , one obtains the equations of motion for the profilefunctions. The field equations for the metric functions f , m and l are derived from the Einstein equations. The explicit form of theequations is essentially same (except for contribution of the mass term) as reported in Ref. [11].The effective coupling constant of the ES system, the only free parameter of the model, is given by(19)α = 4πGF 2π .To solve the equations of motion numerically, the relaxation method with the typical grid size 100 × 30 are performed. We observethat the solution survives at 0 α  0.120. Including the mass term, the range becomes a little narrow. We show examples of ournumerical results for the profile functions F,Θ in Fig. 1 and also for the metric functions f , l, m in Fig. 2.3. The SU(3) collective quantizationWe study the SU(3) extension of solutions in ES system by Yabu–Ando approach. We start by embedded chiral field into SU(3)which is of the form(20)U˜ (r, t) = A(t)(U0(R(t)r) 00 1)A†(t)H. Sato, N. Sawado / Physics Letters B 660 (2008) 72–79 75Fig. 2. The metric functions f , l, m in the cylindrical coordinate with dimensionless, rescaled variables ξ = P/(1 + P), ζ = Z/(1 +Z), P = efπρ, Z = efπ z.where U0 is introduced in Eq. (6). A(t) is time dependent SU(3) rotational matrix and R(t) describes a spatial rotation of thesoliton. We introduce the angular velocities Ωa , ωl which are defined by(21)A†A˙ = i28∑a=1λaΩa,(R˙R†)ik=3∑l=1εiklωl.Substituting the chiral field (20) into the Lagrangians (3)–(5) and after a lengthy calculation, one finally obtain the effective La-grangian of the form:(22)L = −Mclass + 12IN2∑p=1Ω2p +12IJ2∑p=1ω2p +12I3(Ω23 + nω3)2 + 12IS7∑k=4Ω2k −Nc2√3Ω8 + 12γ(1 − D88(A)),where IN , IJ , I3, IS are called the moments of inertia and their explicit forms areIN = πFπe3∫dx dθ[m√l4f 2x2 sin θ sin2 F(1 + cos2 Θ)+√lfsin θ sin2 F{(1 + cos2 Θ)(x2(∂xF )2 + (∂θF )2)(23)+ sin2 F cos2 Θ(x2(∂xΘ)2 + (∂θΘ)2)+ n2ml sin2 θsin2 F sin2 Θ}],76 H. Sato, N. Sawado / Physics Letters B 660 (2008) 72–79I3 = πFπe3∫dx dθ[m√l2f 2x2 sin θ sin2 F sin2 Θ + 2√lfsin θ sin2 F sin2 Θ{x2(∂xF )2 + (∂θF )2(24)+ sin2 F (x2(∂xΘ)2 + (∂θΘ)2)}],IJ = πFπe3∫dx dθ[m√l4f 2x2 sin θ((∂xF )2 + (∂θΘ sinF)2 + (n cot θ sinF sinΘ)2)+√lfx2 sin θ sin2 F{(∂xF∂θΘ − ∂θF∂xΘ)2 + n2((∂xF )2 + (∂xΘ sinF)2)cot2 θ sin2 Θ}(25)+ n2√lf sin θ(cos2 θ + ml)((∂θF )2 + (∂θΘ sinF)2)sin2 F sin2 Θ],IS = πFπe3∫dx dθ (1 − cosF)[m√l4f 2x2 sin θ +√l4fx2 sin θ{(∂xF )2 + (∂xΘ sinF)2}(26)+√l4fsin θ{(∂θF )2 + (∂θΘ sinF)2 + mn2l sin2 θsin2 F sin2 Θ}].12γ (1 − D88) exhibits strength of the symmetry breaking and the explicit form of γ is(27)γ = 2πFπ3e(β2κ − β2π)∫dx dθ x2 sin θ(cosF − 1),where βκ = mκFκeF 2π . D88 is a component of Wigner function which is defined as(28)Dab(A) = 12 Tr(λaA†λbA).From (22) the Hamiltonian reads(29)H = Mclass + J (J + 1)2IJ +12(1IN− 1IS)N(N + 1) + 12(1I3− 1IN− n2IJ)L2 − 38ISB2 + 12ISεSBwhere eigenvalues of diagonal operators are already inserted. Here, the eigenvalue of J is spin, I is isospin, N is right isospinderived from N = 12p0 where (p0, q0) is the minimal irrep and L is the third component of the body fixed spin operator whichdetermine parity P of the state by the relation of P = (−1)L. The eigenvalue of the εSB is derived from following eigenequation(30){C2[SU(3)]+ ISγ (1 − D88)}Ψ = εSBΨwhere C2[SU(3)] is Casimir operator of SU(3).We shall diagonalize (30) via a basis of the SU(3) Wigner functions. We introduce a wave function of the form [19](31)Φ(m)II3Y,NN3YR,JJ3(A,R) =√d(m)(−1) YR2 +N3D(m)∗II3Y,NN3YR(A−1)DJ∗J3,−nN3(R−1)where the dimension of the (p, q) irrep is expressed by d(m) = (p+1)(q +1)(p+q+2)/2, the m is representation of SU(3) group,and subscript of the Y and YR is hypercharge and right hypercharge, respectively. According to Yabu–Ando, the wave function Ψis expanded via (31)(32)Ψ :=∑mβ(m)Φ(m)II3Y,NN3YR,JJ3(A,R).In terms of the basis, the eigenvalue problem (30) can be reduces to a matrix diagonalization problem. In YA, state of baryonappears to be its lowest irrep but contain large admixture of higher irreps. We shall see that such mixing reduces in large gravitylimit.With the operation of the collective quantization, the angular velocity Ω8 appears linear in Eq. (22). Therefore we obtain aconstraint(33)YR = 13NcB,which means that the symmetry UR(1) is redundant. Thus we obtain YR = 2.In terms of Eq. (31), the expectation value of the Casimir invariants C2(SU(3)) in Eq. (30) is easily obtained(34)〈C2(SU(3))〉= 1(p2 + q2 + pq + 3(p + q)).3H. Sato, N. Sawado / Physics Letters B 660 (2008) 72–79 77Fig. 3. The coupling constant dependence of the mass difference from the classical energy for the multiplets {10}, {27}, {35}, {28} ((p, q) = (0,3), (2,2),(4,1), (6,0), respectively) are shown in the unit of GeV. The quantized energy is computed in terms of YA.For the symmetry breaking term ISγ (1 − D88), the estimation of the expectation value can be done by performing the integral ofthe three Wigner rotation matrices [20,21] which is evaluated by the SU(3) Clebsch–Gordan coefficient, or the isoscalar factor(35)∫dAD(m3)∗ν3ν′3(A)D(m1)ν1ν′1(A)D(m2)ν2ν′2(A) = 1d(m)∑μ(m1 m2 m3μν1 ν2 ν3)(m1 m2 m3μν′1 ν′2 ν′3).Computations of the Clebsch–Gordan coefficients can be performed by the numerical algorithm of Ref. [22].4. Numerical resultsFor the numerical analysis, we fix Fπ = 108 MeV, e = 4.84, βπ = 0.263. The kaon decay constant, experimentally, is Fκ ≈√2Fπ , but for the simplicity, we employ Fκ = Fπ . For the kaon mass, we employ the experimental value, i.e., βκ = 0.952.78 H. Sato, N. Sawado / Physics Letters B 660 (2008) 72–79Fig. 4. Mixing probability of the higher representations to the lowest energy state of NN channel in {10}.We estimate mass spectra belonging to SU(3) multiplets {10}, {27}, {35}, {28} (or in the (p, q) representation, (0,3), (2,2),(4,1), (6,0), respectively). The Finkelstein–Rubinstein constraints [23] tells us that for {10}, {35}, J = 1 is chosen for the groundstate, otherwise one can set J = 0 [24]. The eigenvalue of L concerns with the third component of body fixed spin operator [3].Substantially it is related to the orbital angular momentum but no experimental identification has been done. Therefore in ouranalysis we put L = 0 for all multiplets states.In YA treatment, one needs to truncate the base in finite size. We expand the collective wave function with N  3, except forthe states (S, I ) = (−5,1/2) and (−6,0) in {28}, (−4,0), (−4,1) and (−5,1/2) in {35}. In those states, we expand the base withN  4 for obtaining sufficient convergence.In Fig. 3 presents the α dependence of the mass spectra. Actually, we show mass difference between the quantized mass spectraand the classical energy. One easily see that the spectra as well as their differences within each multiplet decrease monotonicallywith increasing α. On the other hand, mass differences between the different multiplet but with same quantum numbers (S, I )increase, which have been already observed in the calculation of SU(2) [11]. In Fig. 4 we illustrate the mixing probability of themultiplet for NN channel in {10} for various α. For increasing α, the mixing of higher representations are significantly decreased;as a result, in a strong gravity region the baryons appear to be their lowest irreps and a naive perturbation treatment is sufficient. Inour analysis, the pion and the kaon mass and the coupling constants are fixed by their experimental values (we simply set Fκ = Fπfor the coupling constant) and if we take into account variations of the mesonic data about change of gravity, exact SU(3) flavorsymmetry will recover at a strong gravity limit.5. ConclusionIn this Letter, we have studied the gravitational effect to the dibaryons in the axially symmetric ES model. In particular, wehave investigated gravity coupling constant dependence of the energy spectra for the SU(3) dibaryons. We have used the collectivequantization in three flavor space. For the symmetry breaking term, we employ the Yabu–Ando treatment. The result have shownthat mass differences between spectra with different strangeness decrease monotonically and increase within different multipletbut with same quantum numbers (S, I ) with increasing α. In the strong gravity limit, the SU(3) flavor symmetry recovers; all thespectra degenerate in each multiplet and no mixing between the multiplets occur. Such symmetry restorations may observe in highenergy experiment at LHC.In this Letter, we treat α as a free parameter. In the ES theory, the Planck mass is related to the pion decay constant fπ andcoupling constant α by Mpl = fπ√4π/α. To realize the realistic value of the Planck mass, the coupling constant should be extremelysmall with α ∼ O(10−39). However, we have shown that the effects of gravity can be observed only in large α. Some theories suchas scalar-tensor gravity theory [25] and “brane world scenario” [26] predict possibility of huge gravitational constant. There mayhave been an epoch in the early universe and may observe at an ultra high energy experiment where the gravitational effects onhadrons are crucial.AcknowledgementsWe would like to thank Rajat K. Bhaduri for drawing our attention to this subject and useful comments. Also we deeply thankNoriko Shiiki for valuable discussions.H. Sato, N. Sawado / Physics Letters B 660 (2008) 72–79 79References[1] T.H.R. Skyrme, Proc. R. Soc. London, Ser. A 260 (1961) 127.[2] G.S. Adkins, C.R. Nappi, E. Witten, Nucl. Phys. B 228 (1983) 552.[3] E. Braaten, L. Carson, Phys. Rev. D 38 (1988) 3525.[4] O.V. Manko, N.S. Manton, S.W. Wood, arXiv: 0707.0868.[5] H. Luckock, I. Moss, Phys. Lett. B 176 (1986) 341.[6] S. Droz, M. Heusler, N. Straumann, Phys. Lett. B 268 (1991) 371.[7] P. Bizon, T. Chmaj, Phys. Lett. B 297 (1992) 55.[8] N. Sawado, N. Shiiki, K. Maeda, T. Torii, Gen. Relativ. Gravit. 36 (2004) 1361.[9] T. Ioannidou, B. Kleihaus, W. Zakrzewski, Phys. Lett. B 600 (2004) 116.[10] N. Shiiki, N. Sawado, S. Oryu, Phys. Rev. D 70 (2004) 114023.[11] H. Sato, N. Sawado, N. Shiiki, Phys. Rev. D 75 (2007) 014011.[12] B. Kleihaus, J. Kunz, A. Sood, Phys. Lett. B 352 (1995) 247.[13] Y. Brihaye, B. 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SU(3) dibaryons in the Einstein–Skyrme model 

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SU(3) dibaryons in the Einstein-Skyrme model

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