We revisit baryons in the Skyrme model. Starting from static baryons in the
helicity eigenstates, we generalize their wavefunctions to the non-static and
relativistic regime. A new representation for gamma matrices in the soliton
collective space is constructed and the corresponding Dirac equation is
obtained. As an example, we draw consideration on how to apply this new
representation on the calculus of vector current vacuum expectation values for
baryon states of spin and isospin half and arbitrary momenta and we show how
elastic form factors can be derived.Comment: 7 pages, 7th Conference Mathematical Methods in Physics - Londrina
  2012, Rio de Janeiro, Brazi

Boschi-Filho, Henrique

Braga, Nelson R. F.

Ihl, Matthias

Torres, Marcus A. C.

English

arXiv

We revisit baryons in the Skyrme model. Starting from static baryons in the helicity eigenstates, we generalize their wavefunctions to the non-static and relativistic regime. A new representation for gamma matrices in the soliton collective space is constructed and the corresponding Dirac equation is obtained. As an example, we draw consideration on how to apply this new representation on the calculus of vector current vacuum expectation values for baryon states of spin and isospin half and arbitrary momenta and we show how elastic form factors can be derived

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arXiv:1211.5783v1  [hep-th]  25 Nov 2012November 27, 2012Towards Relativistic SkyrmionsHENRIQUE BOSCHI-FILHO a , NELSON R. F. BRAGA a , MATTHIAS IHL b ,MARCUS A. C. TORRES aa Instituto de Fı´sica,Universidade Federal do Rio de Janeiro,Caixa Postal 68528,21941-972 Rio de Janeiro, RJ, Brasil.Email: boschi@if.ufrj.br,braga@if.ufrj.br,mtorres@if.ufrj.brb School of Theoretical Physics,Dublin Institute for Advanced Studies,10 Burlington Rd,Dublin 4, Ireland.Email: msihl@stp.dias.ieAbstractWe revisit baryons in the Skyrme model. Starting from static baryons in the helicity eigen-states, we generalize their wavefunctions to the non-static and relativistic regime. A new rep-resentation for gamma matrices in the soliton collective space is constructed and the corre-sponding Dirac equation is obtained. As an example, we draw consideration on how to applythis new representation on the calculus of vector current vacuum expectation values for baryonstates of spin and isospin half and arbitrary momenta and we show how elastic form factorscan be derived.7th Conference Mathematical Methods in Physics - Londrina 2012, Rio de Janeiro, Brazil11. IntroductionSkyrmions are solitons, classical, static solution from NLσM of pions [1] and they are identifiedas baryons in such model. It is a model with chiral Lagrangean and spontaneous chiral symmbreaking. It has attracted renewed interest as a simplified description of baryons in the large Nclimit of QCD. Here we review the work presented in [2] where we show a simple way to overcomeits static or slow moving condition, extending it to a relativistic framework.The static properties of baryons in the Skyrme model (skyrmions) have been studied in [3] ,where baryonic quantum states emerge from a canonical quantization of the soliton moduli. In[4], Sakai and Sugimoto derive the Skyrme model as the 3+1 dimensional pion effective theorydescending from the dynamics of the gauge fields living on flavor D8-branes that probe the D4-brane geometry generated by a stack of color branes.Here, after introducing the Skyrme model and the soliton canonical quantization in section 2,we calculate relativistic baryon wave functions in their helicity eigenstates (section 3) generalizingthe results of [3] by boosting wave functions in the momentum direction. We show that the staterepresentation is in a form compatible with the Dirac spinor representation. In section 4, as anexample we show how to obtain form factors from the Lorentz tensorial decomposition of matrixelements of baryonic currents in our relativistic approach.2. Skyrme ModelSkyrmions are soliton solutions of a nonlinear effective field theory of pions. The action of theSkyrme model is given byS =∫d4x( f 2pi4tr(U−1∂µU)2+132e2 tr[U−1∂µU,U−1∂νU]2) (1)where the pion fields pi(xµ) are encoded in the SU(2) valued Skyrme field,U(xµ) =σ(xµ)I+pi i(xµ) · τ ifpi = eipi(xµ )/ fpi (weakpionfield limit),where σ 2(xµ)+~pi2(xµ) = f 2pi . U transforms as U(xµ)→ gLU(xµ)g−1R under the chiral symmetrySU(2)L×SU(2)R present in the Lagrangian.The Skyrmion is a static solution that goes to unity value at infinity breaking chiral symmetryto the diagonal subgroup. In this case U becomes a mapU : R3 +{∞}= S3 → SU(2)≡ S3 (2)and a topological charge nB ∈ pi3(SU(2)) = Z is identified with the baryon number.2.1. SU(2) collective coordinatesWe begin by reviewing the moduli space approximation method to quantize Skyrmions. The fun-damental idea [3] is to realize that there is a moduli of soliton solution given by SU(2) rotations ofa given one, U0, and define a slowly moving soliton by a time dependent rotation,U(t,xM) =VU0(xM,XM(t))V−1, (3)2where V = V (t,xM) is an SU(2) element and XM, M = {1, . . . ,3}, represents the position of thesoliton in the spatial R3. Introducing the collective coordinates a(t) = a4(t)+ iaa(t)τa as a pointin S3 representing the SU(2) orientation, we note V (t,x)→ a(t) as x → ∞.When a(t)Uoa−1(t) is replaced in the original Lagrangian, a new Lagrangian emerge,L =−M+2λ4∑i=1(a˙i)2 (4)Introducing conjugate momenta pii = ∂L/∂ a˙i =−i ∂∂ai , we find the Hamiltonian:H = M+λ∇2S3 (5)The eigenstates (wavefunctions) are factorized into radial and spherical harmonics components.On the S3, they are scalar spherical harmonics and are known to be homogenous polynomialsT (l)(aI) =CI1···Il aI1 · · ·aIl , (6)where CI1···Il is a traceless symmetric tensor of rank l.The dimension of the tensor C and therefore the dimension of the space of spherical harmonicsof degree l, Hl is (l + 1)2. The space Hl is a representation of the rotation group SO(4) andcorresponds to the (Sl/2,Sl/2) representation of (SU(2)× SU(2))/Z2 ≃ SO(4). Sl/2 denotes thespin l/2 representation of SU(2), with dimSl/2 = l +1. Under group rotation a transforms asa → gI a gJ , gI,J ∈ SU(2)I,J . (7)where SU(2)I and SU(2)J are identified with the isospin rotation and the spatial rotation, respec-tively.3. Static and Relativistic BaryonsThe collective state is quantized considering slowly moving solitons. We will extend it to rela-tivistic baryons by simply treating them as static solitons boosted in a given direction. First, wereview the spherical harmonics tensors to static nucleons. The lowest states are at l = 1 and thetensors become linear in aI coordinates. They correspond to states with spin and isospin 1/2 andwe identify them with protons and neutrons. In spinorial notation we write the particle states as|N,h〉= χN ⊗χh =: χNh , (8)where N = {p,n}, h = {+,−} andχ p = χ+ =(10), χn = χ− =(01). (9)The isospin I3 and spin J3 operators in this representation readIa =i2(a4∂∂aa−aa ∂∂a4 − εabc ab∂∂ac), Ja =i2(−a4 ∂∂aa +aa∂∂a4− εabc ab ∂∂ac), (10)and their eigenstates are given by|p,+〉= a1 + ia2pi, |p,−〉=− i(a4− ia3)pi, |n,+〉= i(a4 + ia3)pi, |n,−〉=−(a1− ia2)pi. (11)33.1. Nucleon relativistic wavefunctionsIn generalizing the expressions aboveto the relativistic case, we initially restrict the discussionto proton and neutron states separately and disregard the isospin information. We introduce arelativistic spinor for a fermion with a given momentum ~p,u(p,h) = 1√2E(f χh~p·~σf χh), with f =√E +mB. (12)Here χh is a helicity eigenstate, which means ~p ·~σ χh = h|~p|χh where h = ±1. This yields theeigenstatesχ+(~p) =1√2|~p|(|~p|+ p3)( |~p|+ p3p1 + ip2), χ−(~p) =1√2|~p|(|~p|+ p3)( −p1 + ip2|~p|+ p3). (13)Notice that χ†h χh = 1 and |~p| =√p21 + p22 + p33. Hence we expect the proton and neutron linear(l = 1) spherical harmonic tensors (equivalent to χh(~p) helicity eigenstates) to beχ p+(ai,~p)=(|~p|+p3)(a1+ia2)−i(p1+ip2)(a4−ia3)pi√2|~p|(|~p|+ p3), χ p−(ai,~p)=(−p1+ip2)(a1+ia2)−i(p3+|~p|)(a4−ia3)pi√2|~p|(|~p|+ p3),χn+(ai,~p)=(|~p|+p3)i(a4+ia3)−(p1+ip2)(a1−ia2)pi√2|~p|(|~p|+ p3), χn−(ai,~p)=(−p1+ip2)i(a4+ia3)−(p3+|~p|)(a1−ia2)pi√2|~p|(|~p|+ p3).(14)The expression above is not valid for p1 = p2 = 0 and p3 = −p and in that case we recall thathelicity changes sign when momentum changes in sign to the opposite direction,χNh (ai,−~p)≡ χN−h(ai,~p), (15)After some algebra we can verify that (14) are eigenfunctions of ~p .~J operators (10):~p .~J χN+(ai,~p) = +12|~p|χN+(ai,~p) and ~p.~J χN−(ai,~p) =−12|~p|χN−(ai,~p). (16)Similarly to Dirac 4-spinors (12),The relativistic nucleon SU(2) wavefunctions areuN(ai,~p ,+) =1√2E(f χN+(ai,~p)|~p|f χN+(ai,~p)), uN(ai,~p ,−) = 1√2E(f χN−(ai,~p)− |~p|f χN−(ai,~p)). (17)The explicit dependence on aI ∈ S3 is not directly observable and the integration in S3 modulirecovers the 4D Dirac spinor with appropriate isospin.3.2. Dirac equation and spin sumIn order to work with relativistic SU(2) wavefunctions instead of Dirac spinorial notation, wedefine the substitutes of gamma matrices in the SU(2) collective space by simply replacing thePauli matrices σ i with 2Ji operators (10). Hence, the new 2×2 gamma matrices areγ0 =−i(1 00 −1), γ i =−i(0 2Ji−2Ji 0). (18)4Such operators act only on spin and we will disregard the isospin index for now. Upon suchsubstitution we can verify the validity of the Dirac equation:(i 6 p+mB)uN(ai,~p ,h) = 0. (19)uN(ai,~p ,h)(i 6 p+mB) = 0. (20)Since Ji operators have real eigenvalues and behave like Pauli matrices, we define their operationto the left by transpose conjugation: ψ†h (~p.~J) = (~p.~Jψh)† = |~p|h2ψ†h .As mentioned before, the spinor normalization is given by an integration of the ai moduli,u(~p ,h′)u(~p ,h) =∫S3u(ai,~p ,h′)u(ai,~p ,h) (21)Working out the integrand,u(ai,~p ,h′)u(ai,~p ,h) =12E( f χ∗h′(ai,~p) |~p |f h′χ∗h′(ai,~p))(1 00 −1)( f χaih (~p)|~p|f hχh(ai,~p))=12E(f 2−h′h |~p |2f 2)χ∗h′(ai,~p)χh(ai,~p) (22)In order to integrate (22) over S3 we write (a1,a2,a3,a4) in spherical coordinates:a1 = sinθ0 sinθ1 sinθ2, a2 = sinθ0 sinθ1 cosθ2, a3 = sinθ0 cosθ1, a4 = cosθ0.The volume element is dΩ3 = sin2 θ0 sinθ1dθ0dθ1dθ2. Using (14), the integration over (22) turnsout to be∫S3u(ai,~p ,h′)u(ai,~p ,h) =12E((E +mB)−h′h(E−mB))∫S3χai∗h′ (~p)χaih (~p) =mBEδh′h. (23)The spin sum is given by∑hu(~p ,h)u(~p ,h) = ∑h∫S3u(ai,~p ,h)u(ai,~p ,h), (24)where the integrand of (24) readsu(ai,~p ,h)u(ai,~p ,h) =12E( f χaih (~p)2~J.~pf χaih (~p))( f χai∗h (~p) − 2~J.~pf χai∗h (~p))=12E(E +mB −2~J.~p2~J.~p −E +mB)|χaih (~p)|2 =12E(−i 6 p+mB)|χaih (~p)|2. (25)Integrating over (25) we get the spin sum (24),∑h∫S3u(ai,~p ,h)u(ai,~p ,h) =12E(−i 6 p+mB)∑h∫S3|χaih (~p)|2 =1E(−i 6 p+mB). (26)54. Application: elastic form factorsThe SU(2) vector current integrated over the unit sphere of spatial R3 [3] behave as∫dΩ2J0,cV ∼ Tr[(∂0a)a−1τc] , and ε i jk∫dΩ2x jJk,cV ∼ Tr[τ ia−1τca], (27)where it should be noted that the SU(2)V currents were decomposed as JµV = Jµ,cV τc. Since thecollective coordinates do not depend on the position in R3, general vector currents readJ0,cV (~k) = e−i~k·~X Ic , Ji,cV (~k) = ie−i~k·~X Λεi jaq j Tr(τcaτaa−1) (28)where q j := p′j − p j. When calculating the vector current proton-proton matrix elements, we areonly interested in the c = 3 components (see above). We define the Dirac and Pauli form factorsaccording to the following decomposition of current matrix elements,〈pX ,BX ,sX |Jµ,aV (0)|p,B,s〉=i(τa)IX3 I32(2pi)3(ηµν − qµqνq2)u¯(pX ,sX)[γνFD,aBBX (q2)+κBσνλ qλ FP,aBBX (q2)]u(p,s)(29)In the elastic case, in the Breit frame, with ~p =−~p ′ =−q2zˆ, the expression (29) becomes〈pX ,BX ,sX |J0,aV (0)|p,B,s〉=12(2pi)3(τa)IX3 I3(mBE)δsX ,−s[FD,aB (q2)− q24m2BFP,aB (q2)](30)〈pX ,BX ,sX |Ji,aV (0)|p,B,s〉=−12(2pi)3(τa)IX3 I3(i2E)ε i jkq j(σk)sX ,−s[FD,aB (q2)+FP,aB (q2)](31)The equations above contain a relativistic correction not present in Skyrme Models.We need to calculate the vevs of Tr(τaa−1τ3a) in (28) in order to find the form factors.In spherical coordinates the traces become (a=3):Tr(τ3a−1τ3a) = 2(cos(θ0)2 + cos(2θ1)sin(θ0)2). (32)The general result for baryon states (a=1,2,3) is〈Tr(τaa−1τ3a)〉 ≡ 〈BX ,~p ′,h′|Tr(τaa−1τ3a)|B0,~p ,h〉=∫S3dΩ3 uX(ai,~p ′,h′)Tr(τaa−1τ3a)u0(ai,~p ,h)= −13√EX E ( f fX−hh′|~p ||~p ′|f fX )χ†h′(~p′)σ aχh(~p) (33)where fX =√EX +mBX and χh′(~p ′) and χh(~p) are the helicity spinors (13). The equationabove plays an important role in the calculation of the electromagnetic proton form factors [4]. Inthe elastic case, in Breit frame, ~p =−~p ′ =−q2zˆ and (33) takes the form〈Tr(τaa−1τ3a)〉= (E +mB−h′h(E−mB))2E(−23)σah′,−h=−23 mBE σ 1h′,−h for a = 1.−23 mBE σ 2h′,−h for a = 2.−23σ 3h′,−h for a = 3.(34)6where we used the substitution (15). This amounts to a relativistic correction in the direction ofmovement zˆ w.r.t. the static case [3], of a factor of mBE .Using (34) in the vector currents (28) , the current matrix elements can be written as〈q2zˆ,BX ,h′|J0,3V (0)|−q2zˆ,B,h〉 = 12(2pi)3(mBE)δ−h′h (35)〈q2zˆ,BX ,h′|Ji,3V (0)|−q2zˆ,B,h〉 = − 12(2pi)3(i2E)8mB3 Λεi3aq3σa−h′h (36)where we utilized eq. (23) and noted that 〈2I3〉+1/2,+1/2 = (τ3)+1/2,+1/2 = 1 for the proton-protonmatrix elements. Employing eqs. (35), (36) and eqs. (30), (31) we arrive atFD,3B (q2) =(1+q24m2B)−1(23Λq2mB+1), FP,3B (q2) =(1+q24m2B)−1(83mBΛ−1). (37)They are proton form factors for τ3 isovector current .5. ConclusionHere we presented a relativistic generalization for non-static baryons in Skyrme models. Thecorrect relativistic normalization for currents and the vacuum expectation values appears quitenaturally in our approach.AcknowledgementsThe authors H.B-F., N.R.F.B. and M.A.C.T. are supported by CAPES and CNPq (Brazilian re-search agencies). The work of M.I. was supported by an IRCSET postdoctoral fellowship.References[1] T. H. R. Skyrme, “A Nonlinear field theory,” Proc. Roy. Soc. Lond. A 260, 127 (1961).[2] H. Boschi-Filho, N. R. F. Braga, M. Ihl and M. A. C. Torres, Phys. Rev. D 85, 085013 (2012)[arXiv:1111.2287 [hep-th]].[3] G. S. Adkins, C. R. Nappi, E. Witten, “Static Properties of Nucleons in the Skyrme Model,”Nucl. Phys. B228, 552 (1983).[4] T. Sakai and S. Sugimoto, “Low energy hadron physics in holographic QCD,” Prog. Theor.Phys. 113, 843 (2005) [arXiv:hep-th/0412141].[5] K. Hashimoto, T. Sakai and S. Sugimoto, “Holographic Baryons: Static Properties and FormFactors from Gauge/String Duality,” Prog. Theor. Phys. 120 (2008) 1093 [arXiv:0806.3122[hep-th]].7

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