Skyrme models- BPS and/or SUSY, at last

Transcription

1 Skyrme models- BPS and/or SUSY, at last J.Sánchez-Guillén University of Santiago de Compostela Dual year Russia-Spain Particle, Nuclear and Astrop. Physics Barcelona, Nov 8-11, 2011

2 Contents The Problem: Skyrme proposal vs. Bogomol nyi (BPS) bounds and Susy extensions. A Hint: Simplified Skyrme models: restricted and planar. Solution: Generalized BPS Skyrme model. Symmetries, Integrability, Bogomolny bound. Exact solutions. Some phenomenology of hadrons and nuclei. Susy extension. Conclusions and outlook. [based on works in : Phys.Lett.B , 2010; Phys.Rev.D82 (2010) , D84(2011) and ; Acta Phys. Pol B41 (2010) and arxives]

3 Skyrme vs. Bogomol nyi (BPS) and Susy Hadrons as soliton solutions of effective field mesonic action balancing the σ term with a quartic geometric contribution topological charge = baryon number. Extensive applications for low energy phenomenology of baryons and nuclei, albeit demanding numerics and problems in large N c QCD connection. Original Skyrme version CANNOT saturate Bogomol nyi bound, which would explain stability and yield solvability (1st. order eqs.). Susy extensions, usually related to BPS, also elusive for Skyrme.

4 Original Skyrme model: Skyrme field U: Lagrangian: x µ U(x) : R 3 R SU(2) Sigma model term: L = L 2 + L 4 + L 0 pion kinetic term L 2 = f 2 π 4 Tr (U µ U U µ U)

5 quartic Skyrme term: L 4 = 1 32e 2 Tr ([U µ U, U ν U] 2 ) necessary to circumvent the Derrick argument L 2, L 4 symmetric under SU(2) SU(2), U VUW Potential term (optional) symmetric under diagonal SU(2): L 0 (= µ 2 V(U, U )) = µ 2 V(tr U) Skyrme CANNOT saturate BPS, no analytical solution.

6 Simplifying Skyrme Lagrangian Planar (d = 2 + 1) constrained vector field version L = λ 2 2 µ φ µ φ λ 4 4 ( µ φ ν φ) 2 λ 0 V(φ 3 ) φ = (φ 1,φ 2,φ 3 ); φ 2 = 1. Potential (λ 0 ) now mandatory. Same topology as ordinary, but more simple ("baby"). Interesting applications in condensed matter (Pokrovsky) For λ 4,0 = 0, meromorphic BPS static solutions (or instantons) (Belavin Polyakov 75). L 4 + L 0 is Derrick stable alternative to conformal L 2 of BP.

7 Numerical analysis of baby Skyrme (bs) shows many interesting numerical solutions, like Q-balls, Q-shells, and Peakons, rather indep. of potential, but... stable multisoliton solutions, exist only for potentials with with exact (compacton) solutions in the restricted action L 4 + L 0 = 1 2 ( µ φ ν φ) 2 + µ 2 V(φ 3 ), which saturate a generalized BPS bound (Ward) of the full bs. Energies of RbS approximate reasonably full bs. Huge Symmetries: Area preserving diffeos on base and Abelian subgroup on target generalized integrable (GI).

8 (GI) is a proposal, (with Alvarez, Ferreira (97),Int. J. Mod. Phys. A 24, 1825 (2009)) to extend to higher dimensions the geometric tools of 2d Integrability, based on holonomies in loop spaces with connections and nonabelian Stokes(Arefeva (80)). GI implies infinite conservation laws and solvability. the full bs has just an integrable submodel defined by the eikonal equation ( µ u) 2 = 0, but the restr. RbS model IS GIntegrable (with the conserved currents) J µ = δg δū K µ δg δu K µ, G = G(uū), K µ = K µ (1 + u 2 ) 2, K µ = (u ν ū ν )ū µ ū 2 ν uµ...and exactly solvable (with compactons)!

9 Corrected Skyrme model: Effective theory higher powers of derivatives expected. L 2 + L 0 - excluded by Derrick and L 4 + L 0 by dynamics: no topological solitons. Consider following Sextic term L 6 = λ2 [ ] Tr (ǫ µνρσ U µ U U ν U U ρ U) Quadratic in time derivatives standard hamiltonian formulation. Remarkable math. and physical properties.

10 Square of the topological (baryon) current where L 6 = λ 2 π 4 B µ B µ B µ = 1 24π 2 Tr (ǫµνρσ U ν U U ρ U U σ U) is the topological (baryon) current Phenomenologically induced by a massive vector meson coupled to the baryon density, improves results of ordinary Skyrme model.

11 BPS Skyrme model: Propose following limit of generalized Skyrme models L 06 = L 6 + L 0 many symmetries, many conservation laws, GIntegrable. BPS (Bogomolny) bound many exact solutions saturating the BPS bound

12 Parametrization for U U = e iξ n σ = cos ξ + i sin ξ n σ n 2 = 1 and stereographic projection n = 1 ( 1 + u 2 u + ū, i(u ū), 1 u 2) L 06 = λ2 sin 4 ξ (1 + u 2 ) 4 (ǫµνρσ ξ ν u ρ ū σ ) 2 µ 2 V(ξ)

13 Symmetries many target space diffeomorphisms L 6 is square of pullback of the volume form in target space SU(2) S 3, sin 2 ξ dv = i (1 + u 2 ) 2 dξdudū has all volume-preserving diffeomorphisms on target space S 3 as symmetries. L 0 = µ 2 V(ξ) respects some of them: the ones that act nontrivially only on u, ū area-preserving diffeos on target space S 2 spanned by u, but may depend on ξ as parameter ξ ξ, u ũ(u, ū, ξ), (1 + ũ 2 ) 2 dξdũd ũ = (1 + u 2 ) 2 dξudū Lie algebra corresponds to N of SU(N) (J. Hoppe 82).

14 Poincare Symmetries Volume-preserving base space diffeos. Energy functional for static fields: E = ( ) d 3 λ 2 sin 4 ξ x (1 + u 2 ) 4(ǫmnl iξ m u n ū l ) 2 + µ 2 V Both d 3 x and ǫ ijk i ξ j u k ū invariant under volume preserving diffeos on base space R 3. NOT a Noether symmetry.

15 Bogomolny (BPS) bound BPS bound B... top. charge (baryon number) ( ) E = d 3 λ 2 sin 4 ξ x (1 + u 2 ) 4 (ǫmnl iξ m u n ū l ) 2 + µ 2 V ( = d 3 λ sin 2 ξ x (1 + u 2 ) 2 ǫmnl iξ m u n ū l ± µ ) 2 V d 3 x 2µλ sin2 ξ V (1 + u 2 ) 2 ǫ mnl iξ m u n ū l [ ±(2λµπ 2 i ) π 2 d 3 x sin2 ξ ] V (1 + u 2 ) 2ǫmnl ξ m u n ū l = 2λµπ 2 B i dξdudū sin 2 ξ V(ξ) π 2 (1 + u 2 ) 2 S 3 2λµπ 2 < V > S 3 B

16 Exact solutions non-trivial configuration: u covers full C, and ξ [0, π] symmetric (hedgehog) ansatz ξ = ξ(r), u(θ,φ) = g(θ)e inφ compatible with field eq. Equation for g(θ) ( ) gg θ θ (1 + g 2 ) 2 = 0 sin θ with solution for all n g(θ) = tan θ 2

17 Equation for ξ or with first integral n 2 λ 2 sin 2 ξ 2r 2 r ( ) sin 2 ξ ξ r r 2 µ 2 V ξ = 0 z = 2µ 3 n λ r 3 sin 2 ξ z ( sin 2 ξ ξ z ) V ξ = 0 dim. reduced version of BPS 1 2 sin4 ξ ξ 2 z = V

18 Example: Skyrme potential V = 1 Tr(1 U) 2 V(ξ) = 1 cos ξ Solution compacton Energy ξ = { linear in B = n 2 arccos 3 3z 4 z [ 0, 4 ] 3 0 z 4 3. E = 64 2π 15 µλ n

19 energy density where E = 8 2µλ(1 n 2 3 r 2 ) for 0 r n 1 3 = 0 for r > n 1 3 ( ) 1 µ r = 4 3 r r 2λ R 0 rescaled radius r R 0... compacton radius for n = 1 baryon density B = sign(n) 4 π 2(1 n 2 3 r 2 ) 1 2 for 0 r n 1 3 = 0 for r > n 1 3

20 Figure: Normalized energy density as a function of the rescaled radius r, for topological charge n=1. For n > 1, the height of the density remains the same, whereas the radius grows like n 1 3

21 Figure: Normalized topological charge density as a function of the rescaled radius r, for topological charge n=1. For n > 1, the height of the density remains the same, whereas the radius grows like n 1 3

22 Some basic phenomenology of nuclei for standard Skyrme potential V = 1 cos ξ

23 A) Classical solitons Linear mass - baryon number relation: BPS bound E = 64 2µλ B /15 E 0 B Masses discrepancy consistently below 0.7%. No binding energy BPS solutions zero binding energy binding energies for physical nuclei small 1% in standard Skyrme model significantly bigger No forces between solitons BPS and compactons no forces physical nuclei: very short range interaction.clustering. radii of nuclei Compacton: definite radius B 1 3 : R = (2 2λ B /µ) (1/3) R 0 3 B, R 0 1, 25fm. Basic relations for nuclei.

24 Incompressible fluid Static energy functional: volume-preserving diffeos (VPDs) Symmetries of an incompressible fluid where surface energy is neglected Physical nuclei do not have this symmetry: have definite shape, deformations cost energy But: volume-preserving deformations cost less energy, liquid drop model of nuclei... cristals in ord. Skyrme classical model reproduces some features of the nuclear liquid drop model (mass prop to volume, strictly finite size.) Alternative starting point quantum and σ corrections (pion interactions, here just from chiral breaking).

25 B) Soliton quantization work in progress; top. charge Q = 1 (nucleon) Collective coordinate quantization (Bogolubov): Quantize symmetry transformations which are NOT symmetries of the soliton (=lightest d.o.f.) Example: rotations (equivalent to isospin = diagonal target space SU(2), because SO(3) b SU(2) diag IS symmetry of hedgehog Rigid rotor quantization (=undeformed rotating soliton, quantized angular momentum) Well-known procedure

26 E.g. for isospin: Insert U(t) = A(t)U 0 A (t) into action (U 0 = Q = 1 soliton (hedgehog) Promote the three time-dependent parameters in A(t) to quantum mechanical variables Result: E = E I J 2 = E I S 2 J... isospin, S... spin, E0... soliton energy (mass) I... moment of inertia of the soliton (rigid rotor) I = 4π 3 dr sin 4 ξ ξ 2 r Classical result: E = E I L 2 L... angular momentum

27 Concretely for Q = 1 soliton (compacton) I = 28 2π 15 7 λµ ( ) 2 λ µ 3, E0 = 64 2π λµ 15 E = E I 2 s(s + 1) Now fit to the proton mass (spin s = 1 2, M p = MeV) and resonance mass (s = 3 2, M = 1232 MeV) and use = MeV fm then λµ = 45.66MeV, λ µ = fm3 may now be used to "predict" other nucleon quantities

28 Charge radii Isoscalar and isovector charge densities (per unit length) ρ 0 = 2 π sin2 ξ ξ r, ρ 1 = 1 I 4π 3 λ2 sin 4 ξ ξ 2 r Resulting isoscalar and isovector mean square electric radii < r 2 > 0 < r 2 > 1 drr 2 ρ 0 = ( λ µ drr 2 ρ 1 = 10 9 ) 2 3 ( ) 2 λ 3 µ Isoscalar magnetic mean square radius < r 2 > m,0 = drr 4 sin 2 ξ ξ r drr 2 sin 2 ξξ r = 5 4 ( ) 2 λ 3 µ

29 Numerical values of charge radii radius BPS massive exp. Skyrme Skyrme compacton electric isoscalar r e, electric isovector r e, magnetic isoscalar r m, r e,1 /r e, r m,0 /r e, r e,1 /r m, Table: Compacton radius and some charge radii and their ratios for the nucleon. All radii are in fm. Radii small by at most 30%; not surprising (no pion cloud), discrepancy of ratios of radii never exceed15%. Magnetic moments better, comparable to standard.

30 Quantum aspects - Q > 1 sector spin and isospin quantum excitations break BPS relation small binding energies, internuclear forces. Bonenfant, Marleau (10) quadratic and quartic small corrections B.E./A (MeV) 5 4 Set Ia Set II 3 Set III Exper. data Baryon number

31 Susy extensions of Baby Skyrmions Baby Skyrme Lagrangian (d = 2 + 1) L = λ 2 2 µ φ µ φ λ 4 4 ( µ φ ν φ) 2 λ 0 V(φ 3 ) SUSY extension: superfield Φ(x, θ) = φ(x) + θ α ψ α (x) θ 2 F(x) susy invar. constraint Φ 2 = 1 φ φ = 1, φ ψ α = 0, φ F = 1 2 ψ α ψ α E. Witten, Phys. Rev. D16, 2991 (1977) Bosonic sector ψ = 0: φ φ = 1, φ F = 0

32 SUSY extension, with D α = α + iθ β αβ O(3) sigma model term (L 2 ) ψ=0 = 1 2 [(Dα Φ i D α Φ i ) ] θ 2 ;ψ=0 = F i F i + µ φ i µ φ i Potential (L 0 ) ψ=0 = [P(Φ 3 ) ] θ 2 ;ψ=0 = F 3P (φ 3 )

33 Quartic (Skyrme) term (L 4 ) ψ=0 = 1 2 ǫ ijkǫ i j k[(d α Φ i D α Φ i D 2 Φ j D 2 Φ j + D α Φ j D α Φ j D 2 Φ i D 2 Φ i ) ] θ 2 ;ψ=0 1 8 ǫ ijkǫ i j k[(d α Φ i D α Φ i D γ D β Φ j D γ D β Φ j + + D α Φ j D α Φ j D γ D β Φ i D γ D β Φ i ) ] θ 2 ;ψ=0 = ǫ ijk ǫ i j k(f i F i F j F j µ φ i µ φ i ν φ j ν φ j ) = ( µ φ νφ) 2

34 Bosonic Lagrangian L b = λ 2 2 [( F) 2 + µ φ µ φ] λ 4 4 ( µ φ ν φ) 2 +λ 0 F 3 P + µ F ( F φ) + µ φ ( φ 2 1) µ F and µ φ... Lagrange multipliers enforcing constraints Field equ. for F i : Solution F i = λ 0 λ 2 (φ 3 φ i δ i3 )P µ F = λ 0 φ 3 P

35 Resulting Lagrangian, Baby Skyrme with potential 0 L b = λ 2 2 µ φ µ φ λ 4 4 ( µ φ ν φ) 2 λ2 0 2λ 2 (1 φ 2 3 )P 2 + µ φ ( φ 2 1) standard (BS) L b with V 0 allow for SUSY extensions, (aswell as just L 2 and L 4 ) BUT NOT the BPS restricted model L 4 + L 0, since the only solution with λ 2 = 0 is λ 0 = 0. Susy extensions of CP 1 version up to now unsuccessful. If finally impossible, our positive result for O(3)-version may point to quantum nature of susy fermions. Our procedure generalizable to higher dimensions. (Khoury, Lehners, Ovrut Phys.Rev. D84 (2011) ).

36 Conclusions Our first results show that Nonlinear theories of Skyrme type still contain unexplored challenges which may lead to relevant physics, with deeper understanding and wide phenomenological applications. useful mathematical tools, like new types of exact topological bounds, generalized integrability, and non standard susy representations. Outlook Improve nuclear, hadronic and condensed phenomenology. Perturbations and time dependence. Generalize Susy extensions to and explore consequences and applications. Connection with QCD and other non-perturbative tools (World-line and RG and GI methods).