Hairy black holes in general Skyrme models

Transcription

1 Hairy black holes in general Skyrme models C. Adam University of Santiago de Compostela SIG5, Krakow, June 2016 based on C. Adam, O. Kichakova, Ya. Shnir, A. Wereszczynski, "Hairy black holes in the general Skyrme model", arxiv: June 21, 2016

2 Contents Skyrme models and Skyrmions Skyrmions & Gravity Einstein eqs. and dimensional reduction Self-gravitating Skyrmions, Neutron Stars, hairy BHs Hairy Black Holes BPS Skyrme model Standard Skyrme model Generalized Skyrme models Summary

3 Skyrme models Non-linear scalar field theories supporting top. solitons ("Skyrmions") Candidate low-energy EFT for QCD; (scalar) Skyrme field mesons Baryons and nuclei realized as top. solitons ("vortices" in "meson fluid") simplest case (two flavors): Skyrme field space = SU(2) (isospin) matrix U (three pions) top. degree of Skyrmion ("winding number " of map R 3 0 S3 SU(2) S 3 ) = baryon number B Syms. of (two-flavor) QCD: (chiral) SU(2) L SU(2) R broken to SU(2) iso

4 Original Skyrme model L = L 2 + L 4 }{{} +L 0, L 0 = µ 2 U(Tr(1 U)), U(0) = 0 L skyrme e.g. U π = Tr(1 U), µ = m π pot L 2 = a g µν Tr (L µl ν), L 4 = b Tr ([L µ, L ν] 2 ), L µ = U µu ( + ++ metric convention) Description of nucleons: 30% level precision Description of nuclei: Some successes: (iso-) spin excitational spectra Main problems: - too large binding energies: topological energy bound E cb, but not saturated (non-bps theory) But may be generalized to (near) BPS theory - Large B: crystals (not liquid)

5 Generalizations Poincare invariance & standard Hamiltonian (quadratic in time derivatives): quite restrictive L = L 2 + L 4 + L 0 + L 6 L 6 = c g 1 g µν B µ B ν, B µ = 1 24π 2 Tr (ɛµλρσ L λ L ρ L σ ) B µ... baryon current with baryon number B = d 3 xb 0 Submodel L 0 + L 6 has BPS property also: perfect fluid EM-tensor; SDiff symmetries Possibility of generalized near-bps Skyrme models

6 Skyrmions and Gravity Promote g µν to dyn. metric by adding EH action S = d 4 x g 1 2 ( 1 ) 16πG R + L Sk In principle: solve to describe self-gravitating Skyrmions, neutrons stars and hairy black holes In practise: needs simplification Static, radially symmetric metric (in Schwarzschild coord.) ds 2 = σ 2 (r)n(r)dt 2 + dr 2 N(r) = 1 2m(r)/r. N(r) + r 2 (dθ 2 + sin 2 θdφ 2 )

7 Symmetry reduction for matter: Either macroscopic: fluid/solid ɛ(r), p(r) with EoS p = p(ɛ) insert into Einstein eqs. Or: sym. red. of field theory Skyrme model: U = cos f + i sin f n τ Hedgehog (B = 1): f = f (r), n = (sin θ cos φ, sin θ sin φ, cos θ) Either calc. T µν, insert into Einstein eqs. 3 ODEs Or sym. red. of action L Sk E Sk(f, f, N) S = dt E, E = d 3 x ( g 1 ) 16πG R + ESk d 3 x g = dωdrr 2 σ simpler derivation of field eqs.

8 E[f, m, σ] = ) drσ ( m G + E r + b.t., E r 4πr 2 E Sk varying σ (Lag. mult.... constraint) varying m m = G E r varying f σ σ = 2G r E r N ( ) f r σ E f r = 0

9 Explicitly (α 2 = 4πG; pion mass potential) m = α2 2 [ ( 1 a 2 r 2 N f 2 + sin 2 f f = b ( ) ( ) + b sin 2 f Nf 2 + sin2 f 2r 2 + cn 2r 2 f 2 sin 4 f + m 2 π r 2 sin 2 f ] 2 σ σ = 1 ( 2 α2 f 2 ar + 2b sin2 f r 1 a + 2b sin2 f r 2 2 cos f sin3 f r 4 N + c sin4 f r 4 ( +m 2 sin f π 2N c 2fr sin4 f r fr sin2 f r 2 { [( 2 a r + N N 4 fr sin f r 4 N N N Generically, 4 integration constants σ 0 (see later) N + c ) sin4 f r 3 + σ ) σ + fr 2 sin(2f ) r 2 + f r sin(2f ) ] r 2 N 2 2fr sin f r 2 + 2fr 2 cos f sin 3 f r 4 + ) σ σ 4 fr sin f r 4 ) } σ σ

10 Regular solutions B = 1: Self-gravitating Skyrmions Boundary conditions f (0) = π; f ( ) = 0; m(0) = 0; σ( ) = 1 discrete # of solutions (0,1,2) B >> 1, spher.sym., e.g., neutron stars. BPS submodel: longitude φ Bφ general Skyrme: NO minimizers (f (0) = Bπ... unstable) macroscop. (fluid/crystal) description: ɛ(r) & p i (r) into E-eqs.

11 Hairy black holes B = 1: Boundary conditions (r h... horizon radius) N(r h ) = 0 : m(r h ) = r h ; f ( ) = 0; σ( ) = 1 2 Only 3 integration constants at r = r h : (Nf )(r = r h ) = N(r h )f (r h ) = 0 f (r h ) = F (f h, r h ) discrete # of solutions (0,1,2) (for fixed r h ) Here: free constants f (r h ) f h, σ(r h ) σ h determined by b.c. at r = B >> 1: hairy BHs of nuclear matter? BPS submodel: φ Bφ (but no hairy BHs) general Skyrme: NO minimizers (f (0) = Bπ... unstable) macroscop. (fluid/crystal) description: ɛ(r) & p i (r)??

12 Black Holes with Skyrmion hair Known results Standard Skyrme model L 2 + L 4 has both regular self-grav. solitons and hairy BHs H. Luckock and I. Moss, Phys. Lett. B 176 (1986) 341 S. Droz, M. Heusler and N. Straumann, PLB 268 (1991) 371 P. Bizon, T. Chmaj, Phys. Lett. B 297 (1992) 55 Recent result: BPS model L 6 + L 0 for specific potential U = 2f sin 2f : NO hairy BH (despite regular solitons with/without gravity) S.B. Gudnason, M. Nitta, N. Sawado, JHEP 1512 (2015) 013?What happens in general? One motivation see also: S.B. Gudnason, M. Nitta, N. Sawado, arxiv:

13 BPS model adapt GNS1 for general one-vacuum potentials U(f = 0) = 0; U,f 0, f [0, π] Static, spher. sym. L 6 + L 0 = c g 1 g 00 B 0 B 0 µ 2 U B 0 = (1/2π 2 ) sin θ sin 2 f f 0 = E r = c π 3 N r 2 sin4 f f 2 + 4πµ 2 r 2 U ( ) f r f σe r = sin 2 f r ( 2c π 3 Nσ r 2 sin2 f f r ( Nσ r 2 sin2 f f ) + 4πµ 2 r 2 U f ) = 2π 4 µ2 c r 2 σu f sin 2 f

14 If f ( ) < (compacton: f (R) < ), integrate 0 = Nσ R r 2 sin2 f f = 2π 4 µ r h c 2 R r h dr r 2 σu f sin 2 f > 0! Compacton: if f (R) = but B 0 (R) f (R) sin 2 f (R) <... proof slightly more complicated The BPS Skyrme model, for one-vacuum potentials, does NOT support hairy BHs (but supports stable top. solitons with/without gravity)... First known case

15 Skyrme model L 2 + L 4 Regular solutions: from Bizon, Chmaj (1992) free parameter f (0). Shooting s.t. f ( ) = 0 two branches (lower, stable and higher, unstable) bifurcate at α max

16 Hairy BH solutions: from Bizon, Chmaj (1992) free parameter f (r h ). Shooting s.t. f ( ) = 0 two branches ("lower", stable and "higher", unstable) bifurcate at rh max approach corresponding reg. sol. for r h 0

17 General Skyrme model L 2 + L 4 + L 6 m 2 πu π Regular solutions: a = b = m π = 1. σ(0) vs. α for some values of c higher α max for higher c (repulsion) two branches ("lower", stable and "higher", unstable) lower approaches corresponding G = 0 sol. higher branch does NOT go back to α = 0 for c c=0 c=0.01 c=0.1 c=1 (0)

18 Regular solutions: a = b = m π = 1. Rescaled ADM mass m( ) = G M ADM vs. α for some values of c lower mass (stable) and higher mass (unstable) branch bifurcation at α max (c). M c=0 c=0.01 c=0.1 c=

19 κ 2 = 1 4 gtt g rr ( r g tt ) 2 r=rh Hairy Black Holes Hairy BH solutions expected, at least for small r h Discrete # of solutions for fixed r h (two free constants f h, σ h, and two conditions f ( ) = 0 and σ( ) = 1) Useful BH quantities: BH entropy S = A h /4 = πr 2 h Black Hole surface gravity κ where Black Hole (Hawking) temperature T T = κ 2π = 1 4π σ(r h)n (r h ) σ(r h ) = 0 T = 0... extremal BH

20 Results: f h vs. r h. α = 0.05, a = b = m π = 1 for some c Upper (unstable) branch, c > 0: f h further decreases, does not approach f h = π. NO solution below certain rh ex (c) f(r h ) 2.8 c=0 c=0.01 c=0.05 c=0.1 c=0.3 c= r h

21 Results: σ h vs. r h. α = 0.05, a = b = m π = 1 for some c Upper (unstable) branch, c > 0: σ h = 0 for some rh ex(c) NO solution for r h < rh ex(c) r h rh ex... extremal limit: extremal (T = 0) BH (r h ) c=0 c=0.01 c=0.05 c= r h

22 Extremal limit, small c ( GNS2): Regular solution a = b = m π = 1 Fixed α: Two solutions for sufficiently small c unstable hairy BH might join unstable regular (r h = 0) solution (0) c=0 c=0.001 c=0.006 c=0.01 c=

23 Extremal limit, small c: hairy BH solution σ h vs. r h for a = b = m π = 1, α = 0.05 Upper branch solution exists for r h 0 for small c, joins unstable regular solution GNS2 "phase transition" for some c = c cr (α) Here c cr (0.05) (0) c=0 c=0.001 c=0.006 c= r h

24 Hairy Black Holes, Temperature a = b = m π = 1, α = 0.05 Upper branch solution approaches T = again for r h 0, for c = 0 and for c < c cr approaches T = 0 at r h = rh ex for c > c cr Here c cr (0.05) r h Schwarzschield c=0 c=0.001 c=0.006 c= T

25 Qualitative understanding of r ex h Expansion about r h N 0 + N 1 (r r h ) +... m r h 2 + m 1(r r h ) +... σ σ h + α2 σ h sin 2 f h J 2 8r h H(r h N 1 ) 2 (r r h) +... f f h + r h sin f h J 2H(r h N 1 ) (r r h) +... ( ) r h N 1 =1 2m 1 1 α2 m 2 2 πrh 2 (1 cos f h ) + 2a sin 2 f h + b r 2 h sin 4 f h J mπr 2 h 4 + ( b + 4arh 2 ) cos fh b cos(3f h ) H ar 4 h + 2br 2 h sin 2 f h + c sin 4 f h

26 Possible sing. r h N 1 0 for small r h For α too large: disaster For α small: may be avoided if f h π sufficiently fast for r h 0: Happens for c < c cr c > c cr : (r h N 1 ) 1 is large close to sing. with σ σ h + σ 1 (r r h ) +..., where σ 1 = α2 σ h sin 2 f h J 2 8r h H(r h N 1 ) 2 and σ(r) < σ( ), large r h N 1 compatible with b.c. σ( ) = 1 ONLY IF σ h is small. b.c. σ( ) = 1 imposes small σ h on numerical int., and σ h 0 at r h = r ex h before r hn 1 0

27 Role of Skyrme term L 4 f h vs. r h, for a = c = m π = 1, α = 0.05 rh max smaller for smaller b, lim b 0 rh max = 0 NO hairy black holes without the Skyrme term (b = 0) although regular Skyrmion solutions exist f(r h ) b=1 b=0.5 b=0.1 b= r h

28 Role of Skyrme term L 4 Rescaled ADM mass m( ) vs. r h, for a = c = m π = 1, α = 0.05 Again, lim b 0 rh max = 0 As always, M unst (r h ) > M st (r h ) 0.09 M b=1 b=0.5 b=0.1 b= r h

29 Role of n.l. Sigma-model term L 2 f h vs. r h, for for b = c = m π = 1, α = 0.05 rh max (a) increases with decreasing a Hairy BHs exist for a = f(r h ) a=1 a=0.5 a=0.1 a= r h

30 Role of n.l. Sigma-model term L 2 f (r) vs. r, for for b = c = m π = 1, α = 0.05, and r h = 0.01 For potential U = U π, a = 0 solution is compacton Compactons for less than quartic approach to vacuum 3 a=0.01 a=0.1 a=0.5 a=1 2.5 f(r) r

31 Summary of Results Skyrme-type models which do possess flat space & self-gravitating top. solitons but don t possess hairy Black Holes Examples: BPS submodel (exact result) Model L 0 + L 2 + L 6 (numerical) General condition for existence of hairy BHs Presence of quartic (Skyrme) term L 4 existence of top. soliton solutions (conjecture) If applicable (for large B) to gravitating nuclear/hadronic matter, interesting repercussions: L 4 might control formation of "hadron black-hole" bound states ("neutron stars" with black hole cores)

32 Influence of sextic term L 6 on hairy BH solutions For c > c cr(α), unstable branch solutions for r h 0 Instead, at r h rh ex, BH temperature T 0 Extremal (T = 0) hairy BH At same c cr(α), regular unstable sol. ceases to exist Future work: Other potentials: results potential-independent? See e.g. GNS2 hairy BHs L 4... deeper reason? Higher-dim Skyrme models: which term required for hairy BHs? Spinning stationary (Kerr-type) solutions: may induce hair even for b = 0? Nonzero cosmological constant?...

33 Backup

34 Detailed calculation of curvature scalar: R = 2 ( 2 m r 2 m + ( 3 σ m m r σ r r r ) + σ σ d 3 x g R = dωdrr 2 σr =... ( 1 2 m r = 2dΩdr [ (rmσ) + (rσ ) (rmσ ) + 2(mσ) 2m σ ]. ) )