Color screening in 2+1 flavor QCD

Color screening in 2+1 flavor QCD J. H. Weber 1 in collaboration with A. Bazavov 2, N. Brambilla 1, P. Petreczky 3 and A. Vairo 1 (TUMQCD collaboration) 1 Technische Universität München 2 Michigan State University 3 Brookhaven National Lab Quarkonium Working Group meeting 217, PKU, 8/11/217 TUM-EFT 81/16; PRD 93 11452 (216); arxiv:161:81 (216) 1 / 17

Introduction First principle description of in-medium quarkonia Ingredients: QCD, no models Analytic approach: non-relativistic EFTs, dimensionally reduced EFTs Need weak coupling (expansion in g = 4πα s) Assume hierarchies between NR and thermal scales p T, p gt,... For T < 1 GeV: α s.4, g 2 Is weak coupling appropriate for realistic temperatures (g 1)? Are hierarchies actually realized (for g 2) and distinguishable? We test weak coupling/hierarchies/regimes using realistic lattice QCD simulations and heavy quarks in the static limit We aim at establishing whether the mechanisms described in EFTs are realized in QCD at the (experimentally) relevant temperatures 2 / 17

Overview Overview & Introduction Correlators of Polyakov loops Comparison to weak coupling Summary Color screening in 2+1 flavor QCD 3 / 17

Overview f bare Q = log P T Two different volumes and two quark masses: Controlled finite volume and quark mass dependence What is new about the TUMQCD lattices? 9 8 7 6 5 4 3 2 1 f Q bare N τ 4 1 86 12 16 6 6.4 6.8 7.2 7.6 8 8.4 8.8 9.2 9.6 TUMQCD collaboration, TUM-EFT 81/16 β Continuum limit with realistic quark masses N τ a(β) = 1/T (N τ, β) T [135, 2325] MeV with at least four N τ N τ =4 16: 12 3+ ens. each, 5.9 β 9.67, a =.85.25 fm. HISQ/Tree action, errors: O(α sa 2, a 4 ); lattice artefacts are reduced. N σ N τ Ensembles: = 4, m l = ms mπ = 161 MeV; β 7.825, a.4 fm 2 most from A. Bazavov et al., PRD 85 5453 (212), PRD 9 9453 (214) [HotQCD] All N τ < 16, m l = ms 5 : 3 5 ensembles each, 3 1 14 TU each, 7.3 β 8.4, a =.25.83 fm; T = lattices available. r 1 scale for β > 8.4 from non-perturbative β function PRD 9 9453 (214) 4 / 17

Free energies Polyakov loops and free energies of static (heavy) quark states The Polyakov loop L is the gauge-invariant expectation value of the traced propagator of a static quark (P) and related to its free energy: L(T ) = P T = Tr S Q (x, x) T = e F b Q /T. L needs renormalization. A. M. Polyakov, PL 72B (1978); L. McLerran, B. Svetitsky, PRD 24 (1981) The Polyakov loop correlator is related to singlet & octet free energies C P (r, T ) = e F b Q Q (r,t ) = 1 9 e F b S /T + 8 9 e F b A /T = 1 9 C S(r, T ) + 8 9 C A(r, T ). S. Nadkarni, PRD 33, 34 (1986) Meaning of gauge-dependent singlet & octet free energies is unclear. C P is also related to the gauge-invariant potentials V S,A of pnrqcd C P (r, T ) = e F b Q Q (r,t ) = 1 b 9 e V S /T + 8 9 Lb A e V b A /T + O(g 6 ) for rt 1. N. Brambilla et al., PRD 82 (21) 5 / 17

Free energies Renormalization of free energies Singlet free energy and potential appear to be related for rm D 1: [ ] e F S (r, T ) = C F α r m D s + m r D + O(g 4 ) = V S (r) + O(g 3 ). N. Brambilla et al., PRD 82 (21) F S and V S share the same renormalization 2C Q, which depends on T only through the lattice spacing: V S = V b S + 2C Q F S = F b S + 2C Q. Use V S at T = : fix r 1 scale & determine 2C Q using static energy. A. Bazavov et al., PRD 85 5453 (212), PRD 9 9453 (214) [HotQCD] Cluster decomposition theorem: F Q Q = F S = 2F Q for r 1/T. renormalize as F Q Q = F b Q Q + 2C Q and F Q = FQ b + C Q. PRD 93 11452 (216) Beyond C Q (β) from T = lattices use direct renormalization of F Q Infer unknown C Q (β) from known C Q (β ref ) using different N τ, Nτ ref C Q (β)= { C Q (β ref ) + FQ(β b ref, Nτ ref ) FQ(β, b N τ ) } S. Gupta et al., PRD 77 3453 (28) 6 / 17

Free energies Renormalization of free energies Singlet free energy and potential appear to be related for rm D 1: [ ] e F S (r, T ) = C F α r m D s + m r D + O(g 4 ) = V S (r) + O(g 3 ). N. Brambilla et al., PRD 82 (21) F S and V S share the same renormalization 2C Q, which depends on T only through the lattice spacing: V S = V b S + 2C Q F S = F b S + 2C Q. Use V S at T = : fix r 1 scale & determine 2C Q using static energy. A. Bazavov et al., PRD 85 5453 (212), PRD 9 9453 (214) [HotQCD] Cluster decomposition theorem: F Q Q = F S = 2F Q for r 1/T. renormalize as F Q Q = F b Q Q + 2C Q and F Q = FQ b + C Q. PRD 93 11452 (216) Beyond C Q (β) from T = lattices use direct renormalization of F Q Infer unknown C Q (β) from known C Q (β ref ) using different N τ, N ref C Q (β)= { C Q (β ref ) + F b Q(β ref, N ref τ ) F b Q(β, N τ ) + Nτ,N ref τ τ } PRD 93 11452 (216) 6 / 17

Color screening for a static quark-antiquark pair periodic boundaries τ 1 2... N x x Q Q r N τ. 2 1 7 / 17

Polyakov loop correlators Polyakov loop correlator and Q Q free energy -1-2 -3 _ F QQ (r,t)-t log 9 [GeV] 14 16 18 2 22 26 32 4 6 8 1 12-4 Preliminary continuum! r [fm].1.2.3.6.1.2.3.6.8 Free energy of a Q Q pair, F Q Q, is also called color-averaged potential: CP ren (r, T ) = P ()P (r) ren T = e F Q Q (r,t ) T = 1 F 9 e S (r,t ) T + 8 F 9 e A (r,t ) T. F Q Q T log 9 is close to the T = static energy V S for very small rt. 8 / 17

Polyakov loop correlators Singlet free energy in Coulomb gauge F S (r,t) [GeV] -1-2 -3-4 -5 Preliminary continuum! 14 16 18 2 22 26 32 4 6 8 1 12 16 22 28 r [fm].1.2.3.6.1.2.3.6 1. ren Singlet free energy: C ren S (r, T ) = 1 3 3 W a()w a (r) a=1 T = e F S (r,t )/T Wilson line correlator requires explicit gauge fixing (Coulomb gauge) F S is numerically close to the T = static energy V S for rt.3. 9 / 17

Polyakov loop correlators Effective coupling: vacuum-like and screening regimes 1.2 1.8.6.4.2 _ α QQ (r,t) 16 18 22 32 6 12 Preliminary continuum! r [fm].1.2.3.6.1.2.3 Effective coupling α Q Q(r, T ) is a proxy for the force between Q and Q. α Q Q α Q Q (r, T ) = r 2 E(r,T ) C F r, E = {F S (r, T ), V S (r)} clearly distinguishes different regimes at small and large r. Is weak coupling appropriate for α Q Q 1? test quantitatively. 1 / 17

Polyakov loop correlators Effective coupling: vacuum-like and screening regimes 1.2 1.8.6.4.2 Running coupling (Coulomb-like) _ α QQ (r,t) 16 18 22 32 6 12 Preliminary continuum! r [fm].1.2.3.6.1.2.3 Vacuum-like regime Screening regime max(α Q Q ) rt.2 rt.3 r maxt.4 Running coupling (string tension-like) Weakly-coupled QGP Effectively 3D (running stalls) Strongly-coupled QGP max(α Q Q) for 3 MeV r max defined through max(α Q), Q which is proxy for the maximal force. For T 3 MeV: max(α Q)(T Q ).5 strongly-coupled QGP. 1 / 17

Polyakov loop correlators 1.2 1.8.6.4.2 _ α QQ (r,t) 16 18 22 32 6 12 Effective coupling: vacuum-like and screening regimes for E = F S Preliminary continuum! r [fm].1.2.3.6.1.2.3 1..8.6.4.2 α QQ _ (r,t) 16 18 22 32 6 12 Finest lattices N τ = 16, 12, 1 r [fm] for E = F Q Q.1.3.6.1.2.3 Vacuum-like regime Color-octet contribution Screening regime rt.1 rt.3 rt.3 Coulomb-like regime dominated by color singlet: very small rt. Much stronger T dependence due to color-octet contribution. 1 / 17

pnrqcd and the vacuum-like regime Static energy and singlet free energy (I) - global features.1 [V S (r)-f S (r,t)]/t.8 251 273 333.6 47 474 562 666.4 Preliminary! N τ = 12.2 -.2 rt.5.1.15.2.25.3.35.4.45.5 V S (r)-f S (r,t) [MeV] 4 2 172 199 251-2 273 333 47 474-4 562 666 Preliminary! N τ = 12 r [fm].2.4.7.1.2.4.7 log scale! V S (T =) F S (T >) up to O(αs 3 ) M. Berwein et al., PRD 96 1425 (217) Cancellations in V S F S smoother for r/a < 3, no renormalization. For rt.1 & T > 3 MeV: V S F S.2T, mild N τ dependence. Mild T dependence for rt.25, sudden onset of medium effects. 11 / 17

pnrqcd and the vacuum-like regime Static energy and singlet free energy (II) weak coupling.3.25.2.15 [V S (r)-f S (r,t)]/t Preliminary! 273 333 47 474 562 666 N τ = 12.1 474.5 562 666 F S (r,t)/t 769 866 974 1162 134 -.5 1494 1666 1938 r [fm] -.1.5.1.15.2.25.3.25.2.15.1.5 [V S (r)-f S (r,t)]/t O(g 5 )+ F S (r,t)/t Preliminary! N τ = 12 47 474 562 666 -.5 rt -.1.5.1.15.2.25 V S (T =) F S (T >) up to O(α 3 s ) M. Berwein et al., PRD 96 1425 (217) Constant term αs 3 T in F S determine with T extrapolation F S [T ] = lim F S [T β, N N τ τ )] F S [T (β, N τ )] using only N τ 12. O(g 5 ) shifted by missing F S term captures main features of data. If V A V S 1/T which corresponds to rt < α s(1/r) thermal effects even exponentially suppressed Brambilla et al., PRD 78 1417 (28) 12 / 17

pnrqcd and the vacuum-like regime Color octet contribution in the Polyakov loop correlator (I).7.6.5.4.3.2 F O (r,t) [GeV] N τ = 8 Preliminary! 188 199 21 226 24 259 277 299 r [fm].1.15.2.25.3.35.4.45.5 pnrqcd: C P is given in terms of potentials V S and V A at T = and of the adjoint Polyakov loop L A at T > N. Brambilla et al., PRD 82 (21) C P (r, T ) = e F Q Q (r,t ) = 1 9 e V S /T + 8 9 L A e V A/T + O(g 6 ) for rt 1. Gauge-invariant decomposition of C P into color singlet and octet is defined assuming weak coupling test if it works for lattice as well. ( Color octet contribution: define e F O /T 9 8 e F Q Q (r,t ) ) 1 9 e V S /T As F O rapidly decreases for higher T, the octet becomes important. 13 / 17

pnrqcd and the vacuum-like regime Color octet contribution in the Polyakov loop correlator (II).2.18.16.14.12.1.8.6.4.2 -r 2 sub _ TF QQ (r,t) _ F QQ, data rec V S, exc. T=172 MeV N τ = 12 Preliminary!.5.1.15.2.25.3.35.4.45 rt.1.9.8.7.6.5.4.3 -r 2 sub _ TF QQ (r,t) F QQ _, data rec V S, cor. N τ = 12 Preliminary!.2 rt.1.5.1.15.2.25.3.35.4.45 Low T = 172 MeV: color singlet V S is enough for reconstructing C P (no sensitivity to color octet due to large statistical errors). High T = 666 MeV: cancellation between color singlet and octet leads to 1/r 2 behavior in F Q Q. We use pnrqcd, i.e. 1 9 e V S /T + 8 9 L A e V A/T. We include Casimir scaling violation: 8V A + V S = 3 α3 s [ π2 3] + r 4 O(α4 s ). F Q Q for rt.4 can be understood in terms of vacuum physics only. 14 / 17

EQCD and the electric screening regime Confronting weak-coupling predictions in the screening regime (I) 1e sub -rf S (r,t)/cf [a.u.] 1e-1 22 26 32 6 12 18 Hashed bands: LO Solid bands: NLO 1e-2 1e-3 Preliminary cont.! rt.2.4.6.8 1 1.2 Scale uncertainty µ = (1 4)πT due to resummation smaller for larger T Λ QCD = 32 MeV F S (r, T ) for rm D 1 at NLO in EQCD M. Laine et al., JHEP 73 54 (27) F sub S F sub S = F S 2F Q = C F α s e rm D (1 + α r s[δz 1(µ) + rt f 1(rm D )]) on the lattice is compatible with EQCD@NLO up to rt.6. Weakly-coupled EQCD is reasonable in electric screening regime of F S. For rt >.8: asymptotic screening is inherently non-perturbative. 15 / 17

EQCD and the electric screening regime Confronting weak-coupling predictions in the screening regime (II) 1e-1 1e-2 N τ = 4 N σ = 6N τ -9r 2 sub _ TF QQ (r,t) [a.u.] 421 1687 5814 Preliminary! Continuum limit with aspect ratio N σ =4N τ Hashed bands: LO Solid bands: NLO 1e-3 1e-4 42 16 Continuum rt.2.4.6.8 1 1.2 Scale uncertainty µ = (1 4)πT due to resummation smaller for larger T Λ QCD = 32 MeV Singlet and octet cancel at LO in F sub Q Q = α2 s 9 F Q Q(r, T ) = α2 s 9 e 2rm D r 2 + C F α sm D. e 2rm D r 2 (1 + α s[δz 1(µ) + rt f 1(rm D )]) S. Nadkarni, PRD 33 (1986) Continuum limit of F sub Q Q agrees with EQCD@NLO up to r 1/m D. Weakly-coupled EQCD is reasonable in the electric screening regime of F Q Q, but non-perturbative (chromo-magnetic) effects are stronger. 16 / 17

Summary We study color screening and deconfinement using the renormalized Polyakov loop correlator and related observables. Continuum limit of static quark correlators in N f = 2+1 QCD up to T 2. GeV and down to r.1 fm. Color-singlet correlators are vacuum-like up to rt.3, exhibit color-electric screening for rm D 1.3 rt.6 compatible with weak coupling and change to asymptotic screening for rt.7. The Polyakov loop correlator C P has a substantial color adjoint contribution for T 2 MeV. For rt.4 weakly-coupled pnrqcd describes C P well in terms of T = potentials and the adjoint Polyakov loop L A. C P has a color-electric screening regime rm D 1. Non-perturbative effects (i.e. the chromo-magnetic sector) are much larger. 17 / 17

Backup slides Continuum extrapolations Continuum extrapolations: singlet free energy rt =.16.5.1.15.2 CL 16 12 1 8 6 N τ 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 18 2 22 26 32 2 22 26 32 22 26 32 26 32 32 f S sub /cont-1 rt =.25.5.1.15.2.25.3.35.4 CL 16121 8 6 4 N τ 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 18 2 22 26 32 2 22 26 32 22 26 32 26 32 32 f S sub /cont-1 rt =.45.2.4.6.8 1 1.2 1.4 CL 16121 8 6 4 N τ 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 18 2 22 26 32 2 22 26 32 22 26 32 26 32 32 f S sub /cont-1.2.4.6.8.1.12.14.16.18 CL 16 12 1 8 6 N τ 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 6 8 12 16 2 8 12 16 2 12 16 2 16 2 2 f S sub /cont-1.5.1.15.2 CL 16121 8 6 4 N τ 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 6 8 12 16 2 8 12 16 2 12 16 2 16 2 2 f S sub /cont-1.1.2.3.4.5.6 CL 16121 8 6 4 N τ 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 6 8 12 16 2 8 12 16 2 12 16 2 16 2 2 f S sub /cont-1 17 / 17

Backup slides Continuum extrapolations.8.7 f sub _ QQ /cont-1 rt =.16 Continuum extrapolations: free energy 1.2 f sub _ QQ /cont-1 rt =.25 1.4 f sub _ QQ /cont-1 rt =.4.6.5.4.3.2.1 16 18 2 22 26 32 CL 16 12 1 8 6 N τ 1.8.6.4.2 16 18 2 22 26 32 CL 16121 8 6 4 N τ 1.2 1.8.6.4.2 18 2 22 26 32 CL 16121 8 6 4 N τ.7 f sub _ QQ /cont-1 1.2 f sub _ QQ /cont-1 1.8 f sub _ QQ /cont-1.6.5.4.3.2.1 4 6 8 12 16 2 1.8.6.4.2 4 6 8 12 16 2 1.6 1.4 1.2 1.8.6.4.2 4 6 8 12 16 CL 16 12 1 8 6 N τ CL 16121 8 6 4 N τ CL 16121 8 6 4 N τ 17 / 17

Backup slides Continuum extrapolations Continuum extrapolations: effective coupling α Q Q [F S] rt =.16.1.2.3.4.5.6.7.8 CL 16 12 1 8 6 N τ 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 18 2 22 26 32 2 22 26 32 22 26 32 26 32 32 α QQ _ /cont-1 rt =.25.1.2.3.4.5.6.7 CL 16121 8 6 4 N τ 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 18 2 22 26 32 2 22 26 32 22 26 32 26 32 32 α QQ _ /cont-1 rt =.45.5 1 1.5 2 CL 16121 8 6 4 N τ 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 16 18 2 22 26 32 18 2 22 26 32 2 22 26 32 22 26 32 26 32 32 α QQ _ /cont-1.1.2.3.4.5.6.7 CL 16 12 1 8 6 N τ 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 6 8 12 16 2 8 12 16 2 12 16 2 16 2 2 α QQ _ /cont-1.1.2.3.4.5 CL 16121 8 6 4 N τ 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 6 8 12 16 2 8 12 16 2 12 16 2 16 2 2 α QQ _ /cont-1.1.2.3.4.5 CL 16121 8 6 4 N τ 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 4 6 8 12 16 2 6 8 12 16 2 8 12 16 2 12 16 2 16 2 2 α QQ _ /cont-1 17 / 17