Phases and facets of 2-colour matter

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1 Phases and facets of 2-colour matter Jon-Ivar Skullerud with Tamer Boz, Seamus Cotter, Leonard Fister Pietro Giudice, Simon Hands Maynooth University New Directions in Subatomic Physics, CSSM, 10 March / 32

2 Outline QC 2 D vs QCD Lattice formulation 2 / 32

3 Tony and lattice QCD 3 / 32

4 QC2 D vs QCD Lattice formulation I I A plethora of phases at high µ, low T Based on models and perturbation theory Indirect approach Study QCD-like theories without a sign problem I Generic features of strongly interacting systems at µ 6= 0 I Check on model calculations, functional methods 4 / 32

5 QC 2 D vs QCD Lattice formulation QC 2 D vs QCD Baryons are bosons (diquarks); superfluid nuclear matter Scalar diquark is pseudo-goldstone (degenerate with pion) Onset transition at µ q = m π /2, not m N /3 5 / 32

6 QC 2 D vs QCD Lattice formulation QC 2 D vs QCD Baryons are bosons (diquarks); superfluid nuclear matter Scalar diquark is pseudo-goldstone (degenerate with pion) Onset transition at µ q = m π /2, not m N /3 Phase diagram Superfluid phase for µ > m π /2: BEC BCS? Exotic phases: quarkyonic, spatially varying? Deconfinement at high density, shape of deconfinement line? 5 / 32

7 QC 2 D vs QCD Lattice formulation Two-colour quarks and gluons Gluodynamics SU(2) and SU(3) very similar? Effects of deconfinement on gluon propagation? Gap equation with effective or one-gluon interaction used to determine superconducting gap more realistic input? Details of phase diagram depend critically on the effective quark mass in the medium. Dynamical quark masses effective strange quark mass? Location of Fermi surface? Direct determination of diquark gap, size of Cooper pairs? 6 / 32

8 QC 2 D vs QCD Lattice formulation Lattice formulation We use Wilson fermions: Correct symmetry breaking pattern, Goldstone spectrum N f < 4 needed to guarantee continuum limit No problems with locality, fourth root trick Chiral symmetry buried at bottom of Fermi sea 7 / 32

9 QC 2 D vs QCD Lattice formulation Lattice formulation We use Wilson fermions: Correct symmetry breaking pattern, Goldstone spectrum N f < 4 needed to guarantee continuum limit No problems with locality, fourth root trick Chiral symmetry buried at bottom of Fermi sea S = ψ 1 M(µ)ψ 1 + ψ 2 M(µ)ψ 2 J ψ 1 (Cγ 5 )τ 2 ψt 2 + Jψ 2 T (Cγ 5 )τ 2 ψ 1 γ 5 M(µ)γ 5 = M ( µ), Cγ 5 τ 2 M(µ)Cγ 5 τ 2 = M (µ) 7 / 32

10 QC 2 D vs QCD Lattice formulation Lattice formulation We use Wilson fermions: Correct symmetry breaking pattern, Goldstone spectrum N f < 4 needed to guarantee continuum limit No problems with locality, fourth root trick Chiral symmetry buried at bottom of Fermi sea S = ψ 1 M(µ)ψ 1 + ψ 2 M(µ)ψ 2 J ψ 1 (Cγ 5 )τ 2 ψt 2 + Jψ T 2 (Cγ 5 )τ 2 ψ 1 γ 5 M(µ)γ 5 = M ( µ), Cγ 5 τ 2 M(µ)Cγ 5 τ 2 = M (µ) Diquark source J κj introduced to lift low-lying eigenmodes in the superfluid phase study diquark condensation without uncontrolled approximations 7 / 32

11 QC 2 D vs QCD Lattice formulation Simulation parameters Name β κ a am π m π /m ρ Coarse fm Light fm Fine fm µ-scans, fixed T Ensemble N s N τ T (MeV) µa ja Coarse , 0.04 (0.03) , , 0.04 Fine , 0.03 (0.01) , , Light , 0.04 ( 0.03) 8 / 32

12 QC 2 D vs QCD Lattice formulation Simulation parameters T -scans, fixed µ (coarse lattice) All simulations done on 16 3 N τ lattices µa ja N τ , , 14, , , , 18, , 14, , 20 In addition, 300 trajectories were generated at ja = 0.03, 0.05 for N τ = 9, 10, 11 at all µ > 0. 9 / 32

13 Diquark condensate µ-scan Coarse lattice Results shown are for linear extrapolation Power law qq = Aj α works for µa 0.4, with α = [Effective field theory predicts α = 1 3 near onset] BCS scaling qq µ 2 for 0.35 µa 0.7 Melted at T = 141MeV (N τ = 8) 10 / 32

14 Diquark condensate µ-scan Coarse lattice Results shown are for linear extrapolation Power law qq = Aj α works for µa 0.4, with α = [Effective field theory predicts α = 1 3 near onset] BCS scaling qq µ 2 for 0.35 µa 0.7 Melted at T = 141MeV (N τ = 8) New transition for µa 0.7? Melting for N τ = 12, µa 0.7? N τ = 16 results are very close to N τ = 24 results. 10 / 32

15 Superfluid to normal transition j 0 extrapolation not fully under control Linear form used here Transition temperatures from inflection points (and j 0) aµ T s (MeV) (27) (9) (6) (7) Remarkably constant! 11 / 32

16 Deconfinement transition Polyakov loop L requires renormalisation, L R = e Fq/T = e (F 0+ F )/T = Z Nτ L L 0 We use two schemes to determine Z L = exp( a F ), Scheme A L R (T = 1 4a, µ = 0) = 1, Scheme B L R (T = 1, µ = 0) = a We determine the deconfinement temperature (crossover region) from the inflection point (linear region) of L R 12 / 32

17 Deconfinement transition Estimates from Scheme B, encompassing Scheme A Scheme dependence broad crossover? µa T d a T d (MeV) (20) 217(23) / 32

18 Number density Normalised to lattice SB or nsb Cont = 4µT 2 + 4µ3 3 3π 2 Close to SB scaling for 0.4 µa 0.7, N τ = 24, 12. Little T -dependence for T 100MeV. Strong thermal effects for N τ = 8 N τ = 16, 24 almost identical 14 / 32

19 Results from fine ensemble BCS scaling of diquark condensate confirmed Similar melting temperature Second transition at high µ a lattice artefact? No deconfinement at low T? 15 / 32

20 Number density on fine ensemble The same qualitative features as on coarse ensemble! 16 / 32

21 Results from light ensemble Deviation from BCS scaling: BEC window? 17 / 32

22 Essential ingredient in gap equation S 1 (p) = S0 1 (p) + Z 2 d 4 qγ µ (p, q)d µν (q)s(p q)γ ν used to determine dynamical fermion mass and superfluid/superconducting gap Link to functional methods (DSE, FRG) Electric gluon may signal deconfinement transition Tensor structure in medium D µν ( q, q 0 ) = P T µνd M ( q 2, q 2 0) + P E µνd E ( q 2, q 2 0) + ξ q µq ν (q 2 ) 2 P M µν( q, q 0 ) = (1 δ 0µ )(1 δ 0ν )(δ µν q µq ν q 2 ), P E µν(q 0, q ) = (δ µν q µq ν q 2 ) PM µν(q 0, q ). 18 / 32

23 results Some finite volume and lattice spacing effects at µ = 0 In-medium modifications, incl. violations of Lorentz symmetry, visible in magnetic gluon at high density 19 / 32

24 : µ-scans, N τ = / 32

25 : µ-scans, N τ = 8 Very little volume dependence No j-dependence observed All modes screened at high µ, low T : Weak-coupling theory says static D M is unscreened Static magnetic gluon not screened at high T 21 / 32

26 : T -scan, µa = / 32

27 fits 2-parameter fit: µ-scans D fit k (q2 ) = Λ2 (q 2 + Λ 2 a k ) b k (q 2 + Λ 2 ) 2 with k = M, E and Λa = 0.999(3) from µ = j = 0 fit T-scan 23 / 32

28 tensor structure S 1 ( p, ω) = i p A( p 2, ω 2 ) + iγ 4 ωc( p 2, ω 2 ) + B( p 2, ω 2 ) + iγ 4 p D( p 2, ω 2 ) S( p, ω) = i p Sa + iγ 4 ωs c + S b + iγ 4 p Sd where ω p 4 iµ. In general the form factors are complex! 24 / 32

29 tensor structure S 1 ( p, ω) = i p A( p 2, ω 2 ) + iγ 4 ωc( p 2, ω 2 ) + B( p 2, ω 2 ) + iγ 4 p D( p 2, ω 2 ) S( p, ω) = i p Sa + iγ 4 ωs c + S b + iγ 4 p Sd where ω p 4 iµ. In general the form factors are complex! The D form factors can be shown to vanish [Rusnak&Furnstahl 1995] 24 / 32

30 Gor kov formalism Quarks and antiquarks are in the same representation. Construct Gor kov spinor ( ) ψ Ψ = ψ T = Ψ(x) Ψ(y) G(x, y) = ( ) SN S A S A S N 25 / 32

31 Gor kov formalism Quarks and antiquarks are in the same representation. Construct Gor kov spinor ( ) ψ Ψ = ψ T = Ψ(x) Ψ(y) G(x, y) = ( ) SN S A S A S N S A contains information about anomalous propagation Self-energies are diquark gaps (superfluid/superconducting) 25 / 32

32 Gor kov formalism Quarks and antiquarks are in the same representation. Construct Gor kov spinor ( ) ψ Ψ = ψ T = Ψ(x) Ψ(y) G(x, y) = ( ) SN S A S A S N S A contains information about anomalous propagation Self-energies are diquark gaps (superfluid/superconducting) General tensor structure is the same as for normal components, scalar diquark gap is b. From isospin and charge conjugation symmetry it follows that S N (x, y) = S N (y, x) T, S A (x, y) = S A (y, x) T 25 / 32

33 Fermi surface and Cooper pairs Fermi surface In a Fermi liquid the Fermi surface is given by det S 1 ( p F, p 4 = 0) = 0 p 2 A 2 + ω 2 C 2 + B 2 = 0 Pole in propagator 26 / 32

34 Fermi surface and Cooper pairs Fermi surface In a Fermi liquid the Fermi surface is given by det S 1 ( p F, p 4 = 0) = 0 p 2 A 2 + ω 2 C 2 + B 2 = 0 Pole in propagator In gapped phase: zero crossing! 26 / 32

35 Fermi surface and Cooper pairs Fermi surface In a Fermi liquid the Fermi surface is given by det S 1 ( p F, p 4 = 0) = 0 p 2 A 2 + ω 2 C 2 + B 2 = 0 Pole in propagator In gapped phase: zero crossing! Size of Cooper pair If we know the anomalous propagator S A (x) we can compute the size of the Cooper pairs: ξ 2 = d 3 x x Tr(S A(x)Λ + ) 2 d 3 x 1 2 Tr(S A(x)Λ + ) 2 26 / 32

36 results in vacuum Large lattice spacing dependence Substantial quark mass dependence for Z(p) Unusual p-dependence in Z(p) infrared suppression recovered in low-mass and continuum limit? M(p) not yet properly corrected! 27 / 32

37 Normal form factors PRELIMINARY Dramatic changes in superfluid phase Vanishing M(0) at µa = 0.65? Zero crossing in S c 28 / 32

38 Anomalous form factors [µa = 0.7, ja = 0.02, n t = 1 2 ] Diquark gap is evident Nonzero tensor form factor A d Evidence of vector form factor A c 29 / 32

39 30 / 32

40 Evidence for three phases/regions Vacuum/hadronic phase below µ o = m π /2, low T BCS/quarkyonic for intermediate µ, low T Deconfined/QGP matter at high T 31 / 32

41 Evidence for three phases/regions Vacuum/hadronic phase below µ o = m π /2, low T BCS/quarkyonic for intermediate µ, low T Deconfined/QGP matter at high T Superfluid to normal T s (µ) remarkably flat above µ o Deconfinement crossover T d (µ) decreasing with µ Situation at very high µ unclear Possible deconfined superfluid region? 31 / 32

42 Evidence for three phases/regions Vacuum/hadronic phase below µ o = m π /2, low T BCS/quarkyonic for intermediate µ, low T Deconfined/QGP matter at high T Superfluid to normal T s (µ) remarkably flat above µ o Deconfinement crossover T d (µ) decreasing with µ Situation at very high µ unclear Possible deconfined superfluid region? Both electric and magnetic gluon screened at high µ, low T Static magnetic gluon not screened at high T Dramatic modification of quarks in superfluid region 31 / 32

43 Outlook Attempt O(2) scaling fit for superfluid to normal transition? Requires several larger lattice volumes Full analysis of quark propagator in progress Analysis of static quark potential in progress Phase scan on light ensemble in progress Temperature scans on fine ensemble Lighter quarks opening of BEC window? 32 / 32

44 Outlook Attempt O(2) scaling fit for superfluid to normal transition? Requires several larger lattice volumes Full analysis of quark propagator in progress Analysis of static quark potential in progress Phase scan on light ensemble in progress Temperature scans on fine ensemble Lighter quarks opening of BEC window? Happy 60th birthday Tony! 32 / 32