QCD matter with isospin-asymmetry. Gergely Endrődi. Goethe University of Frankfurt in collaboration with Bastian Brandt, Sebastian Schmalzbauer

Transcription

1 QCD matter with isospin-asymmetry Gergely Endrődi Goethe University of Frankfurt in collaboration with Bastian Brandt, Sebastian Schmalzbauer SIGN March 2017

2 Outline introduction: QCD with isospin relevant phenomena pion condensation chiral symmetry breaking λ-extrapolation naive method singular value representation leading-order reweighting results summary 1 / 25

3 Introduction isospin density n I = n u n d n I < 0 excess of neutrons over protons excess of π over π + applications neutron stars heavy-ion collisions chemical potentials (3-flavor) µ B = 3(µ u + µ d )/2 µ I = (µ u µ d )/2 µ S = 0 2 / 25

4 Introduction isospin density n I = n u n d n I < 0 excess of neutrons over protons excess of π over π + applications neutron stars heavy-ion collisions chemical potentials (3-flavor) µ B = 3(µ u + µ d )/2 µ I = (µ u µ d )/2 µ S = 0 here: zero baryon number but nonzero isospin µ u = µ I µ d = µ I 2 / 25

5 Introduction QCD at low energies pions on the level of charged pions: µ π = 2µ I at zero temperature µ π < m π vacuum state µ π = m π Bose-Einstein condensation µ π > m π undefined on the level of quarks: lattice simulations no sign problem conceptual analogies to baryon density (Silver Blaze, hadron creation, saturation) technical similarities (proliferation of low eigenvalues) 3 / 25

6 Setup

7 Symmetry breaking QCD with light quark matrix M = /D + m ud 1 + µ I γ 0 τ 3 + iλγ 5 τ 2 chiral symmetry (flavor-nontrivial) SU(2) V U(1) τ3 4 / 25

8 Symmetry breaking QCD with light quark matrix M = /D + m ud 1 + µ I γ 0 τ 3 + iλγ 5 τ 2 chiral symmetry (flavor-nontrivial) SU(2) V U(1) τ3 4 / 25

9 Symmetry breaking QCD with light quark matrix M = /D + m ud 1 + µ I γ 0 τ 3 + iλγ 5 τ 2 chiral symmetry (flavor-nontrivial) SU(2) V U(1) τ3 spontaneously broken by a pion condensate ψγ 5 τ 1,2 ψ a Goldstone mode appears 4 / 25

10 Symmetry breaking QCD with light quark matrix M = /D + m ud 1 + µ I γ 0 τ 3 + iλγ 5 τ 2 chiral symmetry (flavor-nontrivial) SU(2) V U(1) τ3 spontaneously broken by a pion condensate ψγ 5 τ 1,2 ψ a Goldstone mode appears add small explicit breaking 4 / 25

11 Symmetry breaking QCD with light quark matrix M = /D + m ud 1 + µ I γ 0 τ 3 + iλγ 5 τ 2 chiral symmetry (flavor-nontrivial) SU(2) V U(1) τ3 spontaneously broken by a pion condensate ψγ 5 τ 1,2 ψ a Goldstone mode appears add small explicit breaking extrapolate results λ 0 4 / 25

12 Simulation details nx +ny +nz +nt staggered light quark matrix with η 5 = ( 1) ( ) D / M = µ + m λη 5 λη 5 /D µ + m we have γ 5 τ 1 -hermiticity η 5 τ 1 Mτ 1 η 5 = M determinant is real and positive det M = det( /D µ + m 2 + λ 2 ) pioneering studies [Kogut, Sinclair 02] [de Forcrand, Stephanov, Wenger 07] with unimproved action here: N f = rooted stout-smeared staggered quarks + tree-level Symanzik improved gluons 5 / 25

13 Condensates: definition and renormalization ψψ = T V log Z m π = T V log Z λ multiplicative renormalization Z π = Z 1 λ = Z 1 m = Z ψψ convenient normalization Σ ψψ m ψψ 1 m 2 πf 2 π 1 Σ π m π mπf 2 π 2 so that in leading-order chiral PT [Son, Stephanov 00] Σ 2 ψψ (µ I ) + Σ 2 π(µ I ) = 1 6 / 25

14 Condensates: old method

15 Pion condensate: old method traditional method [Kogut, Sinclair 02] measure full operator at nonzero λ (via noisy estimators) Σ π TrM 1 η 5 τ 2 extrapolation very steep 7 / 25

16 Pion condensate: old method traditional method [Kogut, Sinclair 02] measure full operator at nonzero λ (via noisy estimators) Σ π TrM 1 η 5 τ 2 extrapolation very steep 7 / 25

17 Pion condensate: old method traditional method [Kogut, Sinclair 02] measure full operator at nonzero λ (via noisy estimators) Σ π TrM 1 η 5 τ 2 extrapolation very steep 7 / 25

18 Chiral condensate: old method traditional method [Kogut, Sinclair 02] measure full operator at nonzero λ (via noisy estimators) Σ ψψ TrM 1 extrapolation very steep 8 / 25

19 Pion condensate: new method

20 Singular value representation pion condensate π = λ log det( /D µ + m 2 + λ 2 2λ ) = Tr /D µ + m 2 + λ 2 singular values /D µ + m 2 ψ i = ξi 2 ψ i spectral representation π = T V i 2λ ξi 2 + λ 2 = dξ ρ(ξ) 2λ ξ 2 + λ 2 λ 0 πρ(0) first derived in [Kanazawa, Wettig, Yamamoto 11] 9 / 25

21 Singular value representation pion condensate π = λ log det( /D µ + m 2 + λ 2 2λ ) = Tr /D µ + m 2 + λ 2 singular values /D µ + m 2 ψ i = ξi 2 ψ i spectral representation π = T V i 2λ ξi 2 + λ 2 = dξ ρ(ξ) 2λ ξ 2 + λ 2 λ 0 πρ(0) first derived in [Kanazawa, Wettig, Yamamoto 11] compare to Banks-Casher-relation at µ I = 0 9 / 25

22 Singular value density spectral densities at λ/m = / 25

23 Singular value density spectral densities at λ/m = 0.17 read off improved pion condensate 10 / 25

24 Reweighting reweighting factor but λ is small, so expand in it: so we obtain R = det( /D µ + m 2 ) det( /D µ + m 2 + λ 2 ) O rew = OR LO R LO R LO = e λv 4π + higher orders in λ λ = full LOE R vs. R LO on small lattices for ψψ ψψ µ 11 / 25

25 Density at zero scaling with λ is improved drastically 12 / 25

26 Density at zero scaling with λ is improved drastically 12 / 25

27 Density at zero scaling with λ is improved drastically 12 / 25

28 Comparison between old and new methods extrapolation in λ gets almost completely flat 13 / 25

29 New method for other observables

30 Singular value representation chiral condensate ψψ = singular values m log det( /D µ + m 2 +λ 2 ) = Tr ( /D µ + m) + ( /D µ + m) /D µ + m 2 + λ 2 spectral representation /D µ + m 2 ψ i = ξ 2 i ψ i ψψ = T V i 2 Re ψ i /D µ + m ψ i ξ 2 i + λ 2 14 / 25

31 Singular value representation spectral representation at λ = 0 ψψ = T V N i=1 2 Re ψ i /D µ + m ψ i ξ 2 i convergence not visible for N / 25

32 Improvement work instead with the difference δ ψψ ψψ(λ = 0) ψψ(λ) = 2T [ N 1 Re ψ i /D V µ +m ψ i ξ 2 i=1 i ] 1 ξi 2 + λ 2 ψψ(0) = ψψ(λ) + δ ψψ convergence already for small N 16 / 25

33 Improvement work instead with the difference δ ψψ ψψ(λ = 0) ψψ(λ) = 2T [ N 1 Re ψ i /D V µ +m ψ i ξ 2 i=1 i ] 1 ξi 2 + λ 2 ψψ(0) = ψψ(λ) + δ ψψ noisy estimators convergence already for small N 16 / 25

34 Improvement work instead with the difference δ ψψ ψψ(λ = 0) ψψ(λ) = 2T [ N 1 Re ψ i /D V µ +m ψ i ξ 2 i=1 i ] 1 ξi 2 + λ 2 ψψ(0) = ψψ(λ) + δ ψψ noisy estimators singular values convergence already for small N 16 / 25

35 Comparison between old and new methods extrapolation in λ gets almost completely flat 17 / 25

36 Improvement δ ni the same strategy for the isospin density n I = (log Z)/ µ I n I (0) n I (λ) = 2T [ ] N Re ψ i ( /D V µ +m) /D 1 1 µ ψ i ξi 2 + λ 2 i=1 ξ 2 i n I (0) = n I (λ) + δ ni noisy estimators singular values convergence already for small N 18 / 25

37 Comparison between old and new methods extrapolation in λ gets almost completely flat 19 / 25

38 Applications for the results

39 Transition temperature additively renormalized chiral condensate Σ ψψ (T ) Σ ψψ (T = µ I = 0) define T c using the temperature at a constant value (valid at low µ I ) ψψ R [GeV4 ] mr 0-3e-05-6e-05-9e preliminary µ I = 0 MeV µ I = 17 MeV µ I = 34 MeV T [MeV] crossover (T C, µ I,C ) P (T C, µ I,C ) C Pion condensation preliminary T [MeV] µ I /m π 20 / 25

40 Check Taylor-expansion isospin density via Taylor-expansion at µ I = 0 n I (µ I ) = χ I 2 µ I + χ I 4 µ 3 I +... using χ I 2,4 from [BMWc, ] ni /T Taylor exp. O(µ I) O(µ 3 I ) simulation T = 124 MeV m π/2 preliminary ni /T Taylor exp. O(µ I) O(µ 3 I ) simulation preliminary m π/ T = 176 MeV µ I [MeV] µ I [MeV] low T : breakdown of expansion at µ I = m π /2 high T : pin down validity range of LO and NLO expansion 21 / 25

41 Order of the transition finite volume scaling of pion condensate 22 / 25

42 Order of the transition fits fit transition region using chiral perturbation theory [Splittorff et al 02, Endrődi 14] 23 / 25

43 Order of the transition fits fit transition region using chiral perturbation theory [Splittorff et al 02, Endrődi 14] 23 / 25

44 Order of the transition fits fit using O(2) scaling [Ejiri et al 09] 24 / 25

45 Summary improve observables using singular values of /D µ + m flat λ-extrapolation 7 6 Taylor exp. O(µI) O(µ 3 I ) simulation direct check of Taylor-expansion convergence ni /T preliminary mπ/ T = 176 MeV µi [MeV] phase transition of second order 25 / 25