Probing the QCD phase diagram with higher moments

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1 Probing the QCD phase diagram with higher moments in collaboration with: F. Karsch B.-J. Schaefer A. Walther J. Wambach

2 Outline Why higher moments? Algorithmic differentiation Lattice Taylor expansion to finite density extracting the phase boundary locating the critical end point Higher moments as experimental probes precursors of the critical end point

3 QCD Phase diagram not perturbative T Quark-Gluon Plasma Hadron Gas qq > qq > = μ

4 QCD Phase diagram not perturbative Lattice QCD T Quark-Gluon Plasma from first principles Hadron Gas qq > qq > = μ

5 QCD Phase diagram not perturbative Lattice QCD T Quark-Gluon Plasma from first principles Hadron Gas qq > qq > = μ

6 QCD Phase diagram not perturbative Lattice QCD Lattice T~15 MeV T Quark-Gluon Plasma Crossover from first principles Hadron Gas qq functional methods 1st order > renormalisation group Dyson-Schwinger equations qq > = models µ~3 MeV μ effective models

7 QCD Phase diagram not perturbative Lattice QCD Lattice T~15 MeV T Quark-Gluon Plasma Crossover CEP from first principles Hadron Gas qq functional methods 1st order > renormalisation group Dyson-Schwinger equations qq > = models µ~3 MeV μ effective models

8 QCD Phase diagram not perturbative 2 Lattice QCD T, MeV 15 from first principles functional methods 1 CO94 renormalisation group 5 13 LTE4 17 Lattice T~15 MeV HB2 LTE3 LR4 9 5 INJL98 Dyson-Schwinger equations T Quark-Gluon Plasma Crossover LR1 RM98 PNJL6 LSM1 CJT2 2 3NJL5 CEP Hadron Gas NJL1 qq > NJL89a qq 1st order > = models µ~3 MeV NJL89b μ effective models µ B, MeV

9 QCD Phase diagram not perturbative Lattice QCD Lattice T~15 MeV T Quark-Gluon Plasma Crossover CEP from first principles Hadron Gas qq functional methods 1st order > renormalisation group Dyson-Schwinger equations qq > = models µ~3 MeV μ effective models experimental: heavy-ion collisions

Hadron Gas 1 µs Hadron gas (confined) qq > 1st order = CEP Quark gluon plasma (deconfined) Lattice quantum chromodynamics

10 QCD Phase diagram not perturbative Lattice QCD from first principles functional methods renormalisation group Lattice T~15 MeV Dyson-Schwinger equations T c Temperature (MeV) Quark-Gluon Plasma 25 Crossover Early Universe Hadron Gas 1 µs Hadron gas (confined) qq > 1st order = CEP Quark gluon plasma (deconfined) Lattice quantum chromodynamics qq Bag model > models µ~3 MeV μ effective models experimental: heavy-ion collisions 25.1 s , Chemical potential (MeV)

11 Lattice QCD at vanishing density chiral crossover 1.8 Δ l,s figures from L. Levkova (Lattice 211) s fk scale HISQ (HotQCD): T ~ 157 MeV.6 stout (Budapest-Wuppertal): T ~ 147 MeV confinement / deconfinement in the same region (depends on observable) thermodynamic observables equation of state asqtad: N τ =8 N τ =12 HISQ/tree: N τ =6 N τ =8 N τ =12 stout cont. T [MeV] (ε-3p)/t 4 HISQ/tree: N τ =8 N τ =6 p4: N τ =8 asqtad: N τ =12 N τ =8 stout HRG T [MeV]

The fermion sign problem Monte Carlo simulations rely on fermion determinant as probability weight fails at finite chemical potential approaches to circumvent the sign problem: Taylor expansion

12 The fermion sign problem Monte Carlo simulations rely on fermion determinant as probability weight fails at finite chemical potential approaches to circumvent the sign problem: Taylor expansion analytic continuation from imaginary μ reweighting density of state canonical ensemble stochastic quantization (under construction) world line formalism (under construction) complex for μ> Contact your local experts

13 Taylor expansion to finite density Taylor expansion p(t,µ) T 4 = c n (T ) n= µ T n c n (T )= 1 n! n p(t,µ)/t 4 (µ/t ) n µ= coefficients calculated with standard (i.e. μ = ) methods lattice calculations by various groups (e.g. Allton, C. Schmidt, Gavai & Gupta, detar, Ejiri,...) HISQ: RBC-Bielefeld (211) u HISQ: N =6 N = Q HISQ: N = SB. -.5 SB.5-1. T [MeV] T [MeV]

14 Hopes works for small µ/t, i.e., µ/t < 1 only a few coefficients required for convergence calculate thermodynamic observables relevant for heavy-ion collisions might locate the critical endpoint troubles with phase transitions (divergences)

15 Hopes works for small µ/t, i.e., µ/t < 1 Philipsen (CPOD 29) The calculable region of the phase diagram only a few coefficients required for convergence calculate thermodynamic observables relevant for heavy-ion collisions might locate the critical endpoint T T c! QGP troubles with phase transitions (divergences) confined Color superconductor! 21-present: sign problem not solved, circumvented by approximate methods: reweigthing, Taylor expansion, imaginary chem. pot., need µ/t < 1 (µ = µ B /3)

16 Hopes works for small µ/t, i.e., µ/t < 1 only a few coefficients required for convergence calculate thermodynamic observables relevant for heavy-ion collisions might locate the critical endpoint early universe ALICE quark gluon plasma troubles with phase transitions (divergences) Temperature crossover < > > RHIC CBM SPS < > quark matter vacuum hadronic fluid n = n > B B nuclear matter µ crossover superfluid/superconducting phases? 2SC < > > CFL neutron star cores

17 Signals of the critical endpoint? quark number susceptibility convergence radius q (T,µ) T 2 2 = X µ n 2 n(n 1)c n (T ) T n=2,4,...! B / T 2! B /T=. C. Schmidt (28)! B /T=1.5! B /T=2.5 T [MeV] r = lim n!1 r 2n = lim n!1 T / T c () Gavai, Gupta Nt=4 Nt=6 Fodor, Katz Nt=4! 2! 4 c 2n c 2n+2 1/2 C. Schmidt (28)! B / T c () r 2 r 4

susceptibility diverges realistic experiments: finite

18 Signatures of the critical end point second order phase transition correlation length diverges quark number susceptibility diverges realistic experiments: finite volume regular part might dominate higher orders depend on powers with higher

19 Polyakov-Quark-Meson (PQM) model B-J. Schaefer, MW, J. Wambach, Phys. Rev. D 81, 7413 relevant degrees of freedom: (2+1) quarks and mesons PQM model (parameters fitted to meson masses and decay constants) L PQM = q (id/ g 5 ) q + L m U(, ) Polyakov loop variable quarks / antiquarks L m =Tr µ µ m 2 Tr( ) 1 Tr( ) 2 + c det( )+det( ) +Tr H( + ) 2 Tr 2 mesonic fields here: logarithmic Polyakov loop potential (Roessner et al, Phys. Rev. D75, 347) U log T 4 = 1 2 a(t ) h + b(t )ln i

Polyakov-Quark-Meson (PQM) model relevant degrees of freedom: (2+1) quarks and mesons PQM model (parameters 1 fitted to meson masses and decay constants) quarks / antiquarks p/p SB L PQM.

20 Polyakov-Quark-Meson (PQM) model relevant degrees of freedom: (2+1) quarks and mesons PQM model (parameters 1 fitted to meson masses and decay constants) quarks / antiquarks p/p SB L PQM.8= q (id/ g 5 ) q + L m U(, ) B-J. Schaefer, MW, J. Wambach, Phys. Rev. D 81, 7413 Polyakov loop variable.6 QM L m =Tr µ µ m 2 Tr( PQM ) 1 Tr( log ) Tr 2 + c det( )+det( ) +Tr PQM H( + ) pol PQM Fuku.2 p4 mesonic fields asqtad here: logarithmic Polyakov.5 loop potential (Roessner et al, 2 Phys. Rev. D75, ) 3 U log T 4 = 1 2 a(t ) h T/T + b(t )ln i 2

)/T f=u,d,s +ln h1 + 3( io + e (E q,f +µ f )/T )e (E q,f +µ f )/T + e 3(E q,f +µ f )/T equations of motion @ = @ = @ @

21 Explicit and implicit μ-dependence mean-field approximation thermodynamic potential = U ( x, y)+ qq x, y,, + U, quark contribution qq ( x, y,, ) = 2T X Z d 3 p n (2 ) 3 ln h1 + 3( + e i (E q,f µ f )/T )e (E q,f µ f )/T + e 3(E q,f µ f )/T f=u,d,s +ln h1 + 3( io + e (E q,f +µ f )/T )e (E q,f +µ f )/T + e 3(E q,f +µ f )/T equations of @ @ min = explicit μ-dependence implicit μ-dependence global minimum min(µ, T )= x = h x i, y = h y i, = h i, =

22 Why we need a novel technique numerical derivatives / divided differences error prone d 2 dµ 2 = 1 µ 2 [ (µ µ) 2 (µ)+ (µ + µ)] + O( µ2 ) need at least (n+1) function evaluations for n-th derivative analytic derivatives = ( (µ),µ) rapidly increasing number of terms algebra systems can help still a lot of coding required d 2 dµ 2 = 2 µ µ 2 +2 µ 2 µ + 2 µ 2 cannot help with implicit dependencies

23 Algorithmic Differentiation MW, A. Walther, B.-J. Schaefer, Comp. Phys. Commun., 181, pp. 756 idea: differentiate the algorithm using the chain rule no approximations, i.e. machine precision only slight modifications of the code necessary (replace double datatype) arbitrary orders without further coding inverse Taylor expansion to treat implicit derivatives performance (µ) = ) µ AD: 1 evaluation of grand potential (1 minimization) + AD evaluation DD: (n+1) evaluations of grand potential ( (n+1) minimizations)

24 Algorithmic Differentiation idea: differentiate the algorithm using the chain rule no approximations, i.e. 35 machine DD precision only slight modifications of the code necessary (replace double datatype) arbitrary orders without further coding inverse Taylor expansion to treat implicit derivatives performance t [ms] AD highest derivative degree d MW, A. Walther, B.-J. Schaefer, Comp. Phys. Commun., 181, pp. 756 (µ) AD: 1 evaluation Algorithmic of grand Differentiation potential (1 minimization) is a general technique + AD evaluation! DD: (n+1) evaluations of grand potential ( (n+1) minimizations) = ) µ

errors for higher order coefficients convergence properties have obtained little

25 Can we trust the Taylor expansion? only few coefficients extracted from lattice simulations increasing statistical errors for higher order coefficients convergence properties have obtained little attention explore general aspects of Taylor expansion method requires higher coefficients comparison with direct evaluation favorable

26 Higher order coefficients available B.-J. Schaefer, MW, J. Wambach, PoS CPOD29, c c 6-15 c e+7 1e+6 c 12 c 14-1e+6 c 16-5e+7 higher coefficients are oscillating near transition increasing amplitude not negligible for µ/t < 1 3e+9 2e+9 1e+9-1e+9-2e+9-3e+9-4e+9 c 18,98 1 1,2 c 2 5e+1-5e+1,98 1 1,2 T/T c 22,98 1 1,2 2e+12-2e+12-4e+12 small outside transition region no numerical noise singular contribution depends linear on T and quadratically on μ

27 Higher order coefficients available B.-J. Schaefer, MW, J. Wambach, PoS CPOD29, e+9 2e+9 1e+9-1e+9-2e+9-3e+9-4e c c 6-15 c e+7 1e+6 c 12,98 1 1,2 c 14-1e+6 1 c 1 1!/12! (dc 1 /dt) T c 12 5e c 18 c 2 c -5e+1 22 T/T,98 1 1,2 T/T c 16,98 1 1, e+7 2e+12-2e+12-4e+12 higher coefficients are oscillating near transition increasing amplitude not negligible for small outside transition region no numerical noise µ/t < 1 singular contribution depends linear on T and quadratically on μ

28 Thermodynamics at finite μ susceptibility via Taylor expansion 25 2 T [MeV] chiral crossover chiral first order deconfinement CEP µ [MeV] 2 15 µ/t < µ c /T c µ/t = µ c /T c µ/t > µ c /T c PQM Taylor 8th Taylor 16th 2 15 PQM Taylor 8th Taylor 16th 2 15 PQM Taylor 8th Taylor 16th /T 2 1 /T 2 1 /T T/T solid: Padé-approximation; dashed: Taylor expansion at corresponding order (same color, same coefficients) T/T T/T

29 Beyond Taylor: Padé approximation Taylor: t(x) = NX t (i) x i i= Padé: [L/M] R L,M (x) = p(x) q(x) = p + p 1 x + + p L x L 1+q 1 x + + q M x M require: i R L,M (x) x i x= = i t(x) x i x= for i apple N i.e. uses same derivative information as Taylor expansion pole in at, i.e. at pole also used as estimate for phase boundary for general Padé series

30 Beyond Taylor: Padé approximation Taylor: Padé: require: t(x) = [L/M] NX t (i) x i i= i R L,M (x) R L,M (x) = p(x) q(x) = p + p 1 x + + p L x L t 1+q 1 x + + q M x M (L) t (L+1) t (L+M) [L/M] = x i x= i.e. uses same derivative information as Taylor expansion pole in at, i.e. at L = t (L M+1) t (L M+2) t (L+1) t (L M+2) t (L M+3) t (L+2) P M i= t (i) x M+i L PM+1 t (i) x M+i 1 i t(x) x i. x= pole also used as estimate for phase boundary for general Padé series i= t (L M+1) t (L M+2) t (L+1) t (L M+2) t (L M+3) t for i apple N (L+2) t (L) t (L+1) t (L+M) x M x M 1 1 LP i= t (i) x i

31 Thermodynamics at finite μ susceptibility via Taylor expansion 25 and Padé approximation rarely used for μ-extrapolations (M. P. Lombardo PoS LAT25, 168) more suitable for description of singularities, i.e. divergent susceptibility at the CEP T [MeV] chiral crossover chiral first order deconfinement CEP µ [MeV] 2 15 µ/t < µ c /T c µ/t = µ c /T c µ/t > µ c /T c PQM Pade [4/4] Pade [8/8] 2 15 PQM Pade [4/4] Pade [8/8] 2 15 PQM Pade [4/4] Pade [8/8] /T 2 1 /T 2 1 /T T/T solid: Padé-approximation; dashed: Taylor expansion at corresponding order (same color, same coefficients) T/T T/T

32 Thermodynamics at finite μ susceptibility via Taylor expansion 25 and Padé approximation rarely used for μ-extrapolations (M. P. Lombardo PoS LAT25, 168) T [MeV] chiral crossover chiral first order deconfinement CEP more suitable Padé reproduces for description Taylor of singularities, results with fewer coefficients i.e. divergent susceptibility at the CEP µ [MeV] 2 15 µ/t < µ c /T c µ/t = µ c /T c µ/t > µ c /T c PQM Pade [4/4] Pade [8/8] 2 15 PQM Pade [4/4] Pade [8/8] 2 15 PQM Pade [4/4] Pade [8/8] /T 2 1 /T 2 1 /T T/T solid: Padé-approximation; dashed: Taylor expansion at corresponding order (same color, same coefficients) T/T T/T

33 How to extract the phase boundary? nearest singularity in the complex plane limits applicability of Taylor expansion convergence radius for pressure only exact in the limit of infinite expansion F. Karsch, B-J. Schaefer, MW, J. Wambach, Phys. Lett. B698, p. 256 r = lim n!1 r 2n = lim n!1 c 2n c 2n+2 1/2 different observables yield different estimates at finite order r 2n = c 2n c 2n+2 1/2 = (2n + 2)(2n + 1) (2n + 3)(2n + 4) 1/2 r 2n+2 r 2n = r 2n n + O(n 2 ) need apparent convergence to asymptotic value expected deviation from asymptotic value (near to the CEP) r 2n+2 ' r(1 + A/n) r 2n ' r(1 + (A 1)/n)

34 Convergence radii and phase boundary r = lim n!1 r 2n = lim n!1 c 2n c 2n+2 1/2 F. Karsch, B-J. Schaefer, MW, J. Wambach, Phys. Lett. B698, p T/T.6.4 2n= 4 2n= 8 2n=12 T/T.6.4 2n=16 2n=2 2n= µ/t µ/t Red line: chiral crossover (dotted), 1st order (solid) Yellow line: deconfinement crossover Black dot: chiral critical end point

35 Convergence radii and phase boundary r = lim n!1 r 2n = lim n!1 c 2n c 2n+2 1/2 F. Karsch, B-J. Schaefer, MW, J. Wambach, Phys. Lett. B698, p. 256 quark number susceptibility 1 r 2n = c 2n c 2n+2 1/2 = (2n + 2)(2n + 1) (2n + 3)(2n + 4) 1/2 r 2n+2.8 T/T n=16 2n=2 2n= µ/t T/T n= 8 2n=16 2n=24 Red line: chiral crossover (dotted), 1st order (solid) Yellow line: deconfinement crossover Black dot: chiral critical end point µ/t solid lines: convergence radius estimate for the pressure at finite n dashed lines: estimate for the quark number susceptibility

36 Convergence radii and phase boundary F. Karsch, B-J. Schaefer, MW, J. Wambach, Phys. Lett. B698, p. 256 only finite number of coefficients quark number susceptibility apparent convergence r 2n = c 2n c 2n+2 1/2 = (2n + 2)(2n + 1) (2n + 3)(2n + 4) 1/2 r 2n+2 nice reproduction at large T 1 at lower temperatures: 1st order transition conceptual problem T/T n= 8 2n=16 2n=24 expansion works also for μ/t >1 better estimate with susceptibility µ/t solid lines: convergence radius estimate for the pressure at finite n dashed lines: estimate for the quark number susceptibility

37 Comparison of criteria for phase boundary F. Karsch, B-J. Schaefer, MW, J. Wambach, Phys. Lett. B698, p. 256 fixed number of coefficients estimate of phase boundary using convergence radius of pressure µ/t r 2n p [n+1/n+1] p [n+2/n] p [n+3/n-1] r 2n-2 [n/n] [n+1/n-1] and susceptibility.9.8 poles in Padé approximant n uses all coefficients as input r 2n = c 2n c 2n+2 1/2 = (2n + 2)(2n + 1) (2n + 3)(2n + 4) 1/2 r 2n+2 error propagation is more involved; under control with AD

38 Comparison of criteria for phase boundary F. Karsch, B-J. Schaefer, MW, J. Wambach, Phys. Lett. B698, p. 256 fixed number of coefficients estimate of phase boundary using convergence radius of pressure µ/t r 2n p [n+1/n+1] p [n+2/n] p [n+3/n-1] r 2n-2 [n/n] [n+1/n-1] significant improvement for susceptibility and susceptibility and Padé approximation poles in Padé approximant n uses all coefficients as input r 2n = c 2n c 2n+2 1/2 = (2n + 2)(2n + 1) (2n + 3)(2n + 4) 1/2 r 2n+2 error propagation is more involved; under control with AD

39 Locating the critical endpoint convergence radius close to CEP lim n!1 r n(t c )! µ c T c need a way to determine : all expansion coefficients positive: singularity on the real axis! µ Figure: C. Schmidt (29) c 2 c 4 sign of coefficients (Stephanov, Phys.Rev. D73 (26) 9458) c 8 singularity at real chemical potential if all coefficients are positive T CEP T CEP 1 T CEP 8 T RW c 6 at least n=8 required for non-trivial estimate extrapolation required: T c = lim n!1 T c,n asymptotic behavior unknown

40 Locating the critical endpoint phase boundary already estimated still need determine T c : sign of coefficients F. Karsch, B-J. Schaefer, MW, J. Wambach, Phys. Lett. B698, p. 256 T/T n= 8 2n=16 2n=24 c 16 consider T T c,16, i.e. where changes sign µ/t c 16 > c 16 = c 16 < Im(µ/T) Im(µ/T) Im(µ/T) Re(µ/T) Re(µ/T) Re(µ/T)

41 Locating the critical endpoint phase boundary already estimated still need determine : sign of coefficients c 16 consider T T c,16, i.e. where changes sign 1 c 16 > T c T c,n [MeV] c 16 = F. Karsch, B-J. Schaefer, MW, J. Wambach, Phys. Lett. B698, p. 256 T/T n= 8 2n=16 2n= µ/t 1 c 16 < Im(µ/T) Im(µ/T).5 Im(µ/T) n Re(µ/T) Re(µ/T) Re(µ/T)

42 Closer look at first-order transition consider µ/t =3 F. Karsch, B-J. Schaefer, MW, J. Wambach, Phys. Lett. B698, p. 256 first order transition: new global minimum in grand potential not captured by Taylor expansion 1 8 dotted: metastable regime 1.8 x [MeV] th order PQM T/T.6.4 2n= 8 2n=16 2n= T [MeV] µ/t

43 Closer look at first-order transition consider µ/t =3 F. Karsch, B-J. Schaefer, MW, J. Wambach, Phys. Lett. B698, p. 256 first order transition: new global minimum in grand potential not captured by Taylor expansion x [MeV] th order PQM dotted: metastable regime /T th 16th 8th PQM T [MeV] T [MeV]

44 2+1 flavor PQM model at finite density B-J. Schaefer, MW, in preparation (211) Polyakov loop potential fitted to pure gauge 25 2 no matter back-reaction reproduce first order transition at fermions modify the gluonic sector: (Schaefer et al., 27) T [MeV] CEP crossover first, order crossover µ [MeV] coincidence of peaks in chiral condensate and Polyakov loop variables also at larger μ 25 2 with matter back-reaction possible quarkyonic region shrinks beyond MFA (2 flavor): coincidence for all μ (Herbst et al., 211) T [MeV] CEP crossover first, order crossover µ [MeV]

45 Higher moments at finite chemical potential define ratios e.g. kurtosis 2 1 B-J. Schaefer, MW, in preparation (211) 1-1 R 4, T [MeV] µ [MeV]

46 Higher moments at finite chemical potential define ratios B-J. Schaefer, MW, in preparation (211) e.g. kurtosis negative regions around crossover R 4, T [MeV] µ [MeV] what happens towards the CEP with increasing orders? T [MeV] R 4,2 R 6,2 R 8,2 R 1,2 R 12,2 crossover CEP µ [MeV]

Higher moments at finite chemical potential define 21 ratios B-J. Schaefer, MW, in preparation (211) 2 1 1-1 e.g. kurtosis T c,n [MeV] 25 2 negative 195 regions around crossover R 4,2-1 2 23 26 T [MeV] 29 1 2 3 4 5 6 µ [MeV] 19 what happens towards the CEP 185 6 8 1 12 14 16 18 2 with increasing orders?

47 Higher moments at finite chemical potential define 21 ratios B-J. Schaefer, MW, in preparation (211) e.g. kurtosis T c,n [MeV] 25 2 negative 195 regions around crossover R 4, T [MeV] µ [MeV] 19 what happens towards the CEP with increasing orders? n T [MeV] R 4,2 R 6,2 R 8,2 R 1,2 R 12,2 crossover CEP µ [MeV]

48 Using higher moments as precursors of the CEP roots in ratios of odd momenta consider distance to crossover T [MeV] B-J. Schaefer, MW, in preparation (211) -.8 n=3-1 n=5-1.2 n=7 n= µ/t experimentally crossover line is unknown consider distance of roots in subsequent ratios [MeV] n=3 n=5 -.5 n=7 -.6 n= µ/t

Using higher moments as precursors of the CEP roots in ratios of odd momenta consider distance to crossover almost linear behavior in intermediate region extrapolation: experimentally crossover line

49 Using higher moments as precursors of the CEP roots in ratios of odd momenta consider distance to crossover almost linear behavior in intermediate region extrapolation: experimentally crossover line is unknown consider distance of roots in subsequent ratios T [MeV] [MeV] n=3-1 n=5-1.2 n=7 n= B-J. Schaefer, MW, in preparation (211) µ/t -.4 n=3 n=5 -.5 n=7 -.6 n= µ/t

50 μ= good estimate of phase boundary from Taylor expansion better estimate with susceptibility higher coefficients are necessary (obtained with AD) Padé resummation: fewer coefficients required lattice results with few coefficients have to be interpreted carefully i.e. currently no prediction of CEP possible with lattice coefficients thermodynamics from Taylor expansion is reliable also for μ / T > 1 (within r) Taylor expansion can predict phase boundary with an acceptable uncertainty currently no determination of critical temperature

51 μ> calculated higher moments at finite density higher precision using AD sign structure of ratios of higher momenta negative regions merge at CEP estimate the location of the CEP by exploiting the sign structure

52 Outlook extract better signals for the critical end point exploit the Padé approximation combine with results at imaginary chemical potential new criteria for critical temperature? functional studies: include fluctuations: Renormalization group (work in progress with M. Puhr) smoother transition: larger negative areas better signals finite volume effects

Probing the QCD phase diagram with higher moments supported by Bielefelder Nachwuchsfonds. References: F. Karsch, B-J. Schaefer, MW, J. Wambach, Phys. Lett. B698, p. 256; MW, A. Walther, B.-J. Schaefer, Comp.

53 Probing the QCD phase diagram with higher moments supported by Bielefelder Nachwuchsfonds. References: F. Karsch, B-J. Schaefer, MW, J. Wambach, Phys. Lett. B698, p. 256; MW, A. Walther, B.-J. Schaefer, Comp. Phys. Commun., 181 (21), pp ; B-J. Schaefer, MW, J. Wambach, Phys. Rev. D 81, 7413 (21); J. Wambach, B-J. Schaefer, MW, Acta Phys. Pol. B Proc. Suppl. 3, 691; B-J. Schaefer, MW, J. Wambach, PoS CPOD29, 17. B.J. Schaefer, MW, in preparation (211)

Can we locate the QCD critical endpoint with the Taylor expansion?

Can we locate the QCD critical endpoint with the Taylor expansion? in collaboration with: F. Karsch B.-J. Schaefer A. Walther J. Wambach 23.6.21 Mathias Wagner Institut für Kernphysik TU Darmstadt Outline