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

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

2 Outline Taylor expansion higher orders in the PQM model convergence radii and the phase boundary locating the critical endpoint Taylor 2.: Padé-approximation Mathias Wagner Institut für Kernphysik TU Darmstadt

3 Taylor expansion to finite density Taylor expansion p(t,µ) T 4 = n= c n (T ) µ T n c n (T )= 1 n! n p(t,µ)/t 4 (µ/t ) n µ= naive: applicable with few coefficients for! / T < 1 calculations by various lattice groups ( e.g. Allton, C. Schmidt, Gavai & Gupta, detar, Ejiri,...) c 2 u SB n f =2+1, m " =22 MeV n f =2, m " =77 MeV filled: N! = 4 open: N! = 6 T[MeV] c 4 u n f =2+1, m " =22 MeV n f =2, m " =77 MeV.2 filled: N! = 4 2-flavor: Allton (25), 2+1 flavor: Miao (28) open: N! = SB.4 SB T[MeV] -.2 T[MeV] c 6 u n f =2+1, m " =22 MeV n f =2, m " =77 MeV Mathias Wagner Institut für Kernphysik TU Darmstadt

4 Signals of the critical endpoint? χ q (T,µ) T = 2 Ω(T,µ) µ 2 = n(n 1)c n (T ) n=2,4,...! B / T 2! B /T=. C. Schmidt (28)! B /T=1.5! B /T=2.5 T [MeV] µ T n r = lim n r 2n = lim n 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 Mathias Wagner Institut für Kernphysik TU Darmstadt

5 Polyakov-Quark-Meson (PQM) model relevant degrees of freedom: (2+1) quarks and mesons! PQM model (similar to PNJL) (talk by Weise, Fukushima) B-J. Schaefer, MW, J. Wambach, Phys. Rev. D 81, 7413 L PQM = q (id/ gφ 5 ) q + L m U(Φ, Φ) Polyakov loop variable quarks / antiquarks L m =Tr µ φ µ φ m 2 Tr(φ φ) λ 1 Tr(φ φ) 2 λ2 Tr φ φ 2 + c det(φ)+det(φ ) +Tr H(φ + φ ) mesonic fields here: logarithmic Polyakov loop potential (Roessner et al, Phys. Rev. D75, 347) U log T 4 = 1 2 a(t ) ΦΦ + b(t )ln 1 6 ΦΦ +4 Φ 3 + Φ 3 3 ΦΦ Mathias Wagner Institut für Kernphysik TU Darmstadt

6 Polyakov-Quark-Meson (PQM) model relevant degrees of freedom: (2+1) quarks and mesons! PQM model (similar 1 to PNJL) (talk by Weise, Fukushima) B-J. Schaefer, MW, J. Wambach, Phys. Rev. D 81, 7413 L PQM.8= q (id/ gφ 5 ) q + L m U(Φ, Φ) Polyakov loop variable quarks / antiquarks.6 L m =Tr µ φ µ φ QM m 2 Tr(φ φ) λ 1 Tr(φ φ) 2 λ2 Tr φ φ 2 PQM log.4 + c det(φ)+det(φ ) +Tr PQM H(φ + φ ) pol PQM Fuku.2 p4 mesonic fields asqtad p/p SB here: logarithmic Polyakov loop potential (Roessner et al, Phys. Rev. D75, 347) U log T 4 = 1 2 a(t ) ΦΦ T/T + b(t )ln 1 6 ΦΦ +4 Φ 3 + Φ 3 3 ΦΦ Mathias Wagner Institut für Kernphysik TU Darmstadt

7 Explicit and implicit!-dependence mean-field approximation thermodynamic potential Ω = U (σ x, σ y )+Ω qq σx, σ y, Φ, Φ + U Φ, Φ quark contribution Ω qq (σ x, σ y, Φ, Φ) = 2T d 3 p (2π) 3 ln 1 + 3(Φ + Φe (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 1 + 3( Φ + Φe (E q,f +µ f )/T )e (E q,f +µ f )/T + e 3(E q,f +µ f )/T equations of motion Ω = Ω = Ω σ x σ y Φ = Ω Φ = min explicit μ-dependence implicit μ-dependence global minimum min(µ, T )= σ x = σ x, σ y = σ y, Φ = Φ, Φ = Φ Mathias Wagner Institut für Kernphysik TU Darmstadt

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

9 Algorithmic differentiation idea: differentiate the algorithm using the chain rule no approximations, i.e. machine precision only slight modifications of the code necessary MW, A. Walther, B.-J. Schaefer, Comp. Phys. Commun., 181, pp. 756 arbitrary orders without further coding inverse Taylor expansion to treat implicit derivatives performance Ω(µ) σ = σ µ AD: 1 evaluation of grand potential and equations of motion DD: (n+1) evaluations of grand potential (including minimization) Mathias Wagner Institut für Kernphysik TU Darmstadt

10 Algorithmic differentiation idea: differentiate the algorithm using the chain rule no approximations, i.e. machine precision only slight modifications of the code necessary arbitrary orders without 1 further coding inverse Taylor expansion to treat implicit derivatives performance t [ms] AD DD highest derivative degree d MW, A. Walther, B.-J. Schaefer, Comp. Phys. Commun., 181, pp. 756 Ω(µ) σ AD: 1 evaluation of grand potential and equations of motion DD: (n+1) evaluations of grand potential (including minimization) = σ µ Mathias Wagner Institut für Kernphysik TU Darmstadt

11 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 22nd (!) order no numerical noise Mathias Wagner Institut für Kernphysik TU Darmstadt

12 Thermodynamics near the CEP quark number susceptibility 25 diverges at CEP 2 lower orders also in PNJL (Ratti et al. Phys. Lett. B649, 57-6) breakdown of Taylor expansion or signal? T [MeV] chiral crossover chiral first order deconfinement CEP µ [MeV] µ/t < µ c /T c µ/t = µ c /T c µ/t > µ c /T c /T th 16th 8th PQM /T th 16th 8th PQM /T th 16th 8th PQM T/T T/T T/T Mathias Wagner Institut für Kernphysik TU Darmstadt

13 Convergence radii and phase boundary r = lim n r 2n = lim n c 2n c 2n+2 1/2 F. Karsch, B-J. Schaefer, MW, J. Wambach (in preparation) T/T.6.4 n=4 n=8 n=12 T/T.6.4 n=16 n=2 n= !/T!/T Red line: chiral crossover (dotted), 1st order (solid) Yellow line: deconfinement crossover Black dot: chiral critical end point Mathias Wagner Institut für Kernphysik TU Darmstadt

14 Convergence radii and phase boundary F. Karsch, B-J. Schaefer, MW, J. Wambach (in preparation) only finite number of coefficients quark number susceptibility apparent convergence r χ 2n = c χ 2n c χ 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 n=16 n=24 expansion works also for!/t >1 better estimate with susceptibility !/T solid lines: convergence radius estimate for the pressure at finite n dashed lines: estimate for the quark number susceptibility Mathias Wagner Institut für Kernphysik TU Darmstadt

15 Closer look at first-order transition consider µ/t =3 first order transition: new global minimum in grand potential not captured by Taylor expansion x [MeV] /T th 16th 8th PQM 2 PQM 24th order T [MeV] T [MeV] Mathias Wagner Institut für Kernphysik TU Darmstadt

16 Locating the critical endpoint convergence radius close to CEP 1 lim n r n(t c ) µ c T/T n=16 n=2 n=24 need a way to determine : sign of coefficients (talks by Gavai, Stephanov; Stephanov, Phys.Rev. D73 (26) 9458) T c !/T at least n=8 required for non-trivial estimate 21 preliminary 25 extrapolation required: T c = lim n T c,n T c,n [MeV] precise determination and higher orders requires algorithmic differentiation method n Mathias Wagner Institut für Kernphysik TU Darmstadt

17 Padé-improved thermodynamics [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 uses Taylor-coefficients as input i R L,M (x) x i = i t(x) x= x i x= for i N rarely used for!-extrapolations (M. P. Lombardo PoS LAT25, 168) 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 [4/4] Pade [8/8] PQM 2 15 [4/4] [8/8] PQM 2 15 [4/4] [8/8] PQM /T 2 1 /T 2 1 /T T/T T/T T/T Mathias Wagner Institut für Kernphysik TU Darmstadt

18 Padé-improved phase boundary Padé approximation requires at least 3 coefficients roots in denominator as criterion for phase boundary for [N/2] identical to Taylor convergence radius compared to convergence radii of Taylor expansion for pressure and susceptibility at fixed temperature T, fixed highest derivative n! / T F. Karsch, B-J. Schaefer, MW, J. Wambach (in preparation) r n p [N/N] p [N+2/N] p [N+2/N-2] r n [N/N] [N+2/N] r χ 2n = c χ 2n c χ 2n+2 1/2 = n max (2n + 2)(2n + 1) (2n + 3)(2n + 4) comparable results with fewer coefficients 1/2 r 2n Mathias Wagner Institut für Kernphysik TU Darmstadt

19 Summary & Outlook higher coefficients via algorithmic differentiation higher coefficients (>12) are required convergence radius and the phase boundary thermodynamics reliable also for! / T > 1 (within r) Padé resummation: fewer coefficients required possible signals for the critical end point exploit the Padé approximation!-corrections in Polyakov loop potential (Schaefer et al,. Phys.Rev. D76, 7423) include fluctuations (RG)? (e.g. Schaefer et al, Phys.Rev. D75, 8515, talks by Pwalowski, Skokov) applicability of algorithmic differentiation in lattice simulations? Mathias Wagner Institut für Kernphysik TU Darmstadt

20 Can we locate the QCD critical endpoint with the Taylor expansion? Maybe... but it is definitely tough! References: 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, Mathias Wagner Institut für Kernphysik TU Darmstadt