Mapping the phase diagram of strongly interacting matter V. Skokov - PowerPoint PPT Presentation

Mapping the phase diagram of strongly interacting matter V. Skokov in collaboration with B. Friman, K. Morita, and K. Redlich GSI, Darmstadt e-Print: arXiv:1008.4549 EMMI “Strongly Coupled Systems” V. Skokov (GSI, Darmstadt) Mapping the phase diagram 1 / 31

Outline Motivation Simple example of conformal mapping application Analytical structure of thermodynamic functions on the complex µ plane (chiral limit) Location of the second-order phase transition singularity Outlook: Finite size effect V. Skokov (GSI, Darmstadt) Mapping the phase diagram 2 / 31

QCD phase diagram Temperature CEP Quark-gluon phase Hadronic phase Color superconductivity Nuclear matter Baryon chemical potential Experiments Model calclulations Lattice QCD V. Skokov (GSI, Darmstadt) Mapping the phase diagram 3 / 31

QCD phase diagram Temperature CEP Quark-gluon phase Hadronic phase Color superconductivity Nuclear matter Baryon chemical potential Experiments Model calclulations Lattice QCD Functional methods (talk by C. Fischer) V. Skokov (GSI, Darmstadt) Mapping the phase diagram 3 / 31

Lattice QCD and “sign problem” at finite µ The partition function of QCD with integrated out quark degrees of freedom � Z ( µ ) = D A exp( − S [ A ])det[ D ( µ )] with det[ D ( µ )] ∈ Complex . The weight function is not positive definite. The Monte-Carlo technique fails. Indirect approaches to sidestep the sign problem Reweighting technique (Z. Fodor and S. Katz) Imaginary baryon chemical potential: Im µ � = 0, Re µ = 0 (Ph. de Forcand and O. Philipsen; M.-P. Lombardo and M. D’Elia) Taylor expansion (hotQCD collaborations, R. V. Gavai and S. Gupta) V. Skokov (GSI, Darmstadt) Mapping the phase diagram 4 / 31

Thermodynamic function is given by its Taylor expansion at µ = 0 � µ ∂ 2 k ( P / T 4 ) � P � 2 k 1 � � T 4 = c 2 k ( T ) × , c 2 k = � (2 k )! ∂ ( µ/ T ) 2 k T � µ =0 k Radius of convergence R and its estimates R k 1 / 2 1 / (2 k ) � � � � c 2 k 1 � � � � R = k − > ∞ inf R 2 k lim with R 2 k = or R 2 k = � � � � c 2 k +2 c 2 k � � � � Convergence radius is defined by the closest singularity on the complex µ plane Conversely, the convergence properties of a power series provides information on the closest singularity of the original function F. Karsch et. al., arXiv:1009.5211 C. Schmidt(2010) M. Wagner talk 1.2 T/T c 1.15 1 1.1 0.8 R 2( P / T 4) 1.05 n=4 T/T χ 0.6 R 2( χ B ) 1 n=8 R 4( χ B ) open symb. n=12 0.4 0.95 freeze-out curve 0.9 0.2 0.85 0 µ B /T c 0 0.5 1 1.5 2 0.8 0 1 2 3 4 5 µ /T χ V. Skokov (GSI, Darmstadt) Mapping the phase diagram 5 / 31

Ways to improve Taylor series convergence Pade approximant – a ratio of two power series � N i =0 a i x i P N M = 1+ � M i =1 b i x i Pade approximant may be superior to Taylor expansion and may work even beyond a radius of convergence. The drawback of Pade approximation: it is uncontrolled. Conformal mapping Conformal mapping is a transformation ξ = ξ ( z ) that preserves local angles. The main idea is to extend the radius of convergence and to enhance the sensitivity to the properties of the critical point by a non-linear transformation of an original series. E.g. by conformal mapping one can move the physical singularities closer to the expansion point, while taking non-physical singularities as far away as possible. V. Skokov (GSI, Darmstadt) Mapping the phase diagram 6 / 31

Applications In condensed matter physics (3d Ising model and high temperature expansion) A. Danielian and K.W. Stevens, Proc. Phys. Soc. B70, 326 (1957). C. Domb and M.F. Sykes, J. Math. Phys. 2, 63 (1961). C. J. Pearce, Adv. Phys. 27, 89 (1978). In scattering theory to extend applicability of low energy approximations W. R. Frazer, Phys. Rev. 123, 2180 (1961). A. Gasparyan and M. F. M. Lutz, arXiv:1003.3426 [hep-ph]. a pedagogical example for an exactly solvable theory: I. V. Danilkin, A. Gasparyan, and M. F. M. Lutz, [arXiv:1009.5928 [hep-ph]] In quantum field theory for analytic continuation of perturbative results to the strong coupling regime D. I. Kazakov et al. , Theor. Math. Phys. 38, 9 (1979). . . . V. Skokov (GSI, Darmstadt) Mapping the phase diagram 7 / 31

Simple example: Euler transformation ln(1 + z ) = z − 1 2 z 2 + 1 3 z 3 + · · · The radius of convergence of the series is | z | < 1 owing to a branch point singularity at z = − 1. The power series in z → the power series in the variable z ξ = 1+ z . The power series is obtained from the original ln(1 + z ) = − ln(1 − ξ ) = ξ + 1 2 ξ 2 + 1 3 ξ 3 + · · · The radius of convergence of the series is | ξ | < 1. x 2 + y 2 ( x +1) 2 + y 2 < 1 or x > − 1 | ξ | = 2 . V. Skokov (GSI, Darmstadt) Mapping the phase diagram 8 / 31

The original series is known up to m -th order f ( x ) ≈ � m i =0 c i x i Change of the variables: x → ξ/ (1 − ξ ) and series expansion around ξ = 0 � m i ξ i , i =0 c ′ where c ′ i are linear combination of c i with i ≤ m . Back substitution ξ → x / (1 + x ) � m 1+ x ) i x i =0 c ′ i ( V. Skokov (GSI, Darmstadt) Mapping the phase diagram 9 / 31

f ( x ) = ln(1 − x ) ≈ � m i =0 c i x i m = 3 m = 6 2 2 Taylor; 3d order Taylor; 6th order Conf. map; 3d order Conf. map; 6th order 1.5 1.5 Exact Exact f(x) f(x) 1 1 0.5 0.5 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 x x Original Taylor expansion does not describe the exact function for x > 1 even at large m . Taylor expansion + conformal mapping provides systematic improvement with increasing m . V. Skokov (GSI, Darmstadt) Mapping the phase diagram 10 / 31

Higher values of x : m = 3 m = 6 4 2 Taylor; 3d order 3.5 1.5 Conf. map; 3d order 3 1 Exact 2.5 0.5 f(x) f(x) 2 0 1.5 -0.5 Taylor; 6th order Conf. map; 6th order -1 1 Exact -1.5 0.5 -2 0 0 1 2 3 4 5 0 1 2 3 4 5 x x V. Skokov (GSI, Darmstadt) Mapping the phase diagram 11 / 31

Radius of convergence and importance of the analytical structure Temperature CEP Quark-gluon phase Hadronic phase Color superconductivity Nuclear matter Baryon chemical potential Im (Baryon chemical potential) To understand the dependence of convergence radius on temperature and to invent an appropriate conformal mapping, the analytical structure of thermodynamic functions is to be studied. V. Skokov (GSI, Darmstadt) Mapping the phase diagram 12 / 31

Analytic structure of the complex µ plane general structure illustrated by the quark-meson model ( ≈ NJL) in the mean-field approximation. V. Skokov (GSI, Darmstadt) Mapping the phase diagram 13 / 31

Analytic structure of the complex µ plane general structure illustrated by the quark-meson model ( ≈ NJL) in the mean-field approximation. Chiral limit. V. Skokov (GSI, Darmstadt) Mapping the phase diagram 13 / 31

Singular points expected on the complex µ plane Singularities on the complex plane are related to a critical point of a second-order phase transition. The singularity is on the real µ axis. to a crossover transition. The singularity is at some complex µ . See P. C. Hemmer and E. H. Hauge, Phys. Rev. 133 , A1010 (1964); C. Itzykson, R. B. Pearson and J. B. Zuber, Nucl. Phys. B 220 , 415 (1983); M. A. Stephanov, Phys. Rev. D 73 , 094508 (2006). spinodal lines for a first-order phase transition. Singularities are either at the real or complex values of µ . See M. A. Stephanov, Phys. Rev. D 73 , 094508 (2006); “thermal singularities” associated with zeros of the inverse Fermi-Dirac function See F. Karbstein and M. Thies, Phys. Rev. D 75 , 025003 (2007) V. Skokov (GSI, Darmstadt) Mapping the phase diagram 14 / 31

“Thermal” singularities The inverse Fermi-Dirac function has zeros in the complex plane. This leads to ”thermal” singularities of thermodynamic functions. � ω − µ [ f F ( ω )] − 1 = exp � + 1 T � ω − Re [ µ ] � Im µ = i π T � − exp + 1 = 0 T Examples: for massless particles zeros of the Fermi-Dirac functions are located on the lines Re µ = p , Im µ = i π T + 2 i π nT , n = 0 , ± 1 , ± 2 , · · · . m 2 + p 2 , Im µ = i π T + 2 i π nT , � for particles with mass m , Re µ = n = 0 , ± 1 , ± 2 , · · · . See also F. Karbstein and M. Thies, Phys. Rev. D 75 , 025003 (2007) V. Skokov (GSI, Darmstadt) Mapping the phase diagram 15 / 31

Analytic structure of the complex µ plane. Complex µ plane Phase diagram imaginary µ real µ 2.5 T TPC <T<T 0 Restored phase 2 Line of 2-d order PT 1.5 T/T 0 1 0.5 Broken phase TCP ? 0 -10 -8 -6 -4 -2 0 2 4 2 ) | sign( /T 0 | thermal cuts and associated singularities second-order PT line; (branch points) denoted by open dots; possible TPC second-order PT V. Skokov (GSI, Darmstadt) Mapping the phase diagram 16 / 31

Analytic structure of the complex µ plane. Complex µ plane Phase diagram imaginary µ real µ 2.5 T=T 0 Restored phase 2 Line of 2-d order PT 1.5 T/T 0 1 0.5 Broken phase TCP ? 0 -10 -8 -6 -4 -2 0 2 4 2 ) | sign( /T 0 | thermal cuts and associated singularities second-order PT line; (branch points) denoted by open dots; possible TPC second-order PT V. Skokov (GSI, Darmstadt) Mapping the phase diagram 16 / 31