Mass and running effects in the pressure for cold and dense matter Letícia F. Palhares Eduardo S. Fraga Letícia F. Palhares Eduardo S. Fraga

2 nd Rio-Saclay: QCD under extreme conditions 2 OutlineOutline Quark mass and RG corrections in in- medium QCDQuark mass and RG corrections in in- medium QCD The Linear Sigma Model (LSM) at finite densityThe Linear Sigma Model (LSM) at finite density Renormalization of the LSM at finite densityRenormalization of the LSM at finite density Properties of the RG runningProperties of the RG running Conclusions and perspectivesConclusions and perspectives Quark mass and RG corrections in in- medium QCDQuark mass and RG corrections in in- medium QCD The Linear Sigma Model (LSM) at finite densityThe Linear Sigma Model (LSM) at finite density Renormalization of the LSM at finite densityRenormalization of the LSM at finite density Properties of the RG runningProperties of the RG running Conclusions and perspectivesConclusions and perspectives

2 nd Rio-Saclay: QCD under extreme conditions 3 Phase diagram of QCD with massive quarks  Finite quark masses can bring qualitative and quantitative modifications of the phase diagram: – Expected behavior of the critical point with quark mass [Stephanov (2006)]

2 nd Rio-Saclay: QCD under extreme conditions 4 What about the axis? ? Lattice calculations including massive quarks show sensible corrections in the chiral condensate [Bernard et al – MILC collab. (2003)] Within effective models quark mass effects are also relevant for the chiral transition: [Dumitru, Röder & Ruppert (2004)]

2 nd Rio-Saclay: QCD under extreme conditions 5 Consequences on the nature and intensity of the chiral transition  astrophysical implications: new class of stars for a strongly 1st order chiral transition.  Inperturbative QCD:  In perturbative QCD:  2-loop pressure in QCD at finite mu: Quark mass and RG corrections up to ~25% Quark mass and RG corrections up to ~25% In-medium QCD with massive quarks [Fraga, Pisarski & Schaffner-Bielich (2001/2002)] [Fraga & Romatschke (2005)]  Complementary study  Complementary study: Low energy effective models at finite density (e.g. Linear Sigma Model)

2 nd Rio-Saclay: QCD under extreme conditions 6 The Linear Sigma Model at finite density QCD: approximate chiral symmetryQCD: approximate chiral symmetry Spontaneous Symmetry Breaking:Spontaneous Symmetry Breaking: Parameters are fixed to reproduce measured vacuum propertiesParameters are fixed to reproduce measured vacuum properties RenormalizableRenormalizable QCD: approximate chiral symmetryQCD: approximate chiral symmetry Spontaneous Symmetry Breaking:Spontaneous Symmetry Breaking: Parameters are fixed to reproduce measured vacuum propertiesParameters are fixed to reproduce measured vacuum properties RenormalizableRenormalizable Quark-mesoncoupling: Yukawa type Quark-mesoncoupling: Explicit SB However: Truncation of the perturbative series introduces a renormalization scale [even for mean-field!]. implementation of RG running implementation of RG running However: Truncation of the perturbative series introduces a renormalization scale [even for mean-field!]. implementation of RG running implementation of RG running

2 nd Rio-Saclay: QCD under extreme conditions 7 Previous results in the LSM : Previous results in the LSM : –Mean-field calculation of the effective potential at finite T and (no loops, no RG running, zero point (ZPT) logs neglected) (no loops, no RG running, zero point (ZPT) logs neglected) –Mean-field calculation of the effective potential at finite T and (no loops, no RG running, zero point (ZPT) logs neglected) (no loops, no RG running, zero point (ZPT) logs neglected) [Scavenius et al (2001)] –Beyond mean-field for ( resummation needed! ):. ZPT + 2-loop + MSC Resummation [Caldas, Mota & Nemes (2001)]ZPT + 2-loop + MSC Resummation [Caldas, Mota & Nemes (2001)] CJT + Hartree approx. [Röder, Ruppert & Rischke (2003)]CJT + Hartree approx. [Röder, Ruppert & Rischke (2003)] ZPT+ Averaged sigma fluctuations [Mócsy, Mishustin & Ellis (2004)]ZPT+ Averaged sigma fluctuations [Mócsy, Mishustin & Ellis (2004)] –Beyond mean-field for ( resummation needed! ):. ZPT + 2-loop + MSC Resummation [Caldas, Mota & Nemes (2001)]ZPT + 2-loop + MSC Resummation [Caldas, Mota & Nemes (2001)] CJT + Hartree approx. [Röder, Ruppert & Rischke (2003)]CJT + Hartree approx. [Röder, Ruppert & Rischke (2003)] ZPT+ Averaged sigma fluctuations [Mócsy, Mishustin & Ellis (2004)]ZPT+ Averaged sigma fluctuations [Mócsy, Mishustin & Ellis (2004)]......

2 nd Rio-Saclay: QCD under extreme conditions 8 Proposal : Proposal : Study the sigma direction Other condensates don’t affect much the chiral one [Röder, Ruppert & Rischke (2003)] [Röder, Ruppert & Rischke (2003)] ZPT and renormalization in the scheme ZPT and renormalization in the scheme Implementation of RG running Implementation of RG running Study the sigma direction Other condensates don’t affect much the chiral one [Röder, Ruppert & Rischke (2003)] [Röder, Ruppert & Rischke (2003)] ZPT and renormalization in the scheme ZPT and renormalization in the scheme Implementation of RG running Implementation of RG running Beyond mean-field at finite density and zero T (no resummation needed)  Effects on the position of   Consequences on the nature of the chiral PT  astro implications

2 nd Rio-Saclay: QCD under extreme conditions 9 Effective Potential Using a standard procedure [Jackiw], one obtains the loop expansion for the effective potential, neglecting terms of :Using a standard procedure [Jackiw], one obtains the loop expansion for the effective potential, neglecting terms of : Mean-field Exchange contribution with both fields massive:  more involved than QCD

2 nd Rio-Saclay: QCD under extreme conditions 10 Free Terms (quarks [Mean-field] and sigma)Free Terms (quarks [Mean-field] and sigma) ResultsResults Z ero- P oin T terms ExchangeExchangewhere… where… Vacuum (ZPT)

2 nd Rio-Saclay: QCD under extreme conditions 11 exact analytical results (for all values of parameters: ): allows full implementation of RG running exact analytical results (for all values of parameters: ): allows full implementation of RG running

2 nd Rio-Saclay: QCD under extreme conditions 12 Renormalized results ( scheme) R R R Free Terms (quarks [Mean-field] and sigma)Free Terms (quarks [Mean-field] and sigma) R R

2 nd Rio-Saclay: QCD under extreme conditions 13 ExchangeExchange Renormalized results ( scheme) R R R VacuumVacuum R R

2 nd Rio-Saclay: QCD under extreme conditions 14 : mass scale : mass scale m 0 = 0.1 (chosen) m 0 = 0.1 (chosen) large N F flattens running large N F flattens running larger N F  more perturbative larger N F  more perturbative RG running & N F (Yukawa coupling)

2 nd Rio-Saclay: QCD under extreme conditions 15 Conclusion and Perspectives Finite mass effects do play an important role in the phase structure of QCDFinite mass effects do play an important role in the phase structure of QCD We have computed the exchange correction in the LSM at finite mu for arbitrary masses (up to finite barMS corrections)We have computed the exchange correction in the LSM at finite mu for arbitrary masses (up to finite barMS corrections) Next steps: Next steps: –,, nature of the chiral PT... –Corrections to and –Implementation of RG running On the lattice, calculations with realistic quark masses and finite mu (also for effective theories!) [Hands, (2007)] are under way  comparison possibleOn the lattice, calculations with realistic quark masses and finite mu (also for effective theories!) [Hands, (2007)] are under way  comparison possible Finite mass effects do play an important role in the phase structure of QCDFinite mass effects do play an important role in the phase structure of QCD We have computed the exchange correction in the LSM at finite mu for arbitrary masses (up to finite barMS corrections)We have computed the exchange correction in the LSM at finite mu for arbitrary masses (up to finite barMS corrections) Next steps: Next steps: –,, nature of the chiral PT... –Corrections to and –Implementation of RG running On the lattice, calculations with realistic quark masses and finite mu (also for effective theories!) [Hands, (2007)] are under way  comparison possibleOn the lattice, calculations with realistic quark masses and finite mu (also for effective theories!) [Hands, (2007)] are under way  comparison possible