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Lattice QCD at non-zero baryonic density ? E.-M. Ilgenfritz How can lattice QCD describe non-zero Introduction Quantum baryonic density ? statistics and the QCD partition function Chemical potential on the Ernst-Michael Ilgenfritz 1

  1. Outline Lattice QCD at non-zero baryonic density ? Introduction 1 E.-M. Ilgenfritz Quantum statistics and the QCD partition function 2 Introduction Chemical potential on the lattice 3 Quantum statistics and the QCD partition Phase quenching can’t treat the complex-weight problem 4 function The phase boundary at small chemical potential Chemical 5 potential on the lattice Taylor expansion: a general purpose approximation 6 Phase quenching can’t treat the Summary of results up to now 7 complex-weight problem Imaginary chemical potential 8 The phase boundary at Complex Langevin dynamics 9 small chemical potential 10 Complex Langevin dynamics for gauge theories Taylor expansion: a general 11 Other approaches to avoid/cure the sign problem purpose approximation 12 Conclusion Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 2 / 171 chemical

  2. Outline Lattice QCD at non-zero baryonic density ? Introduction 1 E.-M. Ilgenfritz Quantum statistics and the QCD partition function 2 Introduction Chemical potential on the lattice 3 Quantum statistics and the QCD partition Phase quenching can’t treat the complex-weight problem 4 function The phase boundary at small chemical potential Chemical 5 potential on the lattice Taylor expansion: a general purpose approximation 6 Phase quenching can’t treat the Summary of results up to now 7 complex-weight problem Imaginary chemical potential 8 The phase boundary at Complex Langevin dynamics 9 small chemical potential 10 Complex Langevin dynamics for gauge theories Taylor expansion: a general 11 Other approaches to avoid/cure the sign problem purpose approximation 12 Conclusion Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 2 / 171 chemical

  3. Outline Lattice QCD at non-zero baryonic density ? Introduction 1 E.-M. Ilgenfritz Quantum statistics and the QCD partition function 2 Introduction Chemical potential on the lattice 3 Quantum statistics and the QCD partition Phase quenching can’t treat the complex-weight problem 4 function The phase boundary at small chemical potential Chemical 5 potential on the lattice Taylor expansion: a general purpose approximation 6 Phase quenching can’t treat the Summary of results up to now 7 complex-weight problem Imaginary chemical potential 8 The phase boundary at Complex Langevin dynamics 9 small chemical potential 10 Complex Langevin dynamics for gauge theories Taylor expansion: a general 11 Other approaches to avoid/cure the sign problem purpose approximation 12 Conclusion Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 2 / 171 chemical

  4. Introduction Outline Lattice QCD at non-zero baryonic density ? Introduction 1 E.-M. Ilgenfritz Quantum statistics and the QCD partition function 2 Introduction Chemical potential on the lattice 3 Quantum statistics and the QCD partition Phase quenching can’t treat the complex-weight problem 4 function The phase boundary at small chemical potential Chemical 5 potential on the lattice Taylor expansion: a general purpose approximation 6 Phase quenching can’t treat the Summary of results up to now 7 complex-weight problem Imaginary chemical potential 8 The phase boundary at Complex Langevin dynamics 9 small chemical potential 10 Complex Langevin dynamics for gauge theories Taylor expansion: a general 11 Other approaches to avoid/cure the sign problem purpose approximation 12 Conclusion Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 3 / 171 chemical

  5. Introduction The phase diagram of QCD Lattice QCD at non-zero baryonic density ? E.-M. Ilgenfritz Introduction Quantum statistics and the QCD partition function Chemical potential on the lattice Phase quenching can’t treat the complex-weight problem The phase boundary at small chemical potential Taylor expansion: a general purpose Figure: Sketch of the QCD phase diagram in the plane of approximation temperature and net baryon density. Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 4 / 171 chemical

  6. Introduction Is Lattice QCD capable to describe non-zero Lattice QCD at non-zero baryonic density baryonic density ? ? E.-M. Ilgenfritz Introduction In short, the answer is "No, however we try ... and try Quantum to get estimates for the reliability of what we are doing" statistics and the QCD partition function "No", at least in the sense how LQCD has proven to be Chemical an ideal ("easy") machinery at zero baryonic density. potential on the lattice The region of large µ is more or less "terra incognita". Phase quenching can’t treat the It will be the target of heavy ion collisions at energies complex-weight problem of NICA and FAIR. It seems natural that some activity The phase should be directed to this field also in BLTP of JINR. boundary at small chemical potential Finally, if only to describe the equilibrium states in the Taylor expansion: a general phase diagram, something like LQCD adapted to purpose approximation high baryonic density is highly needed. Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 5 / 171 chemical

  7. Introduction Why Lattice QCD has been so successful at Lattice QCD at non-zero baryonic density zero baryonic density ? ? E.-M. Ilgenfritz LQCD at µ = 0 was/is a success story, because it ... Introduction allows straightforward simulations by importance Quantum statistics and the sampling (possible due to choosing the Euclidean QCD partition Lagrangian approach), function Chemical allows a strict separation between positive definite potential on the lattice measure and real-valued configuration space (the Phase quenching lattice field configurations), can’t treat the complex-weight allows to inspect typical (real) lattice fields (in order problem to enquire possible mechanisms by indepth search), The phase boundary at allows to calculate everything; one is not restricted small chemical potential to few particular observables (in some truncation Taylor expansion: they may be related through closed equations like a general purpose SDE (Schwinger-Dyson equations) or similar approximation Summary of continuum approaches like FRG). results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 6 / 171 chemical

  8. Introduction Why Lattice QCD has been so successful at Lattice QCD at non-zero baryonic density zero baryonic density ? ? E.-M. Ilgenfritz LQCD at µ = 0 was/still is a success story, because ... Introduction Quantum a systematic improvement was possible towards statistics and the QCD partition the limit a → 0 (continuum limit), function a gradual improvement is possible towards the Chemical potential on the limit V → ∞ (thermodynamical limit), lattice Phase quenching these limits can be approached also for functions, can’t treat the complex-weight for example for U ( r ) (heavy quark potential), G ( p ) problem (Greens functions), for vertices Γ( p 1 , p 2 , p 3 ) etc. The phase boundary at keeping the physical arguments ( r or p i ) fixed. small chemical potential This made possible a productive interaction with Taylor expansion: a general continuum non-perturbative approaches (SDE purpose approximation and Functional Renormalization Group FRG). Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 7 / 171 chemical

  9. Introduction What is so different in case of µ � = 0 ? Lattice QCD at non-zero baryonic density ? Importance sampling is not possible anymore (due E.-M. Ilgenfritz to the "sign problem", a complex weight problem). Introduction It is impossible to generate and store ensembles for Quantum different fixed densities. statistics and the QCD partition It is impossible to inspect configurations in order to function figure out the microscopic "origin" of different physics Chemical potential on the (that we are alerted of by increasing "non-overlap"). lattice However, particular techniques are available to fight Phase quenching can’t treat the the sign problem for particular observables. complex-weight problem Taylor expansion (in µ ) of the measure at the The phase zero-density limit is a multipurpose method, but has boundary at small chemical a finite convergence radius which is unknown apriori potential (different for different observables, say ∆ p ( µ ) ). Taylor expansion: a general Reweighting is meaningless : overlap problem, this purpose approximation becomes more and more severe beyond µ/ T ≈ 1, Summary of results up to now a barrier that cannot be overcome ! Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 8 / 171 chemical

  10. Introduction The subject of this lecture Lattice QCD at non-zero baryonic density ? E.-M. Ilgenfritz Pointing out the origin of the trouble. Introduction Quantum Different ways to circumvent the problem, statistics and the QCD partition even though things are getting more and more function intricate, much more expensive, less encouraging Chemical potential on the for the freshman, on the other hand more interesting ! lattice Few principally new methods for finite density SU ( 3 ) . Phase quenching can’t treat the complex-weight What I will not discuss here are problem The phase possible side projects that usually may keep particle boundary at small chemical theorists busy in difficult times, particularly suitable potential for countries with a less-developed computing Taylor expansion: a general infrastructure. purpose approximation Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 9 / 171 chemical

  11. Introduction Side projects hypothetically relevant for HIC Lattice QCD at non-zero baryonic density ? Other gauge theories without sign problem: E.-M. Ilgenfritz SU ( 2 ) , G 2 , SO ( 2 N ) .... when considered with µ q Introduction Quantum statistics and the V.V. Braguta (MIPT, ITEP , IHEP and FEFU), E.-M. I. QCD partition function (JINR), A.Yu. Kotov (MIPT and ITEP), A.V. Chemical Molochkov (FEFU), A.A. Nikolaev (ITEP and FEFU), potential on the lattice "Study of the phase diagram of dense two-color QCD Phase quenching within lattice simulation", can’t treat the complex-weight arXiv:1605.04090 problem The phase This collaboration was inofficially founded as a four-sided boundary at small chemical HU Berlin–JINR–ITEP–Vladivostok collaboration by potential Mikhail Polikarpov ( † 2013), Taylor expansion: a general Michael Müller-Preussker ( † 2015) and myself purpose approximation at the "Confinement and Hadron Spectrum X" conference Summary of 2012 in Munich. results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 10 / 171 chemical

  12. Introduction Side projects hypothetically relevant for HIC Lattice QCD at non-zero baryonic density ? Other chemical potentials without sign problem: E.-M. Ilgenfritz isospin chemical potential µ iso , chiral chemical Introduction potential µ 5 (2 papers in 2015) Quantum statistics and the QCD partition V.V. Braguta (ITEP and FEFU), V.A. Goy (FEFU), function E.-M. I. (JINR), A.Yu. Kotov (ITEP), A.V. Molochkov Chemical potential on the (FEFU), M. Müller-Preussker (HU Berlin), "Study of lattice the phase diagram of SU(2) quantum Phase quenching can’t treat the chromodynamics with nonzero chirality", JETP Lett. complex-weight problem 100 (2015) 547 The phase boundary at V.V. Braguta (IHEP and FEFU), V.A. Goy (FEFU), small chemical potential E.-M. I. (JINR), A.Yu. Kotov (ITEP), A.V. Molochkov Taylor expansion: (FEFU), M. Müller-Preussker, B. Petersson (HU a general purpose Berlin), "Two-color QCD with non-zero chiral approximation chemical potential", JHEP 1506 (2015) 094 Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 11 / 171 chemical

  13. Introduction Side projects hypothetically relevant for HIC Lattice QCD at non-zero baryonic density ? Other chemical potentials without sign problem: E.-M. Ilgenfritz isospin chemical potential µ iso , chiral chemical Introduction potential µ 5 (2 papers in 2016) Quantum statistics and the A.Yu. Kotov, V.V. Braguta (ITEP), V.A. Goy (FEFU), QCD partition function E.-M. I. (JINR), A.V. Molochkov (FEFU), Chemical potential on the M. Müller-Preussker, B. Petersson (HU Berlin), lattice S.A. Skinderev (ITEP), "Lattice QCD with chiral Phase quenching can’t treat the chemical potential: from SU(2) to SU(3)", PoS complex-weight problem LATTICE2015 (2016) 185 The phase boundary at V.V. Braguta (MIPT, ITEP , IHEP and FEFU), E.-M. I. small chemical potential (JINR), A. Yu. Kotov (MIPT and ITEP), B. Petersson Taylor expansion: (HU Berlin), S.A. Skinderev (ITEP), "Study of QCD a general purpose phase diagram with non-zero chiral chemical approximation potential", Phys. Rev. D93 (2016) 034509 Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 12 / 171 chemical

  14. Introduction Side projects hypothetically relevant for HIC Lattice QCD at non-zero baryonic density ? E.-M. Ilgenfritz characterizing topological excitations at imaginary chemical potential Introduction Quantum V.G. Bornyakov (ITEP , IHEP and FEFU), D.L. Boyda, statistics and the QCD partition V.A. Goy, A.V. Molochkov, A.A. Nikolaev (ITEP and function FEFU), E.-M. I. (JINR), B.V. Martemyanov (ITEP , Chemical potential on the MEPhI and MIPT), A. Nakamura (Hiroshima U, lattice RIKEN, RCNP Osaka and FEFU Vladivostok) Phase quenching can’t treat the "Dyons and the Roberge-Weiss transition in complex-weight problem lattice QCD" The phase (work in progress) boundary at small chemical Simulations underway with : potential N c = 3 Iwasaki-improved gauge field action Taylor expansion: a general N f = 2 clover-improved Wilson fermion flavors purpose approximation (similar to WHOT-QCD collaboration) Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 13 / 171 chemical

  15. Quantum statistics and the QCD partition function Outline Lattice QCD at non-zero baryonic density ? Introduction 1 E.-M. Ilgenfritz Quantum statistics and the QCD partition function 2 Introduction Chemical potential on the lattice 3 Quantum statistics and the QCD partition Phase quenching can’t treat the complex-weight problem 4 function The phase boundary at small chemical potential Chemical 5 potential on the lattice Taylor expansion: a general purpose approximation 6 Phase quenching can’t treat the Summary of results up to now 7 complex-weight problem Imaginary chemical potential 8 The phase boundary at Complex Langevin dynamics 9 small chemical potential 10 Complex Langevin dynamics for gauge theories Taylor expansion: a general 11 Other approaches to avoid/cure the sign problem purpose approximation 12 Conclusion Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 14 / 171 chemical

  16. Quantum statistics and the QCD partition function The partition function Lattice QCD at non-zero baryonic density ? E.-M. Ilgenfritz Z ( T , V , µ ) = Tr e − ( H − µ N ) / T = e − F / T . Introduction The trace is understood in some basis of eigenstates. Quantum statistics and the From the partition function, or free energy F , other QCD partition function thermodynamic quantities follow by differentiation with Chemical respect to T , µ , V , etc. potential on the lattice � N � = T ∂ � n � = 1 Phase quenching ∂µ ln Z , V � N � , can’t treat the complex-weight problem The phase � χ � = 1 � � N 2 � − � N � 2 � = ∂ � n � boundary at ∂µ . small chemical V potential By studying the behaviour of these and other Taylor expansion: a general thermodynamic quantities while the external parameters purpose approximation like T and µ are changed, the phase structure can be Summary of results up to now scanned in ( T , µ, H .. ) space. (also magnetic field H !) Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 15 / 171 chemical

  17. Quantum statistics and the QCD partition function Other thermodynamical functions derived Lattice QCD at non-zero baryonic density from the partition function ? E.-M. Ilgenfritz From the partition function, all other thermodynamic Introduction equilibrium quantities also follow by taking appropriate Quantum statistics and the derivatives: free energy, pressure, entropy, mean values QCD partition function of charges and the (internal) energy are obtained as Chemical potential on the lattice F = − T ln Z , ∂ ( T ln Z ) Phase quenching ¯ N i = , can’t treat the ∂ ( T ln Z ) ∂µ i complex-weight p = , problem ∂ V − pV + TS + µ i ¯ E = N i . The phase ∂ ( T ln Z ) boundary at S = , small chemical ∂ T potential Taylor expansion: When the partition function is known from any formalism a general purpose (say, a Euclidean lattice calculation), all these relations approximation remain valid. Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 16 / 171 chemical

  18. Quantum statistics and the QCD partition function Conserved charges Lattice QCD at non-zero baryonic density ? In QCD one may consider various conserved charges. E.-M. Ilgenfritz For simplicity, let’s take two flavors, up and down, with Introduction chemical potentials µ u , µ d . Quantum To obtain quark number, we choose the quark chemical statistics and the QCD partition potentials equal, µ u = µ d = µ q , such that function Chemical � n q � = T ∂ potential on the ln Z = � n u � + � n d � . lattice V ∂µ q Phase quenching can’t treat the complex-weight Another possibility is to consider a nonzero isospin problem density. In that case, the chemical potentials are chosen The phase boundary at opposite, µ u = − µ d = µ iso , such that the isospin density small chemical potential equals Taylor expansion: a general � n iso � = T ∂ purpose ln Z = � n u � − � n d � . approximation V ∂µ iso Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 17 / 171 chemical

  19. Quantum statistics and the QCD partition function Conserved charges Lattice QCD at non-zero baryonic density ? E.-M. Ilgenfritz Introduction Quantum Finally, we might be interested in the electrical charge statistics and the QCD partition density and take the chemical potential proportional to function the quarks’ charge, µ u = 2 3 µ Q , µ d = − 1 3 µ Q , such that the Chemical potential on the electrical charge density is given by lattice Phase quenching can’t treat the � n Q � = T ∂ ln Z = 2 3 � n u � − 1 complex-weight 3 � n d � . problem V ∂µ Q The phase boundary at small chemical potential Taylor expansion: a general purpose approximation Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 18 / 171 chemical

  20. Quantum statistics and the QCD partition function The partition function on the lattice Lattice QCD at non-zero baryonic density ? On the lattice, the QCD partition function is written E.-M. Ilgenfritz not as Hilbert space trace over hadrons, but as an Introduction Euclidean path integral in terms of fundamental fields Quantum (quarks, gluons). statistics and the QCD partition function The advantage is not to uncritically anticipate a Chemical particular phase ! (as in the Hadron Resonance Gas potential on the lattice model ! inspired by Hagedorn’s Statistical Bootstrap) Phase quenching can’t treat the It is formulated in terms of the links U x ν = e iaA x ν , with complex-weight problem A x ν the vector potential with a as the lattice spacing. The phase The inverse temperature is given by the extent in the boundary at small chemical potential temporal direction, 1 / T = aN τ , with N τ being the number Taylor expansion: of time slices. a general � � purpose ψ D ψ e − S = DUD ¯ DU e − S YM det M ( U , µ ) . approximation Z = Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 19 / 171 chemical

  21. Quantum statistics and the QCD partition function The lattice action Lattice QCD at non-zero baryonic density ? U denotes the gauge links and ψ, ¯ ψ the quark fields. E.-M. Ilgenfritz Introduction The QCD action has the following schematic form Quantum statistics and the S = S YM + S F QCD partition function Chemical with potential on the lattice � Phase quenching d 4 x ¯ S F = ψ M ( U , µ ) ψ. can’t treat the complex-weight problem The phase S YM is the Yang-Mills action, consisting of closed loops boundary at small chemical formed out of links U x µ (e.g. plaquettes, see later). potential M ( U , µ ) denotes the fermion matrix of a bilinear form, Taylor expansion: a general depending on all links U x µ and the chemical potential(s). purpose approximation Integrating over the quark fields yields the above form, a Summary of result, which contains the determinant det M . results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 20 / 171 chemical

  22. Quantum statistics and the QCD partition function Simulation by importance sampling of gauge Lattice QCD at non-zero baryonic density link configurations ? E.-M. Ilgenfritz Now, in numerical simulations the integrand, Introduction ρ ( U ) ∼ e − S YM det M ( U , µ ) , Quantum statistics and the QCD partition would be a (usually real and positive) probability weight function such that configurations of gauge links can be generated, Chemical potential on the relying on importance sampling. Thus, some version of lattice importance sampling (Hybrid Monte Carlo etc.) can be Phase quenching can’t treat the used. complex-weight problem At non-zero baryonic chemical potential, however, The phase the fermion determinant turns out to be complex, boundary at small chemical [ det M ( U , µ )] ∗ = det M ( U , − µ ∗ ) ∈ C . ( ∗ ) potential Taylor expansion: a general As a result, the weight ρ ( U ) in total is complex and purpose approximation standard numerical algorithms based on importance Summary of sampling are not applicable. results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 21 / 171 chemical

  23. Quantum statistics and the QCD partition function The emergence of the "sign problem" Lattice QCD at non-zero baryonic density ? This is sometimes referred to as the "sign problem", E.-M. Ilgenfritz even though "complex-phase problem" would be Introduction more appropriate. Quantum statistics and the In particle physics, it appears not only in QCD. QCD partition function It appears also, if one goes to Minkowski space Chemical (formulating real-time quantum dynamics). potential on the lattice Phase quenching It appears in other branches of theoretical physics can’t treat the as well (for example, condensed matter and polymer complex-weight problem physics). The phase boundary at Nowadays, it is recognized as one central problem small chemical potential in mathematical and computational physics Taylor expansion: (Topical Workshops, Topical Task Force Programs ...). a general purpose approximation It is closely related to "Resurgence Field Theory" ..., Summary of which unifies perturbative and non-perturbative physics. results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 22 / 171 chemical

  24. Quantum statistics and the QCD partition function Chemical potential for fermion fields in the Lattice QCD at non-zero baryonic density continuum ? E.-M. Ilgenfritz The presence of the complex-phase problem is NOT Introduction restricted to (induced exclusively by) fermions ! Quantum Discussing fermions first, here is the Euclidean action statistics and the QCD partition for non-interacting fermions : function Chemical potential on the � 1 / T lattice � d 3 x ¯ S = d τ ψ ( γ ν ∂ ν + m ) ψ. Phase quenching can’t treat the 0 complex-weight problem Due to the global symmetry The phase boundary at ψ → e i α ψ, ψ → ¯ ¯ ψ e − i α , small chemical potential fermion number is a conserved charge, Taylor expansion: a general � � purpose d 3 x ¯ d 3 x ψ † ψ approximation N = ψγ 4 ψ = ⇒ ∂ τ N = 0 . Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 23 / 171 chemical

  25. Quantum statistics and the QCD partition function Introducing chemical potential of fermion Lattice QCD at non-zero baryonic density fields into the action ? E.-M. Ilgenfritz To obtain the grand canonical partition function in the Introduction Euclidean path integral formulation, one adds the Quantum following term to the action, statistics and the QCD partition � 1 / T function � � µ N = µ d 3 x ¯ d 3 x µ ¯ ψγ 4 ψ = d τ ψγ 4 ψ, Chemical T T potential on the 0 lattice � � Phase quenching can’t treat the d 3 x ¯ d 3 x ψ † ψ N = ψγ 4 ψ = ⇒ ∂ τ N = 0 . complex-weight problem The phase which reads, after inclusion of an Abelian gauge field A ν boundary at small chemical � 1 / T � potential d 3 x ¯ S = d τ ψ [ γ ν ( ∂ ν + iA ν ) + µγ 4 + m ] ψ Taylor expansion: 0 a general � purpose d 4 x ¯ approximation = ψ M ( A , µ ) ψ. Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 24 / 171 chemical

  26. Quantum statistics and the QCD partition function A few observations: Lattice QCD at non-zero baryonic density ? µ appears in the same way as iA 4 , i.e. as the E.-M. Ilgenfritz imaginary part of the four-component of an abelian Introduction vector field. This will be important when chemical Quantum potential is introduced in the lattice formulation. statistics and the QCD partition Generically, the action is complex. This can be function seen by the absence of " γ 5 hermiticity". Chemical potential on the At µ = 0 it is easy to see that lattice ( γ 5 M ) † = γ 5 M , Phase quenching M † = γ 5 M γ 5 , can’t treat the complex-weight problem leading to The phase det M † = det ( γ 5 M γ 5 ) = det M = ( det M ) ∗ , boundary at small chemical potential i.e. the determinant is real. Otherwise, for µ � = 0 Taylor expansion: a general M † ( µ ) = γ 5 M ( − µ ∗ ) γ 5 , purpose approximation resulting in Eq. (*), therefore a complex determinant. Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 25 / 171 chemical

  27. Quantum statistics and the QCD partition function Few more observations: Lattice QCD at non-zero baryonic density ? E.-M. Ilgenfritz When the chemical potential is purely imaginary, the determinant is real again. This has been Introduction exploited extensively and will be discussed later. Quantum statistics and the For Abelian gauge theories, the chemical potential QCD partition function can be removed by a simple gauge transformation Chemical of A 4 (choose µ imaginary and use analyticity). potential on the lattice This is no longer true in Non-Abelian SU ( N ) theories Phase quenching or for theories with more than one chemical potential. can’t treat the complex-weight problem The sign problem is not specific for fermions. In The phase particular, it is not due to the Grassmann nature of boundary at small chemical fermionic fields. potential Taylor expansion: The sign problem arises from the complexity of the a general determinant (in case of fermions) or complexity of the purpose approximation action in general, in any path integral weight. Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 26 / 171 chemical

  28. Quantum statistics and the QCD partition function Chemical potential for bosonic fields in the Lattice QCD at non-zero baryonic density continuum ? E.-M. Ilgenfritz Consider a complex scalar field with a global symmetry Introduction φ → e i α φ . The action is Quantum � statistics and the � | ∂ ν φ | 2 + m 2 | φ | 2 + λ | φ | 4 � d 4 x QCD partition S = , function Chemical and the conserved charge is written potential on the lattice � Phase quenching d 3 x i [ φ ∗ ∂ 4 φ − ( ∂ 4 φ ∗ ) φ ] . N = can’t treat the complex-weight problem The partition function in its Hilbert space form is again The phase Z = Tr e − ( H − µ N ) / T . boundary at small chemical potential Before one expresses this in path integral form, the Taylor expansion: a general Hamiltonian and the conserved charge (densities) purpose approximation must be expressed in terms of the canonical momenta √ Summary of π 1 = ∂ 4 φ 1 , π 2 = ∂ 4 φ 2 , where φ = ( φ 1 + i φ 2 ) / 2. results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 27 / 171 chemical

  29. Quantum statistics and the QCD partition function Chemical potential for bosonic fields in path Lattice QCD at non-zero baryonic density integral form ? E.-M. Ilgenfritz For example, the charge now takes the form Introduction � Quantum d 3 x ( φ 2 π 1 − φ 1 π 2 ) . N = statistics and the QCD partition function The partition function reads in the Euclidean phase space Chemical potential on the path integral form lattice Phase quenching Z = Tr e − ( H − µ N ) / T can’t treat the � � complex-weight problem = D φ 1 D φ 2 D π 1 D π 2 The phase � boundary at � � small chemical d 4 x × exp i π 1 ∂ 4 φ 1 + i π 2 ∂ 4 φ 2 − H + µ ( φ 2 π 1 − φ 1 π 2 ) potential Taylor expansion: a general After integrating out the momenta (done as usual), one purpose approximation finds the Euclidean action in the path integral (over φ Summary of alone is now integrated, no integration over π is left). results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 28 / 171 chemical

  30. Quantum statistics and the QCD partition function Chemical potential for bosonic fields in the Lattice QCD at non-zero baryonic density Euclidean action ? E.-M. Ilgenfritz Introduction � � ( ∂ 4 + µ ) φ ∗ ( ∂ 4 − µ ) φ + | ∂ i φ | 2 + m 2 | φ | 2 + λ | φ | 4 � Quantum d 4 x S = . statistics and the QCD partition function S = Chemical potential on the � � | ∂ ν φ | 2 + ( m 2 − µ 2 ) | φ | 2 + µ ( φ ∗ ∂ 4 φ − ∂ 4 φ ∗ φ ) + λ | φ | 4 � lattice d 4 x Phase quenching can’t treat the complex-weight problem The chemical potential appears again as an The phase boundary at imaginary vector potential. small chemical potential The term linear in µ is purely imaginary, resulting Taylor expansion: in a complex action S ∗ ( µ ) = S ( − µ ∗ ) . a general purpose approximation The term quadratic in µ arose from integrating out Summary of the momenta. This is absent in fermionic theories. results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 29 / 171 chemical

  31. Quantum statistics and the QCD partition function The Silver Blaze problem: a miraculous Lattice QCD at non-zero baryonic density µ -independence for low T at µ < µ onset ? ? E.-M. Ilgenfritz Consider a particle with mass m and a conserved charge Introduction at low temperature: as usual, µ is the change in free Quantum statistics and the energy (work) when a particle carrying the conserved QCD partition function charge is added, i.e. the energy reservoir for adding one Chemical particle. Hence it is plausible that potential on the lattice if µ < m : not enough energy available to create a Phase quenching can’t treat the new particle ⇒ no change in the groundstate; complex-weight problem if µ > m : plenty of energy available ⇒ now the The phase groundstate acquires a nonzero density of particles. boundary at small chemical potential Hence it follows from simple statistical mechanics that at Taylor expansion: zero temperature the density becomes nonzero (a.k.a. a general purpose "onset") only for µ > µ onset ≡ m . This will be approximation demonstrated for free fermions. Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 30 / 171 chemical

  32. Quantum statistics and the QCD partition function The Essence of the Silver Blaze problem Lattice QCD at non-zero baryonic density ? In general, the term "Silver Blaze" denotes a miraculous E.-M. Ilgenfritz (almost-) independence of µ at low enough T throughout Introduction the interval 0 < µ < µ onset , Quantum where µ onset = O ( some characteristic mass of the theory ) . statistics and the QCD partition function This (almost-) independence has its origin in Chemical potential on the cancellations related to the sign problem. lattice Phase quenching can’t treat the These cancellations are eventually absent in a not complex-weight adequately substituted theory (e.g. the "phase problem The phase quenched theory"). This one is simply misleading boundary at small chemical (no approximation at all!) because it actually potential represents other, "wrong" physics ! Taylor expansion: a general purpose The complex phase problem is not a minor defect ! approximation Summary of It is necessary to reproduce the correct physics. results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 31 / 171 chemical

  33. Quantum statistics and the QCD partition function There would be no free energy difference Lattice QCD at non-zero baryonic density between quark and antiquark without ? E.-M. Ilgenfritz Im ( det M ) � = 0 ! (Ph. de Forcrand) Introduction Quantum statistics and the � Tr Polyakov � = exp ( − 1 QCD partition T F q ) function Chemical � potential on the [ Re ( P ) × Re ( det M ) − Im ( P ) × Im ( det M )] e − S YM DU = lattice Phase quenching can’t treat the complex-weight � Tr Polyakov + � = exp ( − 1 problem T F ¯ q ) The phase � boundary at small chemical [ Re ( P ) × Re ( det M ) + Im ( P ) × Im ( det M )] e − S YM DU = potential Taylor expansion: a general For SU ( 2 ) , N f = 2, the square of the determinant remains purpose approximation real positive even when µ � = 0. But the µ q can be turned Summary of into µ iso by a redefinition of the quark fields. results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 32 / 171 chemical

  34. Quantum statistics and the QCD partition function The Silver Blaze problem is not unfamiliar Lattice QCD at non-zero baryonic density from standard thermodynamics with mass m ? E.-M. Ilgenfritz The standard expression for the logarithm of the partition Introduction function for a free relativistic fermion gas with mass m is Quantum statistics and the � d 3 p � � 1 + e − β ( ω p − µ ) � QCD partition ln Z = 2 V βω p + ln function ( 2 π ) 3 Chemical � 1 + e − β ( ω p + µ ) �� potential on the + ln , lattice Phase quenching � p 2 + m 2 and β = 1 / T . can’t treat the where ω p = complex-weight problem 2 is the spin factor, the first term is the zero-point energy The phase and the other terms represent particles and anti-particles boundary at small chemical at nonzero temperature and chemical potential. potential The fermion minus antifermion density is Taylor expansion: a general � � � d 3 p purpose � n � = T ∂ ln Z 1 1 approximation = 2 e β ( ω p − µ ) + 1 − . e β ( ω p + µ ) + 1 ( 2 π ) 3 V ∂µ Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 33 / 171 chemical

  35. Quantum statistics and the QCD partition function The two cases, below and above onset m Lattice QCD at non-zero baryonic density ? We consider the low-temperature limit, T → 0. E.-M. Ilgenfritz We distinguish two cases (separated by m ): Introduction µ < m : the ‘1’ in the denominator of the Fermi-Dirac Quantum distribution can be ignored and statistics and the QCD partition � d 3 p � e − β ( ω p − µ ) − e − β ( ω p + µ ) � function � n � ∼ 2 → 0 . Chemical ( 2 π ) 3 potential on the lattice Particles and antiparticles are only thermally excited Phase quenching and therefore Boltzmann suppressed. can’t treat the complex-weight µ > m : in this case µ can be larger than ω p , the problem The phase Fermi-Dirac distribution becomes a step function boundary at at T = 0, further rising like ∼ µ 3 : small chemical potential � µ 2 − m 2 � 3 / 2 � Taylor expansion: d 3 p a general � n � ∼ 2 ( 2 π ) 3 Θ( µ − ω p ) = Θ( µ − m ) . purpose 3 π 2 approximation Summary of As expected, nonzero density for µ > m (i.e. "onset"). results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 34 / 171 chemical

  36. Chemical potential on the lattice Outline Lattice QCD at non-zero baryonic density ? Introduction 1 E.-M. Ilgenfritz Quantum statistics and the QCD partition function 2 Introduction Chemical potential on the lattice 3 Quantum statistics and the QCD partition Phase quenching can’t treat the complex-weight problem 4 function The phase boundary at small chemical potential Chemical 5 potential on the lattice Taylor expansion: a general purpose approximation 6 Phase quenching can’t treat the Summary of results up to now 7 complex-weight problem Imaginary chemical potential 8 The phase boundary at Complex Langevin dynamics 9 small chemical potential 10 Complex Langevin dynamics for gauge theories Taylor expansion: a general 11 Other approaches to avoid/cure the sign problem purpose approximation 12 Conclusion Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 35 / 171 chemical

  37. Chemical potential on the lattice Introducing a chemical potential for lattice Lattice QCD at non-zero baryonic density fermions ? E.-M. Ilgenfritz The naive way, adding µ ¯ ψγ 4 ψ to the action, leads to Introduction µ -dependent ultraviolet divergences like what appears Quantum � µ � 2 . statistics and the in the energy density, ǫ ( µ ) − ǫ ( 0 ) ∼ a QCD partition function Instead, we better follow the observations made in the Chemical continuum : potential on the lattice the chemical potential couples to the 4-th component Phase quenching of the corresponding conserved point-split current; can’t treat the complex-weight it appears as the imaginary part of the fourth problem component of an Abelian vector field. The phase boundary at small chemical The terms in the action from which the conserved lattice potential current follows, the so-called hopping terms, are Taylor expansion: a general S ∼ ¯ ψ x U x ν γ ν ψ x + ν − ¯ ψ x + ν U † purpose x ν γ ν ψ x , approximation Summary of for all directions ν = 1 , 2 , 3 , 4. results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 36 / 171 chemical

  38. Chemical potential on the lattice Introducing a chemical potential for lattice Lattice QCD at non-zero baryonic density fermions ? E.-M. Ilgenfritz The exactly conserved (point-split) current reads then Introduction ψ x + ν U † j ν ∼ ¯ ψ x U x ν γ ν ψ x + ν + ¯ x ν γ ν ψ x . Quantum statistics and the QCD partition Chemical potential is now introduced as an imaginary function Abelian vector field in the 4-direction, i.e. multiplying Chemical potential on the the (non-Abelian) links by Abelian factors exp ( ± a µ ) lattice Phase quenching e a µ U x 4 , U x 4 = e iA 4 x can’t treat the forward hopping: ⇒ complex-weight U † e − a µ U † problem x 4 = e − iA 4 x backward hopping: ⇒ x 4 . The phase boundary at Features of this construction : small chemical potential the correct (naive) continuum limit is preserved, Taylor expansion: µ couples to the exactly conserved charge, even a general purpose at finite lattice spacing a , approximation Summary of no additional ultraviolet divergences are generated. results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 37 / 171 chemical

  39. Chemical potential on the lattice Consequence of chemical potential of lattice Lattice QCD at non-zero baryonic density fermions: forward and backward hopping get ? E.-M. Ilgenfritz different weight in the determinant Introduction Quantum statistics and the QCD partition function Chemical potential on the e µ N τ = e µ/ T e − µ N τ = e − µ/ T lattice Phase quenching can’t treat the (a) (b) complex-weight problem Figure: (a) Forward (backward) hopping is (dis)favoured by The phase boundary at e µ n τ ( e − µ n τ ), while closed loops are µ -independent. (b) Loops small chemical wrapping around the temporal direction contribute e ± µ/ T . This potential Taylor expansion: interpretation is useful for the hopping parameter expansion or a general any decomposition (say, by reduction formulae) of the fermion purpose approximation determinant ! It suggests that imaginary µ is equivalent to Summary of phase-rotated boundary conditions for wrapping in 4-direction. results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 38 / 171 chemical

  40. Chemical potential on the lattice Consequence of chemical potential of lattice Lattice QCD at non-zero baryonic density bosons : a closed solution for λ = 0 ? E.-M. Ilgenfritz Consider a self-interacting complex scalar field in the Introduction presence of a chemical potential µ , with the continuum Quantum action S = statistics and the QCD partition � � | ∂ ν φ | 2 + ( m 2 − µ 2 ) | φ | 2 + µ ( φ ∗ ∂ 4 φ − ∂ 4 φ ∗ φ ) + λ | φ | 4 � function d 4 x Chemical potential on the lattice The Euclidean action is complex and satisfies Phase quenching S ∗ ( µ ) = S ( − µ ∗ ) . Take m 2 > 0, such that at vanishing µ can’t treat the complex-weight and small µ the theory is in the symmetric phase. problem The lattice action (lattice spacing a lat put equal 1) is The phase boundary at � � small chemical 2 d + m 2 � � φ ∗ x φ x + λ ( φ ∗ x φ x ) 2 potential S = Taylor expansion: x a general purpose 4 � � � approximation � φ ∗ x e − µδ ν, 4 φ x +ˆ ν + φ ∗ ν e µδ ν, 4 φ x − . x +ˆ Summary of results up to now ν = 1 Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 39 / 171 Number of Euclidean dimensions d = 4. chemical

  41. Chemical potential on the lattice Solving the lattice boson problem with Lattice QCD at non-zero baryonic density non-zero chemical potential ? E.-M. Ilgenfritz The complex field is written in terms of two real fields φ a Introduction 1 ( a = 1 , 2) as φ = 2 ( φ 1 + i φ 2 ) . The lattice action reads √ Quantum statistics and the QCD partition � 3 � 2 d + m 2 � a , x + λ � � 2 � � function 1 φ 2 φ 2 S = − φ a , x φ a , x +ˆ a , x 2 i Chemical 4 potential on the x i = 1 lattice � Phase quenching − cosh µ φ a , x φ a , x +ˆ 4 + i sinh µ ε ab φ a , x φ b , x +ˆ . can’t treat the 4 complex-weight problem ε ab = antisymmetric tensor with ǫ 12 = 1 The phase (a "hopping term" interchanging 1 ←→ 2). boundary at small chemical potential The sinh µ term is the imaginary part of the action. Taylor expansion: a general From now on the self-interaction is ignored and we take purpose approximation λ = 0. The action is now reduced to bilinear form (which Summary of renders the problem directly solvable). results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 40 / 171 chemical

  42. Chemical potential on the lattice Consequence of chemical potential of lattice Lattice QCD at non-zero baryonic density bosons: Gaussian path integral, has a closed ? E.-M. Ilgenfritz solution Introduction In momentum space the action reads Quantum statistics and the � � QCD partition 1 1 S = 2 φ a , − p ( δ ab A p − ε ab B p ) φ b , p = 2 φ a , − p M ab , p φ b , p , function p p Chemical potential on the lattice where � A p Phase quenching � can’t treat the − B p M p = , complex-weight B p A p problem The phase and boundary at small chemical potential 3 sin 2 p i � m 2 + 4 Taylor expansion: A p = 2 + 2 ( 1 − cosh µ cos p 4 ) , a general purpose i = 1 approximation B p = 2 sinh µ sin p 4 . Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 41 / 171 chemical

  43. Chemical potential on the lattice Consequence of chemical potential of lattice Lattice QCD at non-zero baryonic density bosons: Gaussian path integral, closed ? E.-M. Ilgenfritz solution Introduction The propagator corresponding to the action is Quantum statistics and the QCD partition G ab , p = δ ab A p + ε ab B p function . A 2 p + B 2 Chemical p potential on the lattice The dispersion relation that follows from the poles of the Phase quenching propagator, taking p 4 = iE p , reads can’t treat the complex-weight problem � � � 1 + 1 The phase 1 + 1 ω 2 ω 2 cosh E p ( µ ) = cosh µ 2 ˆ ± sinh µ 4 ˆ p , boundary at p small chemical potential where Taylor expansion: a general sin 2 p i � p = m 2 + 4 ω 2 purpose ˆ 2 . approximation i Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 42 / 171 chemical

  44. Chemical potential on the lattice Lattice QCD at non-zero baryonic density ? E.-M. Ilgenfritz Introduction Quantum This can be written (thanks to the addition theorem statistics and the QCD partition for the hyperbolic cosh) as function Chemical potential on the cosh E p ( µ ) = cosh [ E p ( 0 ) ± µ ] , lattice Phase quenching can’t treat the complex-weight problem The phase boundary at small chemical potential Taylor expansion: a general purpose approximation Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 43 / 171 chemical

  45. Chemical potential on the lattice Comparison of the spectrum between full and Lattice QCD at non-zero baryonic density phase-quenched theory ? E.-M. Ilgenfritz Thus, the (positive energy) solutions in the theory are Introduction E p ( µ ) = E p ( µ = 0 ) ± µ. Quantum statistics and the QCD partition The critical µ value for onset is µ c = E 0 ( 0 ) , so that one function mode becomes exactly massless at the transition µ c Chemical potential on the (Goldstone boson). lattice The phase-quenched theory, in contrast, corresponds to Phase quenching can’t treat the putting sinh µ = B p = 0 (removal of the imaginary part complex-weight problem of action). The dispersion relation in the phase-quenched The phase theory is then completely different: boundary at small chemical potential 1 � � 1 + 1 ω 2 cosh E p ( µ ) = 2 ˆ , Taylor expansion: p cosh µ a general purpose p ( µ ) = m 2 − µ 2 + p 2 = E 2 approximation corresponding to E 2 p ( µ = 0 ) − µ 2 Summary of in the continuum limit. results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 44 / 171 chemical

  46. Chemical potential on the lattice Exercises I Lattice QCD at non-zero baryonic density ? E.-M. Ilgenfritz Introduction Quantum statistics and the Compare the spectrum of the full and the QCD partition phase-quenched theory, when µ < µ c . function Chemical At larger µ , it is necessary to include the potential on the lattice self-interaction λ to stabilize the theory. Phase quenching can’t treat the Based on what you know about symmetry complex-weight problem breaking, sketch the spectrum in the full and The phase the phase-quenched theory also for larger µ . boundary at small chemical potential Taylor expansion: a general purpose approximation Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 45 / 171 chemical

  47. Chemical potential on the lattice Are the thermodynamic quantities Lattice QCD at non-zero baryonic density independent of µ at vanishing temperature ? ? E.-M. Ilgenfritz Although the spectrum depends on µ , thermodynamic Introduction quantities do not. Up to an irrelevant constant, the Quantum logarithm of the partition function is statistics and the QCD partition � � function ln Z = − 1 ln det M = − 1 ln ( A 2 p + B 2 p ) , 2 2 Chemical potential on the p p lattice and some observables are given by Phase quenching can’t treat the ∂ ln Z A p �| φ | 2 � = − 1 ∂ m 2 = 1 � complex-weight , problem Ω Ω A 2 p + B 2 p The phase p boundary at small chemical and potential A p A ′ p + B p B ′ � n � = 1 ∂ ln Z = − 1 � Taylor expansion: p , a general A 2 p + B 2 Ω ∂µ Ω purpose p p approximation Summary of where Ω = N 3 σ N τ and A ′ p = ∂ A p /∂µ , B ′ p = ∂ B p /∂µ . results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 46 / 171 chemical

  48. Chemical potential on the lattice Exercises II Lattice QCD at non-zero baryonic density ? E.-M. Ilgenfritz Introduction Quantum Difference compared to the phase-quenched theory ? statistics and the QCD partition function Evaluate the sums (e.g. numerically) to demonstrate Chemical that thermodynamic quantities are independent of µ potential on the lattice in the thermodynamic limit at vanishing temperature, Phase quenching but this holds only in the full theory ! can’t treat the complex-weight problem These exercises are based on G. Aarts, The phase JHEP 0905 (2009) 052, arXiv:0902.4686 boundary at small chemical potential Taylor expansion: a general purpose approximation Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 47 / 171 chemical

  49. Phase quenching can’t treat the complex-weight problem Outline Lattice QCD at non-zero baryonic density ? Introduction 1 E.-M. Ilgenfritz Quantum statistics and the QCD partition function 2 Introduction Chemical potential on the lattice 3 Quantum statistics and the QCD partition Phase quenching can’t treat the complex-weight problem 4 function The phase boundary at small chemical potential Chemical 5 potential on the lattice Taylor expansion: a general purpose approximation 6 Phase quenching can’t treat the Summary of results up to now 7 complex-weight problem Imaginary chemical potential 8 The phase boundary at Complex Langevin dynamics 9 small chemical potential 10 Complex Langevin dynamics for gauge theories Taylor expansion: a general 11 Other approaches to avoid/cure the sign problem purpose approximation 12 Conclusion Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 48 / 171 chemical

  50. Phase quenching can’t treat the complex-weight problem How to deal with the complex weight in Lattice QCD at non-zero baryonic density practical simulations ? ? E.-M. Ilgenfritz A prompt (but naive !) answer would be: Introduction simplify the weight for sampling, just neglecting the Quantum phase which usually is preventing the sampling; statistics and the QCD partition account for the phase factor later by reweighting function Chemical (in the moment when calculating observables). potential on the lattice Let us consider again the partition function Phase quenching � � can’t treat the ψ D ψ e − S = DUD ¯ DU e − S B det M , Z = (1) complex-weight problem The phase with a complex determinant, boundary at small chemical det M = | det M | e i ϕ . potential (2) Taylor expansion: a general An seemingly straightforward solution to the complex- purpose approximation phase problem is to "absorb" the phase factor into the Summary of observable, just as a reweighting factor. results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 49 / 171 chemical

  51. Phase quenching can’t treat the complex-weight problem Phase quenching Lattice QCD at non-zero baryonic density ? E.-M. Ilgenfritz � Introduction DU e − S B det M O � O � full = Quantum � DU e − S B det M statistics and the QCD partition � function DU e − S B | det M | e i ϕ O = Chemical � DU e − S B | det M | e i ϕ potential on the lattice � e i ϕ O � pq Phase quenching = . can’t treat the � e i ϕ � pq complex-weight problem The phase �·� full denotes expectation values taken with respect boundary at small chemical to the original, complex weight ρ ( U ) ∝ det M , potential Taylor expansion: a general �·� pq denotes expectation values with respect to the purpose approximation "phase-quenched" weight, i.e. using ρ ( U ) ∝ | det M | . Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 50 / 171 chemical

  52. Phase quenching can’t treat the complex-weight problem Why is phase quenching useless closer to Lattice QCD at non-zero baryonic density the thermodynamical limit ? ? E.-M. Ilgenfritz Look at the "average phase factor" � e i ϕ � pq . Introduction This has the form of a ratio of two partition functions : Quantum � statistics and the DU e − S B | det M | e i ϕ = Z full QCD partition � e i ϕ � pq = = e − Ω∆ f , function � DU e − S B | det M | Z pq Chemical potential on the where we have expressed the partition functions in terms lattice of the free energy densities, Phase quenching can’t treat the Z ≡ Z full = e − F / T = e − Ω f full , Z pq = e − F pq / T = e − Ω f pq , complex-weight problem The phase with Ω the spacetime volume ( Ω = V / T in physical units boundary at small chemical or N τ N 3 s in lattice units), and potential Taylor expansion: ∆ f = f full − f pq > 0 a general purpose is the difference of the free energy densities. approximation Summary of Obviously, the following inequality holds: Z full ≤ Z pq . results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 51 / 171 chemical

  53. Phase quenching can’t treat the complex-weight problem Overlap problem Lattice QCD at non-zero baryonic density ? The expectation value, that one is seeking for, E.-M. Ilgenfritz � O � full = � e i ϕ O � pq Introduction � e i ϕ � pq Quantum statistics and the is of exponentially undefined type "0 / 0" in the limit QCD partition function V → ∞ . Chemical potential on the lattice One says: "The sign problem is exponentially hard." Phase quenching can’t treat the Physics of the two ensembles differs in an essential way: complex-weight problem if they share (only few) configurations at all, these are The phase possessing strongly different weight in the respective boundary at small chemical ensembles. potential Taylor expansion: a general What different physics corresponds to the purpose approximation phase-quenched ensemble compared to the Summary of fixed-baryon-density ensemble ? results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 52 / 171 chemical

  54. Phase quenching can’t treat the complex-weight problem Overlap problem = missing overlap between Lattice QCD at non-zero baryonic density ensembles ? E.-M. Ilgenfritz Consider two mass-degenerate flavors. | det M | not easy ! Introduction ρ ( U ) ∝ [ det M ( µ )] 2 fixed quark density ensemble Quantum statistics and the QCD partition whereas the function | det M ( µ ) | 2 Chemical phase quenched ensemble ρ ( U ) ∝ potential on the lattice det M † ( µ ) det M ( µ ) ∝ Phase quenching can’t treat the ∝ det M ( − µ ) det M ( µ ) , complex-weight problem is actually corresponding to an isospin chemical potential The phase with a value µ iso = µ coinciding with µ . boundary at small chemical Difference of phase structure (in µ q vs. µ iso = µ u = − µ d potential Taylor expansion: is easy to understand physically (but difficult to a general purpose understand in terms of gauge configurations !). approximation Undiscovered topological features ? Summary of results up to now "Disoriented" condensates ? Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 53 / 171 chemical

  55. Phase quenching can’t treat the complex-weight problem A severe sign problem exists due to pion Lattice QCD at non-zero baryonic density condensation ? E.-M. Ilgenfritz Introduction 1 1 T T/T c = 0.76 T/T c = 0.90 * <exp(2i θ )> 1+1 Quantum CPT Allton statistics and the 0.5 0.5 QCD partition < +> = 0 π function / 0 0 0 0.5 1 0 0.5 1 Chemical 1 1 potential on the severe T/T c = 1.00 T/T c = 1.11 lattice * <exp(2i θ )> 1+1 sign problem 0.5 0.5 Phase quenching can’t treat the complex-weight 0 0 problem mu 0 m /2 mN /3 π 0 0.5 1 0 0.5 1 2 µ /m π 2 µ /m π The phase boundary at Figure: Left : Sketch of the QCD pseudo-critical line T c ( µ ) (in red), starting from ∼ m N / 3 at T = 0, small chemical superimposed with the phase transition line (in blue) of the phase-quenched theory ( alias isospin potential chemical potential), starting pion condensation from m π / 2 at T = 0. Right : Comparison of values of the “average phase factor” � exp ( 2 i θ ) � , measured in lattice simulations and predicted by one-loop χ PT Taylor expansion: (Splittorff 2007). Good agreement with χ PT persists up to T / T c ∼ 0 . 90. a general purpose approximation Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 54 / 171 chemical

  56. Phase quenching can’t treat the complex-weight problem Average phase factor in the phase-quenched Lattice QCD at non-zero baryonic density theory at T = 0 ? E.-M. Ilgenfritz 1 Introduction Quantum i φ > pq <e statistics and the QCD partition function Chemical potential on the lattice Phase quenching can’t treat the 0 m π /2 µ 0 complex-weight problem Figure: Average phase factor in the thermodynamic limit The phase boundary at V → ∞ in the phase-quenched theory at T = 0. In other small chemical potential words, throughout the interval 0 < µ < m π / 2 phase quenching Taylor expansion: is not misleading at T = 0 ! But no interesting physics is a general happening there in both theories ! In the interval purpose approximation 0 < µ < m N / 3, however, strong cancellations are required to Summary of cancel the unwanted µ -dependence from the full theory. results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 55 / 171 chemical

  57. Phase quenching can’t treat the complex-weight problem Silver blaze problem from the Dirac operator’s Lattice QCD at non-zero baryonic density eigenvalue point of view ? E.-M. Ilgenfritz Consider the Dirac operator as Introduction M = D + m with D = D / + µγ 4 . Quantum statistics and the QCD partition The partition function is written as function � Chemical DU det ( D + m ) e − S YM = �� det ( D + m ) �� YM , Z = potential on the lattice Phase quenching where the subscript YM indicates the average over can’t treat the complex-weight the gluonic field only. (The brackets ��·�� YM are not problem normalized like expectation values !) The phase boundary at The determinant is the product of the eigenvalues, small chemical potential � det ( D + m ) = ( λ k + m ) D ψ k = λ k ψ k . Taylor expansion: a general k purpose approximation Note that since D is not γ 5 hermitian at nonzero µ , the Summary of eigenvalues are complex (a cloud in complex plane). results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 56 / 171 chemical

  58. Phase quenching can’t treat the complex-weight problem Silver blaze problem for the chiral condensate Lattice QCD at non-zero baryonic density ? Look at the chiral condensate, which is expressed as E.-M. Ilgenfritz �� �� ψψ � = 1 ∂ ln Z = 1 1 1 � � � ¯ Introduction ( λ j + m ) , Ω ∂ m Z Ω λ k + m Quantum k j statistics and the YM QCD partition since the derivative with respect to m removes every function factor λ k + m from the determinant once. This can be Chemical potential on the written in terms of the density of eigenvalues, defined as lattice � Phase quenching ρ ( z ; µ ) = 1 DU det ( D + m ) e − S YM 1 � can’t treat the δ 2 ( z − λ k ) complex-weight Z Ω problem k �� �� The phase = 1 det ( D + m ) 1 boundary at � δ 2 ( z − λ k ) small chemical . potential Z Ω k YM Taylor expansion: a general Writing the condensate as integral over the density: purpose approximation � d 2 z ρ ( z ; µ ) � ¯ Summary of ψψ � = z + m . results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 57 / 171 chemical

  59. Phase quenching can’t treat the complex-weight problem How can the Silver Blaze effect traced back Lattice QCD at non-zero baryonic density to the spectral function ? ? E.-M. Ilgenfritz For every fixed configuration at µ � = 0, the spectral density Introduction explicitely depends on µ . Quantum When the gauge average ist taken, the average spectral statistics and the QCD partition density and any integral over it looses dependence on µ function as long as 0 < µ < m B / 3. This is the Silver Blaze region. Chemical potential on the The average spectral density is a complicated (weird !!!) lattice function oscillating with amplitude ∝ e Ω µ , very rapidly with Phase quenching can’t treat the complex-weight a period 1 / Ω (inverse space-time voume). Only when all problem is absolutely correctly integrated, the unwanted (wrong) The phase boundary at µ -dependence will be cancelled. small chemical potential This singular behavior has been studied by Osborn, Taylor expansion: Splittorff, Verbaarschot (in the years 2005 to 2008). a general purpose This is illustrated by a 0 + 1 dimensional toy-model that approximation can be followed in G. Aarts and K. Splittorff, JHEP 1008 Summary of results up to now (2010), arXiv:1006.0332 Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 58 / 171 chemical

  60. The phase boundary at small chemical potential Outline Lattice QCD at non-zero baryonic density ? Introduction 1 E.-M. Ilgenfritz Quantum statistics and the QCD partition function 2 Introduction Chemical potential on the lattice 3 Quantum statistics and the QCD partition Phase quenching can’t treat the complex-weight problem 4 function The phase boundary at small chemical potential Chemical 5 potential on the lattice Taylor expansion: a general purpose approximation 6 Phase quenching can’t treat the Summary of results up to now 7 complex-weight problem Imaginary chemical potential 8 The phase boundary at Complex Langevin dynamics 9 small chemical potential 10 Complex Langevin dynamics for gauge theories Taylor expansion: a general 11 Other approaches to avoid/cure the sign problem purpose approximation 12 Conclusion Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 59 / 171 chemical

  61. The phase boundary at small chemical potential Return to the more realistic problem of phase Lattice QCD at non-zero baryonic density diagram of 4-dimensional QCD ? E.-M. Ilgenfritz Introduction Quantum statistics and the QCD partition function Chemical potential on the lattice Phase quenching can’t treat the complex-weight problem The phase boundary at small chemical potential Taylor expansion: a general purpose Figure: Conjectured phase diagram of QCD as a function of approximation quark chemical potential µ and temperature T , from Wikipedia. Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 60 / 171 chemical

  62. The phase boundary at small chemical potential Minimalistic phase diagram Lattice QCD at non-zero baryonic density ? E.-M. Ilgenfritz QGP endpoint (second order) Introduction Quantum T statistics and the first order QCD partition crossover function Chemical potential on the lattice Phase quenching can’t treat the confined complex-weight problem The phase µ boundary at small chemical potential Figure: “Standard” phase diagram. Taylor expansion: a general purpose approximation Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 61 / 171 chemical

  63. The phase boundary at small chemical potential Curvature of the phase boundary near µ = 0 Lattice QCD at non-zero baryonic density ? For small chemical potential, the pseudo-critical E.-M. Ilgenfritz temperature of the phase boundary at small nonzero µ Introduction can be written as a series in µ/ T , for instance as Quantum statistics and the � � 2 � � 4 QCD partition T c ( µ ) µ µ function T c ( 0 ) = 1 + a 2 + a 4 + . . . T c ( 0 ) T c ( 0 ) Chemical potential on the lattice Since the partition function is an even function of µ , Phase quenching only even powers of µ appear. can’t treat the complex-weight problem FAQ : Curvature of the phase (crossover) boundary ? The phase boundary at small chemical Eventually not identical to the chemical freeze-out curve !! potential Taylor expansion: a general purpose The "sign problem" is hoped to be less severe for small µ approximation Summary of and T close to the crossover at T c ( 0 ) ! results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 62 / 171 chemical

  64. The phase boundary at small chemical potential Once again about Reweighting Lattice QCD at non-zero baryonic density ? The general strategy in reweighting was already E.-M. Ilgenfritz discussed above. The partition function is now written as Introduction � Quantum Z w = DU w ( U ) , w ( U ) ∈ C , statistics and the QCD partition function and observables are expressed as Chemical potential on the lattice � DU O ( U ) w ( U ) Phase quenching � O � w = . � can’t treat the DU w ( U ) complex-weight problem Let us now introduce a new weight r ( U ) (" r " resembling The phase boundary at "reweighting" or "real"), which is chosen at will, such that small chemical potential � DU O ( U ) w ( U ) Taylor expansion: r ( U ) r ( U ) = � O w r � r a general � O � w = . purpose � � w DU w ( U ) r � r approximation r ( U ) r ( U ) Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 63 / 171 chemical

  65. The phase boundary at small chemical potential Once again about reweighting Lattice QCD at non-zero baryonic density ? E.-M. Ilgenfritz The average reweighting factor (w.r.t. the r ensemble) indicates the severity of the overlap problem. Introduction Quantum � w � r = Z w statistics and the = e − Ω∆ f , ∆ f = f w − f r ≥ 0 , QCD partition r Z r function Chemical potential on the where Ω denotes again the spacetime volume. lattice There is considerable freedom in choosing the new Phase quenching can’t treat the weight r ( U ) , provided that it has the interpretation complex-weight problem of a probability weight, such that sampling (for the The phase purpose of numerical simulation) is possible. boundary at small chemical potential One may adapt the "model" r more successfully to Taylor expansion: a general the problem at hand, avoiding previous mistakes like purpose approximation in the phase-quenching case ! Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 64 / 171 chemical

  66. The phase boundary at small chemical potential Two examples of reweighting strategies: Lattice QCD at non-zero baryonic density Glasgow vs Budapest ? E.-M. Ilgenfritz Glasgow reweighting: works at a fixed temperature Introduction (same lattice coupling β ) and jumps in µ directly from Quantum statistics and the 0 to the target µ , QCD partition function w det M ( U , µ ) r ∼ det M ( U , µ = 0 ) , Chemical potential on the lattice as illustrated in next Figure (left). Phase quenching can’t treat the However, this choice has a severe overlap problem, complex-weight problem since the high-density phase is probed with a typical The phase confinement ensemble at µ = 0, just at the same boundary at small chemical temperature T < T c ( µ = 0 ) below deconfinement. potential The onset is not observed at m baryon / 3 where it Taylor expansion: a general should be, but at m π / 2, similar to phase quenched purpose approximation simulations (i.e. there is no improvement over the Summary of previous quenched studies in valence approx.) results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 65 / 171 chemical

  67. The phase boundary at small chemical potential The Glasgow strategy fails for reasonable Lattice QCD at non-zero baryonic density volumes ? E.-M. Ilgenfritz Introduction Quantum statistics and the QCD partition function Chemical potential on the One expects ∆ f to be large and hence the overlap lattice problem will appear already on very small volumes Phase quenching can’t treat the (for example, a lattice volume 4 4 ). complex-weight problem The phase boundary at small chemical potential Taylor expansion: a general purpose approximation Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 66 / 171 chemical

  68. The phase boundary at small chemical potential The two reweighting strategies: Glasgow vs. Lattice QCD at non-zero baryonic density Budapest ? E.-M. Ilgenfritz Introduction T T Quantum statistics and the QCD partition function Chemical potential on the lattice µ µ Phase quenching can’t treat the complex-weight Figure: Reweighting at Fixed Temperature (Glasgow) (left) and problem Multiparameter Reweighting (Budapest), which is aiming to The phase maximise the overlap as good as possible (right). Sampling of boundary at small chemical Budapest style proceeds at the reference point on the potential temperature axis and (very importantly !!!) successfully Taylor expansion: a general captures there a mixture of confining and deconfining purpose configurations ! approximation Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 67 / 171 chemical

  69. The phase boundary at small chemical potential More precise about Budapest reweighting Lattice QCD at non-zero baryonic density ? Multiparameter/overlap preserving reweighting : here the E.-M. Ilgenfritz temperature (or lattice coupling β ) is adapted as well (see Introduction the last Figure, right). Hence Quantum w det M ( U , µ ) statistics and the det M ( U , µ = 0 ) e − ∆ S YM , r ∼ QCD partition function Chemical potential on the ∆ S YM = S YM ( U , β ) − S YM ( U , β c ( µ = 0 )) lattice Phase quenching is the difference between gauge actions at the actual ( T ) can’t treat the complex-weight and the reference temperature T ref = T c ( µ = 0 ) . problem The main idea here is the attempt to stay on the The phase boundary at pseudo-critical line T c ( µ ) , improving overlap, since small chemical potential both the confined phase and the quark-gluon plasma Taylor expansion: are sampled, albeit at higher T than really needed. a general purpose T c ( µ ) is found by a T -scan (max. of susceptibility ?) approximation Summary of at any fixed µ , regardless whether µ < µ E or µ > µ E . results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 68 / 171 chemical

  70. The phase boundary at small chemical potential Imaginary chemical potential µ = i µ I shifts Lattice QCD at non-zero baryonic density the quark condensate oppositely to real µ ? E.-M. Ilgenfritz Introduction Quantum statistics and the QCD partition function Chemical potential on the lattice Phase quenching can’t treat the complex-weight problem The phase boundary at small chemical potential Figure: At imaginary chemical potential one can simulate ! Taylor expansion: Immediate simulation results (squares) can be compared with a general purpose results of Glasgow-type (dots) and Budapest-type (crosses) approximation reweighting. Glasgow reweighting is by far insufficient ! Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 69 / 171 chemical

  71. The phase boundary at small chemical potential Critical endpoint from Budapest reweighting Lattice QCD at non-zero baryonic density ? µ E and T E = T c ( µ E ) denote the critical endpoint where E.-M. Ilgenfritz the crossover line goes over into a first order line. Introduction First, one has to find the line of maximal susceptibility Quantum (in other words, the ridge of the overlap measure). statistics and the QCD partition The endpoint is fixed along the line of maximal function susceptibility β max ( µ ) by an analysis of Lee-Yang Chemical potential on the zeroes: when to β max an imaginary part β I is added, lattice the partition function develops a pattern of zeroes. Phase quenching can’t treat the complex-weight If the location of the Lee-Yang-zero closest to the real problem axis moves towards the real axis in the limit V → ∞ , The phase boundary at this tells us that one is sitting in the µ region small chemical potential related to the first order transition. Taylor expansion: If the location stays away from the real axis a general purpose (independent of V ), this is telling us that one is approximation sitting in the µ region related to the crossover. Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 70 / 171 chemical

  72. The phase boundary at small chemical potential Lee-Yang zeros determining the endpoint of Lattice QCD at non-zero baryonic density the first oder electroweak phase transition in ? E.-M. Ilgenfritz a gauge-Higgs model Introduction Quantum statistics and the QCD partition function 1.0 Chemical potential on the lattice Phase quenching » Z norm » can’t treat the complex-weight problem 0 The phase 0.00001 0.00001 0.00002 0.00002 boundary at small chemical Im b H potential Taylor expansion: Figure: 3 d view of | Z norm | embracing the first zeroes found by a general adding Im β G to real β G = 12, at Higgs mass M ∗ purpose H = 70 GeV and approximation for a volume 80 3 (Gürtler, Schiller, E.-M. I., 1997). Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 71 / 171 chemical

  73. The phase boundary at small chemical potential The Lee-Yang pattern locating the CEP at Lattice QCD at non-zero baryonic density µ � = 0 ? E.-M. Ilgenfritz 2 x36x4 lattice Introduction 24 0.01 Quantum statistics and the 0.01 0.008 QCD partition function 0.05 Chemical 0.006 0.1 potential on the β Im lattice 0.004 Phase quenching 0.5 can’t treat the complex-weight 0.002 problem The phase 0 boundary at 5.688 5.69 5.692 5.694 5.696 β Re small chemical potential Figure: Lee-Yang Zeroes in the complex β plane, in the case of Taylor expansion: a general pure SU ( 3 ) gauge theory (Ejiri 2006) (left) and the distance of purpose approximation the smallest Lee-Yang zero from the real axis as function of the Summary of chemical potential, in the case of full QCD (Fodor 2004) (right). results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 72 / 171 chemical

  74. The phase boundary at small chemical potential Fixing the critical endpoint CEP Lattice QCD at non-zero baryonic density ? E.-M. Ilgenfritz This approach has led to a determination of the location of the critical endpoint for realistic quark masses Introduction (Fodor 2004): Quantum (notice that quark number chemical potential µ q = µ B / 3) statistics and the QCD partition function Chemical µ q E = 120 ( 13 ) MeV, T E = 162 ( 2 ) MeV, whereas potential on the lattice T c ( µ q = 0 ) = 164 ( 3 ) MeV (see next Figure). Phase quenching can’t treat the complex-weight An earlier analysis with 3 × bigger quark masses and problem 3 × smaller volume (resulting in much heavier baryons !) The phase boundary at had given (Fodor 2003): (with twice as high µ q small chemical E !) potential Taylor expansion: µ q a general E = 241 ( 31 ) MeV, T E = 160 ( 4 ) MeV, whereas purpose T c ( µ q = 0 ) = 172 ( 3 ) MeV. approximation Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 73 / 171 chemical

  75. The phase boundary at small chemical potential Status of the Multiparameter Reweighting Lattice QCD at non-zero baryonic density strategy ? E.-M. Ilgenfritz Introduction Quantum The final multiparameter reweighting result was obtained statistics and the QCD partition using N f = 2 + 1 quark flavors with physical quark function masses on a coarse lattice with only N τ = 4 time slices. Chemical potential on the a ≈ 1 / ( 4 × 160 MeV ) ≈ 0 . 25 fm . lattice Phase quenching can’t treat the Unfortunately, this method is very expensive to extend complex-weight problem to smaller lattice spacing (larger N τ ) and it has not been The phase repeated attempting to approach the continuum limit. boundary at small chemical potential A critical analysis has been presented by Splittorff (2006). Taylor expansion: a general purpose approximation Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 74 / 171 chemical

  76. The phase boundary at small chemical potential Critical endpoint ( µ E , T E ) Lattice QCD at non-zero baryonic density ? E.-M. Ilgenfritz Introduction Quantum statistics and the QCD partition function Chemical potential on the lattice Phase quenching can’t treat the complex-weight problem The phase boundary at small chemical potential Taylor expansion: Figure: Left: Location of the critical endpoint for N f = 2 + 1 a general purpose using multi-parameter/overlap preserving reweighting, on a approximation lattice with N τ = 4 (Fodor 2004). Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 75 / 171 chemical

  77. The phase boundary at small chemical potential The use of multiparameter reweighting Lattice QCD at non-zero baryonic density ? E.-M. Ilgenfritz the whole story 0.003 courtesy Z. Fodor Introduction 0.002 Quantum Im β 0 statistics and the 0.001 QCD partition 0 function -0.001 Chemical potential on the 0 0.05 0.1 0.15 0.2 lattice µ Phase quenching can’t treat the Figure: Left : QCD phase diagram from Fodor (2004) obtained complex-weight problem by combined reweighting in µ and β of the µ = 0 , β = β c reference ensemble (blue dot). Right : improved data illustrating The phase boundary at the insensitivity of Im β LY relative to µ , followed by an abrupt small chemical potential change. Taylor expansion: a general purpose approximation Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 76 / 171 chemical

  78. The phase boundary at small chemical potential Height lines of the average phase factor in Lattice QCD at non-zero baryonic density the matrix model of Han and Stephanov ? E.-M. Ilgenfritz 2 2 0.8 0.8 m � 0.07 m � 0 Introduction 1.75 1.75 1.5 0.6 1.5 0.6 Quantum 1.25 1.25 statistics and the 0.4 0.4 T 1 T 1 QCD partition 2nd order 0.75 0.75 TCP CP 0.2 function 0.2 0.5 0 0.5 1st order 0 1st order Chemical 0.25 0.25 potential on the 0.25 0.5 0.75 1 1.25 1.5 1.75 2 0.25 0.5 0.75 1 1.25 1.5 1.75 2 lattice Μ Μ Phase quenching Figure: Height lines of the average sign in the µ – T plane for can’t treat the complex-weight the random matrix model of Han and Stephanov (2008) problem designed to describe the transition and the sign problem The phase (2008). Left: Contours of average phase factor for m = 0 . 07 boundary at small chemical with first-order line and critical endpoint CEP . Right: Contours potential of average phase factor for the chiral limit m = 0. First-order Taylor expansion: a general line, chiral symmetry second-order transition line and tricritical purpose point TCP are shown. approximation Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 77 / 171 chemical

  79. The phase boundary at small chemical potential The matrix model of Han and Stephanov for Lattice QCD at non-zero baryonic density dense QCD (arXiv:0805.1939) ? E.-M. Ilgenfritz Introduction � Quantum D X e − N Tr XX † det N f D = � det N f D � X , statistics and the Z N f = QCD partition function Chemical where D is the 2 N × 2 N matrix approximating the Dirac potential on the lattice operator: � � Phase quenching m iX + C can’t treat the D = , complex-weight iX † + C problem m The phase with boundary at � � small chemical 1 1 N / 2 0 potential C = µ 1 1 N + iT . Taylor expansion: 0 − 1 1 N / 2 a general purpose approximation Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 78 / 171 chemical

  80. The phase boundary at small chemical potential The scenarios predicted by the model of Han Lattice QCD at non-zero baryonic density and Stephanov ? E.-M. Ilgenfritz Introduction Quantum Two scenarios are predicted by the model: statistics and the QCD partition function first order line with critical endpoint for quark mass Chemical away from chiral limit potential on the lattice first order line separated from second order line Phase quenching (extending to µ = 0) for the chiral limit (zero quark can’t treat the complex-weight mass) problem The phase In each case, the height line of R = 0 keeps the phase boundary at small chemical transitions separated (not accessible by extrapolation) potential from the rest of the µ – T plane. Taylor expansion: a general purpose approximation Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 79 / 171 chemical

  81. The phase boundary at small chemical potential The average phase factor in the matrix model Lattice QCD at non-zero baryonic density of Stephanov ? E.-M. Ilgenfritz The leading exponential behavior of the partition functions Introduction Z 1 + 1 and Z 1 + 1 ∗ is the same, it will cancel in the ratio R . Quantum Taking into account preexponential factors (determined statistics and the QCD partition by the second order derivatives of the potential function function Ω 1 + 1 ( A ) and Ω 1 + 1 ∗ ( A ) with respect to all elements of Chemical potential on the flavor matrix A ) : lattice Phase quenching � � 2 π � 4 � can’t treat the � − 1 � 2 e − N Ω Q ( A ) N →∞ ′′ complex-weight � Z Q → det Ω , problem Q � N � A = A saddle The phase boundary at small chemical where Q indicates the respective quark content of the potential theory, 1 + 1 or 1 + 1 ∗ , and Taylor expansion: � ∂ 2 Ω Q a general � purpose ′′ det Ω Q ≡ det . approximation ∂ A α ∂ A β Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 80 / 171 chemical

  82. The phase boundary at small chemical potential The average phase factor in the matrix model Lattice QCD at non-zero baryonic density of Stephanov ? E.-M. Ilgenfritz Introduction Quantum statistics and the QCD partition Finally, the average phase factor is given by function Chemical � − 1 potential on the � � ′′ = u 2 − v 2 2 det Ω � lattice R ≡ � e 2 i θ � 1 + 1 ∗ = Z 1 + 1 1 + 1 � Z 1 + 1 ∗ = Phase quenching x 2 − y 2 � ′′ det Ω can’t treat the � 1 + 1 ∗ A = A saddle complex-weight problem with u , v , x and y depending on m , T , µ and the The phase boundary at saddlepoint equation. small chemical potential Taylor expansion: a general purpose approximation Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 81 / 171 chemical

  83. The phase boundary at small chemical potential A more intuitive "overlap measure": α Lattice QCD at non-zero baryonic density ? α is defined as the fraction of sampled configurations E.-M. Ilgenfritz that contributes the biggest contributions (amounting Introduction to a fraction 1 − α ) to the average sign (total weight Quantum contributed to the target ensemble). statistics and the QCD partition The reweighting step should not be too small and not function too big ! Therefore the optimal overlap is α = 50 percent. Chemical potential on the The height lines of the overlap measure α in the β – µ lattice plane show clearly, where one can rely on reweighting. Phase quenching can’t treat the The grey area is not accessible by reweighting from the complex-weight problem reference point located at β = β c ( µ = 0 ) at µ = 0. The phase The ridge of the susceptibility (usually locating the boundary at small chemical crossover line) falls on top of the ridge of the overlap potential measure α . Taylor expansion: a general The half width in µ of the ridge, µ 1 / 2 , defined by α = 0 . 5, purpose shrinks with increasing volume like µ 1 / 2 ∼ V − γ with approximation Summary of γ ≈ 1 / 3. results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 82 / 171 chemical

  84. The phase boundary at small chemical potential Height lines of the overlap measure α Lattice QCD at non-zero baryonic density ? E.-M. Ilgenfritz 6 8 10 Introduction 12 Quantum statistics and the QCD partition function Chemical potential on the lattice Phase quenching can’t treat the complex-weight problem Figure: (a) The left panel shows the real µ – β plane. 33000 configurations were simulated at the The phase boundary at parameter set: β = 5 . 274, m u , d = 0 . 096, m s = 2 . 08 m u , d on a 4 · 8 3 size lattice. This is β c ( µ = 0 ) in small chemical the N f = 2 + 1 case. The dotted lines are contours of constant overlap. The dotted area is the unknown potential territory where the overlap vanishes. The solid line is the phase transition/crossover line determined by the peaks of susceptibility. (b) In the right panel the volume and the µ dependence of the overlap α is Taylor expansion: shown. Upper curves correspond to smaller lattice sizes, 4 · 6 3 , 4 · 8 3 , 4 · 10 3 and 4 · 12 3 respectively. a general purpose The half width µ 1 / 2 scales as indicated. approximation Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 83 / 171 chemical

  85. The phase boundary at small chemical potential Phase boundary obtained by different Lattice QCD at non-zero baryonic density methods ? E.-M. Ilgenfritz a µ 0 0.1 0.2 0.3 0.4 0.5 5.06 Introduction 1.0 5.04 QGP <sign> ~ 0.85(1) Quantum 5.02 statistics and the <sign> ~ 0.45(5) 0.95 5 QCD partition function 4.98 <sign> ~ 0.1(1) 0.90 4.96 Chemical potential on the 4.94 T/T c 0.85 β lattice confined 4.92 Phase quenching 4.9 0.80 can’t treat the 4.88 3 imaginary µ D’Elia, Lombardo 16 complex-weight 3 2 param. imag. µ Azcoiti et al., 8 4.86 problem 3 dble reweighting, LY zeros Fodor, Katz, 6 0.75 4.84 3 Same, susceptibilities Our reweighting, 6 3 The phase canonical deForcrand, Kratochvila, 6 4.82 boundary at 0.70 4.8 small chemical 0 0.5 1 1.5 2 potential µ /T Taylor expansion: Figure: Pseudo-critical temperature determined by various a general purpose approaches for the same lattice theory (4-flavor staggered approximation quarks with mass am = 0 . 05 on an N t = 4 lattice) (Kratochvila Summary of 2005). All approaches agree among each other for µ/ T � 1. results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 84 / 171 chemical

  86. Taylor expansion: a general purpose approximation Outline Lattice QCD at non-zero baryonic density ? Introduction 1 E.-M. Ilgenfritz Quantum statistics and the QCD partition function 2 Introduction Chemical potential on the lattice 3 Quantum statistics and the QCD partition Phase quenching can’t treat the complex-weight problem 4 function The phase boundary at small chemical potential Chemical 5 potential on the lattice Taylor expansion: a general purpose approximation 6 Phase quenching can’t treat the Summary of results up to now 7 complex-weight problem Imaginary chemical potential 8 The phase boundary at Complex Langevin dynamics 9 small chemical potential 10 Complex Langevin dynamics for gauge theories Taylor expansion: a general 11 Other approaches to avoid/cure the sign problem purpose approximation 12 Conclusion Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 85 / 171 chemical

  87. Taylor expansion: a general purpose approximation Taylor expansion of log det M Lattice QCD at non-zero baryonic density ? E.-M. Ilgenfritz An alternative, and more modest, idea relies on a Taylor Introduction series expansion of the logarithm of the determinant in Quantum statistics and the µ/ T around µ = 0. It applies to the full interior of the QCD partition function phase diagram. The coefficients of the expansion Chemical can be calculated using conventional simulations potential on the lattice at µ = 0, where the sign problem is fortunately absent. Phase quenching This approach is continuously pursued by several groups: can’t treat the complex-weight Allton (2002), Gavai (2004), Allton (2005), problem Kaczmarek (2011), Endrodi (2011), Borsanyi (2012). The phase boundary at small chemical potential A recent review can be found in S. Borsanyi, Taylor expansion: arXiv:1511.06541 (Lattice 2015) a general purpose approximation Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 86 / 171 chemical

  88. Taylor expansion: a general purpose approximation Taylor expansion of the pressure Lattice QCD at non-zero baryonic density ? E.-M. Ilgenfritz We start from the grand-canonical ensemble, considering Introduction the pressure, Quantum statistics and the QCD partition p ( T , µ ) = T function V ln Z . Chemical potential on the lattice Since the pressure is an even function of µ , one can write Phase quenching can’t treat the complex-weight ∆ p ( T , µ ) ≡ p ( T , µ ) − p ( T , 0 ) = µ 2 ∂ 2 p µ = 0 + µ 4 ∂ 4 p � � problem � � µ = 0 + . � � ∂µ 2 ∂µ 4 2 ! 4 ! The phase boundary at small chemical potential p ( T , µ = 0 ) is obtained from the "interaction measure" Taylor expansion: a.k.a. the "trace anomaly", evaluated at µ = 0. a general purpose approximation Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 87 / 171 chemical

  89. Taylor expansion: a general purpose approximation The pressure evaluated at µ = 0 Lattice QCD at non-zero baryonic density ? E.-M. Ilgenfritz The quantity I ( T ) = ǫ ( T , µ = 0 ) − 3 p ( T , µ = 0 ) is related to a total derivative w.r.t. T : Introduction Quantum statistics and the QCD partition I ( T ) T 5 = d p ( T , µ = 0 ) function . Chemical T 4 dT potential on the lattice The l.h.s. quantity is called "trace anomaly", alias Phase quenching can’t treat the "interaction measure". This relation can be integrated complex-weight problem giving the EoS at µ = 0 The phase boundary at small chemical � T potential dT ′ I ( T ′ ) p ( T , µ = 0 ) − p ( T 0 , µ = 0 ) = Taylor expansion: T 4 T 4 T ′ 5 a general T 0 0 purpose approximation Summary of results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 88 / 171 chemical

  90. Taylor expansion: a general purpose approximation Trace anomaly as (subtracted) lattice Lattice QCD at non-zero baryonic density expectation value ? E.-M. Ilgenfritz The integrand (trace anomaly) expresses the Introduction lattice-scale-dependence of the lattice action. Quantum statistics and the � d ln Z �� I ( T ) 1 QCD partition � T 4 = − function � T 3 V d ln a sub Chemical potential on the with action with different coupling parameters b i lattice � Phase quenching S = b i S i (3) can’t treat the complex-weight problem � ∂ S �� I ( T ) 1 db i � The phase � T 4 = boundary at � T 3 V ∂ b i da small chemical sub potential i Taylor expansion: with subtracted expectation values a general purpose � �� � �� � �� approximation � � � ... sub = ... finite T lattice − ... � � � Summary of T = 0 lattice results up to now Imaginary E.-M. Ilgenfritz (BLTP, JINR, Dubna) Lattice QCD at non-zero baryonic density ? 10/17 August 2016 89 / 171 chemical

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