Lattice QCD at finite temperature and density Massimo D’Elia Genoa University & INFN GGI - April 23, 2008
1 – OUTLINE • QCD phenomenology at finite temperature and baryon density, lat- tice simulations and the sign problem. • Existing methods to partially circumvent the sign problem. • Some lattice results in QCD and QCD-like theories.
QCD at finite density and the sign problem Perturbative Regime Deconfined phase Chiral Symmetry Restored Axial U(1) effectively restored T sQGP ? T c nature of the critical line? T ? E st 1 order Non−Perturbative Regime QCD Color superconductivity Confinement Perturbative Regime vacuum state Chiral Symmetry Breaking deconfined quark matter ? Axial U(1) broken µ µ C The study of QCD at finite temperature and baryon density is relevant to fundamental phenomenological issues: 1) experimental search for the deconfinement transition in heavy ion collisions; 2) properties of compact astrophysical objects. Several questions need to be clarified: location and nature of the transition lines; properties of strongly interacting matter, specially close to the transition.
QCD at finite density and the sign problem Perturbative Regime Deconfined phase Chiral Symmetry Restored Axial U(1) effectively restored T sQGP ? T c nature of the critical line? T ? E st 1 order Non−Perturbative Regime QCD Color superconductivity Confinement Perturbative Regime vacuum state Chiral Symmetry Breaking deconfined quark matter ? Axial U(1) broken µ µ C As a matter of fact, our knowledge of the QCD phase diagram and of the dynamics of strongly interacting matter at finite temperature and baryon density is still partial. Lattice QCD simulations, which are the best non-perturbative computational tool based only on QCD first principles, are hindered at finite baryonic density by the sign problem.
2 – QCD at finite temperature and density H QCD � � e − The finite temperature QCD partition function, Z ( V, T ) = Tr , can be T written as a functional integral over euclidean space-time with finite temporal extent τ = 1 /T � R 1 /T D A D ψ D ¯ d 3 x L QCD R ψe − dt Z = 0 1 µν + ¯ 4 g 2 G a µν G a γ E µ ( ∂ µ + iA a µ T a ) + m � � L QCD = ψ ψ Periodic boundary condition in time direction are taken for gauge fields, antiperiodic for fermionic fields.
The partition function can be written on a discretized space-time (lattice): � � ψe − ( S G + S F ) = D Ue − S G det M [ U ] D U D ψ D ¯ Z = R x +ˆ “ ” µ U µ ( x ) = P exp i dy ν A ν ( y ) is the gauge link variable x “ ” is the pure gauge action, β = 2 N c 1 S G = β P 1 − N c TrΠ µν ( x ) x,µ<ν g 2 µ ) U † ν ) U † Π µν ( x ) = U µ ( x ) U ν ( x + ˆ µ ( x + ˆ ν ( x ) is the plaquette variable S F = ¯ ψ i M ij ψ j is the fermionic action with M the fermionic matrix, na¨ ıvely h i S f = 1 x,µ ¯ µ ) − U † x m ¯ ψ ( x ) γ E P U µ ( x ) ψ ( x + ˆ µ ( x − ˆ µ ) ψ ( x − ˆ µ ) + P ψ ( x ) ψ ( x ) µ 2 Dynamical fermion contributions are encoded in the fermion determinant det M [ U ] which appears after integration of fermion variables. T = 1 1 τ = N t a ( β, m ) a → 0 as β → ∞ , therefore T is a monotonic increasing function of β .
The thermal expectation value of a generic operator O is written as DU det M [ U ] e − S g [ U ] O [ U ] � � O � = (1) � DU det M [ U ] e − S g [ U ] if det M [ U ] e − S g [ U ] > 0 this has a probabilistic interpretation and Monte Carlo meth- ods can be applied to numerically determine it: only few gauge field configurations give sensible contribution to the functional integral and one looks for a good algo- rithm to sample them (importance sampling). The reality of the fermion determinant is guaranteed by γ 5 hermiticity ⇒ (det M ) ∗ = det M γ 5 M † γ 5 = M =
A finite baryonic density can be introduced by adding a finite chemical potential H QCD − µN � � e − Z ( µ ) = Tr T d 3 x ¯ d 3 xψ † ψ = � � where N = ψγ 0 ψ is the quark (baryonic) number operator. In the euclidean path integral formulation the fermionic part of L QCD is modified as follows: ψ ( γ µ ( ∂ µ + iA µ ) + m ) ψ → ¯ ¯ ψ ( γ µ ( ∂ µ + iA µ ) + m + µγ 0 ) ψ The added baryon chemical potential can be viewed as the temporal component of a constant U (1) imaginary background field. This is also the way it is usually imple- mented on the lattice to avoid perturbative divergences ( P. Hasenfratz F. Karsch, Phys. Lett. B125 (1983) 308 J.B. Kogut et al., Nucl. Phys. B225 (1983) 93 R. V. Gavai Phys. Rev. D32 (1985) 519): t − U † t − e − aµ U † t ( n − ˆ → e aµ U t ( n ) δ m,n +ˆ t ( n − ˆ D t ( n, m ) = U t ( n ) δ m,n +ˆ t ) δ m,n − ˆ t ) δ m,n − ˆ t t Where U t is the temporal link (elementary parallel transport)
In this way the hermiticity properties of the fermion matrix in general are lost and the residual surviving symmetry is (det M ( µ )) ∗ = det M ( − µ ) det M is in general complex = ⇒ Monte Carlo simulations are unfeasibile. This is usually known as the sign problem. It is a problem of technical nature but is strictly related to the fact that we want to create a net unbalance between parti- cles and antiparticles, i.e. suppress antiquark and enhance quark propagation.The problem is known also in other contexts dealing with fermionic systems.
Consider for instance the expectation value of the Polyakov loop and of its hermitian conjugate: • It is known that L = � Tr P � and ¯ L = � Tr P † � are both real at real chemical poten- tial, but L � = ¯ L , and in particular L ( µ ) = ¯ L ( − µ ) (see e.g. F. Karsch and H. W. Wyld, 1985). This is indeed what we expect since L and ¯ L describe propagation of a static quark (antiquark) respectively, so their values must be different in presence of a baryon chemical potential. • However, configuration by configuration Tr P and Tr P † are always the complex conjugate of each other, the fact that Re � Tr P � µ � = Re � Tr P † � µ is strictly related to the complex nature of the “measure” which is meant �·� µ A complex measure is necessary in order to have L � = ¯ L . Notice that the partition function is, of course, still real: different complex contribu- tions cancels out. It could be possible in principle to find a smart change of variables to perform the sum avoiding the sign problem. If it exists, it has not yet been found, so we have to deal with partial solutions to the problem.
3 – Trying to evade the sign problem The importance of studying the QCD phase diagram from first principles lattice gauge theory simulations has been worth a lot of efforts in trying to evade the sign problem. Most methods try to extract information from simulations where the sign problem is absent (like at µ = 0 ) and some results have been obtained in the last few years, mostly in the region of small µ and high T , which is the one relevant for heavy ion collisions. As an alternative one can study sign problem free theories, like two color QCD (real sign) or QCD in presence of a finite isospin density. We will review these methods, then focussing on results obtained via the imaginary chemical potential approach and on results regarding the theory within two-color QCD.
Theories without the sign problem QCD at finite isospin density (det M ( µ )) ∗ = det M ( − µ ) , that means that a theory with a finite isospin density µ is (i.e. µ u = − µ d = µ is / 2 ) has no sign problem (assuming m u = m d ) det M ( µ ) det M ( − µ ) = det M ( µ )(det M ( µ )) ∗ = | det M ( µ ) | 2 > 0 This is like ignoring the determinant phase, det M ( µ ) = | det M ( µ ) | e iθ (phase quenched QCD). QCD with two colours N c = 2 is a special case of gauge theory: the fundamental representation for quarks is real ( σ 2 U µ σ 2 = U ∗ µ ). = ⇒ Traces over closed loops are always real = ⇒ det M [ U ] , being expressible in terms of traces over closed loops, is real. Such theories are interesting laboratories for exploring some properties or methods, however one should always be aware that they are different physical systems (mesons ∼ baryons in 2-color QCD, meson matter instead of baryon matter in finite isospin QCD)
Reweighting Gauge configurations sampled at µ = 0 can in principle be used to obtain expecta- tion values at µ > 0 , using the following identity: � � det( M ( µ )) DU e − S g [ U ] det M (0) det M ( µ ) det( M (0)) O DU e − S g [ U ] det M ( µ ) O [ U ] � det M (0) O [ U ] � µ =0 � O � = = = DU e − S g [ U ] det M ( µ ) � DU e − S g [ U ] det M (0) det M ( µ ) � � � det( M ( µ )) det M (0) det( M (0)) µ =0 The method, proposed in the 80’s by the Glasgow group, fails. Major problems are: • Bad sampling: configurations sampled ad µ = 0 give poor sampling of the inte- gral at µ � = 0 . The two statistical ensembles at µ � = 0 and µ = 0 , being related to different physical situations, may have very poor overlap. The problem worsens as volume increases. • Large errors coming from the oscillating factor det M ( µ ) det M (0) (which is also not so eas- ily computable): � det M ( µ ) / det M (0) � ∼ 0 at large µ ’s, the statistics required for a given accuracy grow exponentially with the volume
Recommend
More recommend
