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We summarise recent progress in simulating QCD at nonzero baryon density
using complex Langevin dynamics. After a brief outline of the main idea, we
discuss gauge cooling as a means to control the evolution. Subsequently we
present a status report for heavy dense QCD and its phase structure, full QCD
with staggered quarks, and full QCD with Wilson...
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... compact astrophysical objects, such as neutron stars, from first principles. Furthermore, QCD describes one of the fundamental forces in Nature and hence there is an intrinsic desire to understand it under extreme conditions, such that the usual QCD vacuum is replaced by new phases of matter. A possible sketch of the QCD phase diagram is given in Fig. ...
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These lecture notes contain an elementary introduction to lattice QCD at
nonzero chemical potential. Topics discussed include chemical potential in the
continuum and on the lattice; the sign, overlap and Silver Blaze problems; the
phase boundary at small chemical potential; imaginary chemical potential; and
complex Langevin dynamics. An incomplete...
Citations
... A sketch of the QCD phase diagram[5] ...
Jets, defined as collimated sprays of high-momentum particles, are experimental signatures of hard-scattered quarks and gluons produced in hadronic interactions. The jet production cross section is calculable within perturbative Quantum ChromoDynamics (pQCD), and therefore jet measurements provide stringent tests of pQCD predictions. In relativistic heavy-ion collisions, jets are well calibrated probes of the Quark-Gluon Plasma (QGP). Under extreme conditions of temperature and/or pressure, partons are deconfined and form a strongly interacting QCD medium. The initial hard scattered partons lose energy while traversing this medium due to radiative and collisional energy loss. Consequently, jet properties get modified in comparison with the vacuum case, phenomenon named jet quenching. QGP transport properties can be studied by measuring jet quenching.The charged jet production cross sections in pp collisions at and were measured by the ALICE experiment and compared to Leading-Order (LO) pQCD predictions. In Pb-Pb collisions, the strength of jet suppression was quantitatively assessed at and via the measurement of the nuclear modification factors (). The strength of charged jet suppression was quantified as a function of in-medium parton path-length based on the measured . The jet elliptic flow , defined as the jet azimuthal distribution relative to the order event plane, which is sensitive to the difference of the in-medium parton path-length in-plane and out-of-plane, was measured at . The measured jet in mid-central Pb-Pb collisions was consistent with model predictions. The medium response has been studied through jet-track correlations at as a function of centrality. The result suggested that the in-medium suppressed energy was re-distributed to large angles with respect to the jet axis. The phenomenon was described by a phenomenological calculation taking into account hydrodynamical evolution of the medium.In this thesis, two complementary aspects of jet measurements with the ALICE detector at the LHC were studied. First, the upgrade of the ALICE electromagnetic calorimeter trigger system is presented. The Di-jet Calorimeter (DCal) has been installed during LHC Long Shutdown 1 (LS1) to extend the azimuthal coverage of the existing ElectroMagnetic Calorimeter (EMCal) and PHOton Spectrometer (PHOS). The trigger system has been upgraded to account for this new detector configuration. The firmware for the Summary Trigger Unit (STU), which is the electronics of the trigger system, was upgraded to implement a brand new algorithm combining information from the calorimeters. Second, the measurement of the production cross section of charged jets reconstructed with cone resolution parameter R=0.2, 0.3, 0.4, and 0.6 in pp collisions at is outlined. A comparison of the production cross section to LO and Next-Leading-Order (NLO) pQCD predictions is shown. Good agreement of the production cross section with NLO pQCD calculations is found for . The measurement of charged jet in mid-central (30-50%) Pb-Pb collisions at is also presented. The results are compared with a toy-model Glauber simulation based on the measured path-length dependence of jet suppression. Finally, the measurement of charged jet-hadron correlations in mid-central (30-50%) Pb-Pb collisions at with respect to the order event plane is also presented in order to study initial collision geometry dependence of jet modification in Pb-Pb collisions.
... It is possible to extend lattice-QCD predictions to small values of µ/T using a Taylor expansion on the T -axis [18, 19], and so the lattice can probe both the T -axis and slightly off of it at high T , but the rest of the phase plane remains inaccessible to direct simulation because of this sign problem. There are many proposals to extend direct simulations to non-zero density, including Lefschetz thimbles [20], complex Langevin [21, 22, 23, 24], and strong coupling expansions [25]. For an introductory survey of non-zero density lattice-QCD ideas, see Ref. [26]. ...
The equation of state (EoS) of quantum chromodynamics (QCD) at zero temperature can be calculated in two different perturbative regimes: for small values of the baryon chemical potential , one may use chiral perturbation theory (ChEFT); and for large values of , one may use perturbative QCD (pQCD). There is, however, a gap for , where these theories becomes non-perturbative, and where there is currently no known microscopic description of QCD matter. Unfortunately, this interval obscures the values of found within the cores of neutron stars (NSs). In this thesis, we argue that thermodynamic matching of the ChEFT and pQCD EoSs is a legitimate way to obtain quantitative constraints on the non-pertubative QCD EoS. Moreover, we argue that this method is effective, verifiable, and systematically improvable. First, we carry out a simplified matching procedure in QCD-like theories that can be simulated on the lattice without a sign problem. Our calculated pressure band serves as a prediction for lattice-QCD practitioners and will allow them to verify or refute the simplified procedure. Second, we apply the state-of-the-art matched EoS of Kurkela et al. (2014) to rotating NSs. This allows us to obtain bounds on observable NS properties, as well as point towards future observations that would more tightly constrain the current state-of-the-art EoS band. Finally, as evidence of the ability to improve the procedure, we carry out calculations in pQCD to improve the zero-temperature pressure. We calculate the full contribution to the pQCD pressure for massless quarks, as well as a significant portion of the piece and even some of the piece.
... from which the quarks have been integrated out leaving us with the quark determinant. In the so-called heavy-dense limit for large quark mass m and simultaneously large chemical potential µ, the quark determinant factorises into [12][13][14][15][16]: ...
During the last 40 years, Monte Carlo calculations based upon Importance Sampling have matured into the most widely employed method for determinig first principle results in QCD. Nevertheless, Importance Sampling leads to spectacular failures in situations in which certain rare configurations play a non-secondary role as it is the case for Yang-Mills theories near a first order phase transition or quantum field theories at finite matter density when studied with the re-weighting method. The density-of-states method in its LLR formulation has the potential to solve such overlap or sign problems by means of an exponential error suppression. We here introduce the LLR approach and its generalisation to complex action systems. Applications include U(1), SU(2) and SU(3) gauge theories as well as the Z3 spin model at finite densities and heavy-dense QCD.
... While established techniques fail, several recent techniques have been studied that at least in principle could permit one to calculate thermodynamic properties from first principles in QCD at large density. These techniques include Lefschetz thimbles [4], complex Langevin [5][6][7][8], strong coupling expansion [9], and hadron resonance gas plus perturbative QCD ("HRG+pQCD" in the following) [10][11][12]. In this work, we propose a series of 'control studies' in QCD-like theories (in particular two-color QCD with two fundamental flavors and four-color QCD with two flavors in the two-index, antisymmetric representation), which-despite not corresponding to the actual theory of strong interactions realized in nature-have the advantage of not suffering from a sign problem, and are thus amenable to direct simulations using established lattice-QCD techniques. ...
We calculate the equation of state at non-zero temperature and density from
first principles in two-, three- and four-color QCD with two fermion flavors in
the fundamental and two-index, antisymmetric representation. By matching
low-energy results (from a `hadron resonance gas') to high-energy results from
(resummed) perturbative QCD, we obtain results for the pressure and trace
anomaly that are in quantitative agreement with full lattice-QCD studies for
three colors at zero chemical potential. Our results for non-zero chemical
potential at zero temperature constitute predictions for the equation of state
in QCD-like theories that can be tested by traditional lattice studies for
two-color QCD with two fundamental fermions and four-color QCD with two
two-index, antisymmetric fermions. We find that the speed of sound squared at
zero temperature can exceed one third, which may be relevant for the
phenomenology of high-mass neutron stars.
Complex Langevin simulations allow numerical studies of theories that exhibit a sign problem, such as QCD, and are thereby potentially suitable to determine the QCD phase diagram from first principles. Here we study QCD in the limit of heavy quarks for a wide range of temperatures and chemical potentials. Our results include an analysis of the adaptive gauge cooling technique, which prevents large excursions into the non-compact directions of the SL() manifold. We find that such excursions may appear spontaneously and change the statistical distribution of physical observables, which leads to disagreement with known results. Results whose excursions are sufficiently small are used to map the boundary line between confined and deconfined quark phases.
We consider the one-dimensional massive Thirring model formulated on the
lattice with staggered fermions and an auxiliary compact vector (link) field,
which is exactly solvable and shows a phase transition with increasing the
chemical potential of fermion number: the crossover at a finite temperature and
the first order transition at zero temperature. We complexify its
path-integration on Lefschetz thimbles and examine its phase transition by
hybrid Monte Carlo simulations on the single dominant thimble. We observe a
discrepancy between the numerical and exact results in the crossover region for
small inverse coupling and/or large lattice size L, while they are in
good agreement at the lower and higher density regions. We also observe that
the discrepancy persists in the continuum limit keeping the temperature finite
and it becomes more significant toward the low-temperature limit. This
numerical result is consistent with our analytical study of the model's thimble
structure. And these results imply that the contributions of subdominant
thimbles should be summed up in order to reproduce the first order transition
in the low-temperature limit.
We investigate Lefschetz thimble structure of the complexified
path-integration in the one-dimensional lattice massive Thirring model with
finite chemical potential. The lattice model is formulated with staggered
fermions and a compact auxiliary vector boson (a link field), and the whole set
of the critical points (the complex saddle points) are sorted out, where each
critical point turns out to be in a one-to-one correspondence with a singular
point of the effective action (or a zero point of the fermion determinant). For
a subset of critical point solutions in the uniform-field subspace, we examine
the upward and downward cycles and the Stokes phenomenon with varying the
chemical potential, and we identify the intersection numbers to determine the
thimbles contributing to the path-integration of the partition function. We
show that the original integration path becomes equivalent to a single
Lefschetz thimble at small and large chemical potentials, while in the
crossover region multi thimbles must contribute to the path integration.
Finally, reducing the model to a uniform field space, we study the relative
importance of multiple thimble contributions and their behavior toward
continuum and low-temperature limits quantitatively, and see how the rapid
crossover behavior is recovered by adding the multi thimble contributions at
low temperatures. Those findings will be useful for performing Monte-Carlo
simulations on the Lefschetz thimbles.
For several crucial microseconds of its early history, the Universe consisted
of a Quark-Gluon Plasma. As it cooled during this era, it traced out a
trajectory in the quark matter phase diagram. The form taken by this trajectory
is not known with certainty, but is of great importance: it determines, for
example, whether the cosmic plasma passed through a first-order phase change
during the transition to the hadron era, as has recently been suggested by
advocates of the "Little Inflation" model. Just before this transition, the
plasma was strongly coupled and therefore can be studied by holographic
techniques. We show that holography imposes a strong constraint (taking the
form of a bound on the baryonic chemical potential relative to the temperature)
on the domain through which the cosmic plasma could pass as it cooled, with
important consequences for Little Inflation. In fact, we find that holography
applied to Little Inflation implies that the cosmic plasma must have passed
quite close to the quark matter critical point, and might therefore have been
affected by the associated fluctuation phenomena.
