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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...
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... freedom, confinement, chiral symmetry breaking, phase transitions, etc. Because QCD is strongly interacting at low energies, many interesting questions are not easily answerable. The questions relevant for these lectures concern the QCD phase diagram, i.e. the phase structure of strongly interacting matter, of which a sketch is presented in Fig. 1. Several phases are shown: the hadronic phase at low temperature and density, the quark-gluon plasma at high temperature, possible colour-superconducting phases for cold dense matter, and perhaps there exist other phases not indicated in this version of the phase ...
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... centre symmetry is broken. Note that the perturbative vacuum corresponds to U 4 = 1 1 (A 4 = 0) and therefore P = 1. Hence centre symmetry is broken perturbatively and there are in fact N equivalent vacua. For N = 3, this is illustrated in Fig. 10 (z = 1, e ±2πi/3 ). Since perturbation theory is relevant at high temperature, we may already expect that at high temperature the centre symmetry is (spontaneously) broken. P Figure 10. Equivalent vacua in SU(3) gauge theory, in the case of broken centre symmetry Z 3 ...
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... perturbation theory is relevant at high temperature, we may already expect that at high temperature the centre symmetry is (spontaneously) broken. P Figure 10. Equivalent vacua in SU(3) gauge theory, in the case of broken centre symmetry Z 3 . ...
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... In the confined phase P = 0 and this observation is not relevant. However, in the deconfined phase, P = 0, and the direction of symmetry breaking changes. This is illustrated in Fig. 11 (left) by the symbols with the three little arrows. The remarkable consequence of this is that exactly at the boundaries, given by µ I /T = (2r + 1)π/N (r = 0, 1, 2, . . .), we find again a proper first-order phase transition, with the Polyakov loop as order parameter, even in the presence of quarks! To continue, let us now include the ...
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... this line increases quadratically with µ I , as indicated with the dotted line in Fig. 11 (left). It is natural to connect the thermal transition line with the vertical Roberge-Weiss line at µ I /T = π/3. The point where the lines meet is known as the Roberge-Weiss endpoint. For larger µ I , the phase structure is determined by the periodicity. To combine the findings for real and imaginary chemical potential in one ...
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... line with the vertical Roberge-Weiss line at µ I /T = π/3. The point where the lines meet is known as the Roberge-Weiss endpoint. For larger µ I , the phase structure is determined by the periodicity. To combine the findings for real and imaginary chemical potential in one diagram, we show the resulting phase structure in the µ 2 − T plane in Fig. 11 (right). The thermal transition line decreases linearly in µ 2 around µ 2 ∼ 0 and connects to the Roberge-Weiss endpoint on the ...
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... heavy or light quarks: the first-order transition remains first order for all imaginary µ. At the Roberge-Weiss endpoint, three first-order lines come together, making it triple point. This is illustrated in Fig. 12 (left); • quarks with intermediate mass: the crossover at µ 2 = 0 turns into a first-order transition at some value of µ I and possibly also at some value of real µ. The point(s) where this occurs are second-order critical endpoints (CEP). The Roberge-Weiss point is still a triple point, see Fig. 12 (middle); • adapting the quark mass ...
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... making it triple point. This is illustrated in Fig. 12 (left); • quarks with intermediate mass: the crossover at µ 2 = 0 turns into a first-order transition at some value of µ I and possibly also at some value of real µ. The point(s) where this occurs are second-order critical endpoints (CEP). The Roberge-Weiss point is still a triple point, see Fig. 12 (middle); • adapting the quark mass even more: the CEP at imaginary µ coincides with the Roberge- Weiss point. The transition is a crossover for all values of µ I . There might also still be a CEP for real µ, see Fig. 12 ...
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... The point(s) where this occurs are second-order critical endpoints (CEP). The Roberge-Weiss point is still a triple point, see Fig. 12 (middle); • adapting the quark mass even more: the CEP at imaginary µ coincides with the Roberge- Weiss point. The transition is a crossover for all values of µ I . There might also still be a CEP for real µ, see Fig. 12 ...
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... first-order lines come together (for heavy and light quarks), or a second-order critical endpoint (for intermediate masses). Note that the temperature of the Roberge-Weiss endpoint depends on the quark mass as well: it increases with quark mass, just as the critical temperature at µ = 0 increases with quark mass. This leads to the result shown in Fig. 13 (left): the critical temperature T RW as a function of the quark mass, for N f = 3. Since the Roberge-Weiss point is second order for intermediate quark masses and first order for larger and smaller masses, there are two tricritical points on this diagram, namely where the first and second-order lines ...
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... Recall that at µ = 0 this plot indicates the order (first, second or crossover) of the thermal transition. First let us consider the case of µ I /T = π/3. As argued above, the transition takes place at T RW and it is either first order or second order, depending on the quark masses. Hence a conjectured Columbia plot at µ I /T = π/3 is as shown in Fig. 13 (right). We remind the reader that the entire plot is critical and that the boundaries where the first-and second-order transitions meet are tricritial. This should be compared with the Columbia plot at µ = 0, where the central region indicates a crossover and the boundaries are second-order lines. Note that Fig. 13 (left) is the N f = 3 ...
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... at µ I /T = π/3 is as shown in Fig. 13 (right). We remind the reader that the entire plot is critical and that the boundaries where the first-and second-order transitions meet are tricritial. This should be compared with the Columbia plot at µ = 0, where the central region indicates a crossover and the boundaries are second-order lines. Note that Fig. 13 (left) is the N f = 3 (diagonal) cut through the plot on the right. The tricritical lines can be determined numerically by varying the quark masses, since there is no sign problem. This amounts to a detailed study of the properties of the Roberge-Weiss endpoint [45,46]. Finally, we can properly extend the Columbia plot with the chemical ...
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... the lower boundary comes from the Roberge-Weiss periodicity. The result is shown in Fig. 14 (left). The red fishnets indicate second-order surfaces, inside of which the transition is a crossover, while near the m q = 0 and m q → ∞ axes, the transition is first order. The plane µ = 0 is the original Columbia plot, while the plane (µ/T ) 2 = −(π/3) 2 was already shown in Fig. 13 (right). By considering degenerate quark masses (N f = ...
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... from the Roberge-Weiss periodicity. The result is shown in Fig. 14 (left). The red fishnets indicate second-order surfaces, inside of which the transition is a crossover, while near the m q = 0 and m q → ∞ axes, the transition is first order. The plane µ = 0 is the original Columbia plot, while the plane (µ/T ) 2 = −(π/3) 2 was already shown in Fig. 13 (right). By considering degenerate quark masses (N f = 3), we get the cut through the three-dimensional Columbia plot as shown in Fig. 14 (right). All features of this plot should now be familiar. In all known cases, it appears that the first-order regions shrink as (µ/T ) 2 is increased. This can be made very precise for heavy quarks ...
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... the transition is a crossover, while near the m q = 0 and m q → ∞ axes, the transition is first order. The plane µ = 0 is the original Columbia plot, while the plane (µ/T ) 2 = −(π/3) 2 was already shown in Fig. 13 (right). By considering degenerate quark masses (N f = 3), we get the cut through the three-dimensional Columbia plot as shown in Fig. 14 (right). All features of this plot should now be familiar. In all known cases, it appears that the first-order regions shrink as (µ/T ) 2 is increased. This can be made very precise for heavy quarks [46,51,52]. To do this, let us take the blue/dashed line on the right-hand side of indicates the boundary between the crossover and first-order ...
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... works in practice, i.e. how far the scaling region extends away from x * , has been tested in models where the sign problem is milder than in full QCD, namely in the three-state Potts model [51], an effective model for QCD with heavy quarks, and in QCD in a combined strong coupling and hopping parameter expansion [52]. The results are shown in Fig. 16. In both models the sign problem is sufficiently mild such that simulations for real µ are possible (in the Potts model the sign problem can be eliminated completely via a reformulation [53] and the results for QCD actually come from semi-analytical considerations). This allows us to see that tricritical scaling works extremely well: ...
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... the three-dimensional Columbia plot the regions of first-order transitions were shown to shrink as (µ/T ) 2 increases. The astute reader may wonder what this implies for the critical endpoint at real chemical potential, discussed in Sec. 6. Here several scenarios are possible, illustrated in Fig. 17. Note that the position of physical quark masses is indicated with the vertical blue line. The standard scenario is sketched on the left: the surface bends away from the m q = 0 axis and the critical endpoint is located at the intersection of the (red) surface and the (blue) line. If on the other hand the first-order region shrinks, as ...
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... on the other hand the first-order region shrinks, as is the case for heavy quarks, there is no critical endpoint related to the second-order surface (centre). Finally, it is possible that Figure 17. Possible scenarios for the curvature of the second-order surface for light quarks and the critical endpoint for physical quark masses [4]. ...
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... way from those obtained at µ = 0 or using the absolute value of the determinant. Given the excessive cancelation between configurations with 'positive' and 'negative' weight, one may wonder whether it is possible to give a sensible meaning to the notion of dominant configurations, e.g. by extending the configuration space, as illustrated in Fig. 18. Here we discuss the answer according to complex Langevin ...
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... i.e. by going into complex plane. The lesson is therefore to analytically continue ("complexify") the degrees of freedom, x → z = x + iy, which enlarges the configuration space and gives new directions to explore. In particular, it might be possible to find a real and positive distribution P (x, y), which is amenable to numerical approaches, see Fig. 18. In complex Langevin dynamics, it is proposed that this distribution is constructed as the solution of a stochastic process [55,56]. To motivate this, consider again the Gaussian integral (8.1). We note that the action satisfies S * (b) = S(−b * ) and we take a > 0 and real, such ...
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... is illustrated in Fig. 19 and hence x and y are not decoupled. The resulting probability distribution P (x, y) is therefore a proper two-dimensional, real and positive, distribution, as demonstrated in Fig. 20. The Langevin process finds this distribution, giving an explicit realisation of the sketch in Fig. 18. ...
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... is illustrated in Fig. 19 and hence x and y are not decoupled. The resulting probability distribution P (x, y) is therefore a proper two-dimensional, real and positive, distribution, as demonstrated in Fig. 20. The Langevin process finds this distribution, giving an explicit realisation of the sketch in Fig. 18. ...
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... the fermion determinant here). For instance, the original statement of unitarity, U U † = 1 1, is now replaced with U U −1 = 1 1, which still holds of course. Similarly, physical observables should be written as functions of U and U −1 , such that they are holomorphic. This is similar to the discussion for the Gaussian models above, see e.g. Eqs. (8.16, ...
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... where the second inequality is obvious and the first inequality follows from the polar decomposition of U ∈ SL(N, C): U = V P , with V ∈ SU(N ) and P a positive semidefinite hermitian matrix with det P = 1 [77]. Indeed, during a complex Langevin simulation these norms become nonzero, as demonstrated in Fig. 21 for QCD in the presence of static quarks for two values of the chemical potential µ. Intuition says that during a simulation the evolution should be controlled in the following way: configurations should stay close to the SU(N ) ...
Citations
... However, the matter formed in heavy-ion collisions in general involve a finite net baryon number. In the presence of a nonzero baryon chemical potential (µ B ), lattice QCD suffers from the fermion sign problem [27,28], which severely limits its applicability in studying the properties and phase structure of QCD matter at finite baryon density. ...
This thesis aims to elucidate the role of initial baryon stopping and its diffusion in heavy-ion collisions (HIC) using hydrodynamic model. In this regard, we have studied the observable-directed flow () of identified hadrons, particularly the of baryons and antibaryons, as well as the splitting observed between them in detail. We propose a new ansatz for the initial baryon distribution. By employing this initial baryon deposition model alongside a tilted energy distribution as inputs to a hybrid framework, we successfully describe the rapidity-odd of identified hadrons, including the elusive baryon-antibaryon splitting of across a wide range of . Our model, incorporating baryon stopping and it's subsequent diffusion within a relativistic hydrodynamic framework and employing a crossover equation of state derived from lattice QCD calculations, establishes a non-critical baryonic baseline. Moreover, we demonstrate that recent STAR measurements of the centrality and system-size dependence of splitting between oppositely charged hadrons-attributed to electromagnetic field effects-are significantly influenced by background contributions from baryon stopping and its diffusion. Furthermore, we show that the rapidity dependence of the splitting of the rapidity-even component of between p and is highly sensitive to the initial baryon deposition scheme. If measured experimentally, this could constraint the rapidity dependence of the initial baryon deposition profile. Moreover, it could offer valuable phenomenological insights into the baryon junction picture and help refine constraints on the baryon diffusion coefficient of the medium. Notably, utilizing this phenomenologically successful baryon deposition model, we present the first estimation of the baryon diffusion coefficient for the strongly interacting QCD matter created in HIC.
... First-principles lattice evaluations of the quantity fail in this regime due to the sign problem of lattice quantum chromodynamics (QCD); see Refs. [1][2][3][4][5] for reviews. Nevertheless, due to the recent observations of binary neutron-star (NS) merger events [6][7][8][9], as well as other astrophysical measurements of NSs [10][11][12][13][14][15][16][17][18][19][20][21], the investigation of the thermodynamic behavior of strongly interacting matter at large densities and low temperatures is an active area of research. ...
The equation of state of deconfined strongly interacting matter at high densities remains an open question, with effects from quark pairing in the preferred color-flavor-locked (CFL) ground state possibly playing an important role. Recent studies suggest that at least large pairing gaps in the CFL phase are incompatible with current astrophysical observations of neutron stars. At the same time, it has recently been shown that in two-flavor quark matter, subleading corrections from pairing effects can be much larger than would be na\"ively expected, even for comparatively small gaps. In the present Letter, we compute next-to-leading-order corrections to the pressure of quark matter in the CFL phase arising from the gap and the strong coupling constant, incorporating neutron-star equilibrium conditions and current state-of-the-art perturbative QCD results. We find that the corrections are again quite sizable, and they allow us to constrain the CFL gap in the quark energy spectrum to at a baryon chemical potential , even when allowing for a wide range of possible behaviors for the dependence of the gap on the chemical potential.
... To complement this work, a simultaneous theoretical effort is targeting QCD phase structure from first principles [8]; however, at present we lack rigorous and systematically improvable methods to probe much of µ B -T plane. On the numerical front, lattice QCD calculations at µ B ≳ T are obstructed by a sign problem [9][10][11][12]. While many approaches are under development to overcome sign problems , they cannot fully handle QCD yet. ...
A bstract
Lattice ℤ 3 theories with complex actions share many key features with finite- density QCD including a sign problem and CK symmetry. Complex ℤ 3 spin and gauge models exhibit a generalized Kramers-Wannier duality mapping them onto chiral ℤ 3 spin and gauge models, which are simulatable with standard lattice methods in large regions of parameter space. The Migdal-Kadanoff real-space renormalization group (RG) preserves this duality, and we use it to compute the approximate phase diagram of both spin and gauge ℤ 3 models in dimensions one through four. Chiral ℤ 3 spin models are known to exhibit a Devil’s Flower phase structure, with inhomogeneous phases that can be thought of as ℤ 3 analogues of chiral spirals. Out of the large class of models we study, we find that only chiral spin models and their duals have a Devil’s Flower structure with an infinite set of inhomogeneous phases, a result we attribute to Elitzur’s theorem. We also find that different forms of the Migdal-Kadanoff RG produce different numbers of phases, a violation of the expectation for universal behavior from a real-space RG. We discuss extensions of our work to ℤ N models, SU( N ) models and nonzero temperature.
... To elucidate the lattice QCD observations, the effective model analysis based on the chiral symmetry of the underlying QCD theory has also been employed, such as the Nambu-Jona-Lasinio (NJL) model [14][15][16] and the chiral perturbation theory [17]. On the other hand, turning our focus to finite dense systems related to neutron stars, the lattice QCD simulations are difficult to apply at large quark chemical potentials due to the sign problem [18,19]. Compared to finite temperature systems, our understanding of magnetic effects on the high-density matters is limited. ...
We discuss the effect of the quark anomalous magnetic moment (AMM) on the neutral dense quark matter under magnetic fields based on the Nambu–Jona–Lasinio (NJL) model at finite baryon density. To address its correlation with the chiral symmetry, we consider a simplified situation: the model includes the two-quark flavors under constant magnetic fields, and incorporates the effective interaction of the quark AMM linked to the spontaneous chiral symmetry breaking. It has been found that the magnetization is affected by the presence of the quark AMM, which can lead to alter the sign of the magnetization, particularly immediately after the phase transition with relatively large magnetic fields. We then examine the equation of state (EoS) in cases with and without magnetization for anatomizing the thermodynamic quantities. Without the magnetization, a small magnetic field stiffens the EoS, but with increasing the magnetic field, the EoS tends to soften. The stiffness of the EoS is found to be influenced by the magnetic effect on the critical chemical potential of the chiral phase transition and the quark number density at this critical point. As a result, the mass and radius of the neutron star composed of quark matter increase with the small magnetic field but turn to decrease as the magnetic field further increases. By including the quark AMM, the critical chemical potential is decreased and the quark number density takes a smaller value. Thus, for the stronger magnetic fields, the quark AMM suppresses the softening effect of the magnetic field on the EoS, leading to increased mass and radius compared to when the quark AMM is absent. In contrast, for the small magnetic field, the contribution of the quark AMM to the EoS is marginal. When the magnetization is taken into account, the magnetic effect on the stiffness of the EoS is overshadowed by the contribution of the magnetization. However, this overshadowing occurs regardless of whether the magnetization affected by the quark AMM takes negative or positive values. As a result, the effect of the quark AMM is not evident in the mass-radius relation.
... One of the most powerful tools to shed light on the QCD problem is the first-principles lattice QCD simulation. But lattice simulations with a chemical potential at lower temperature are not straightforward, due to the so-called sign problem of the Monte-Carlo computation [2,3]. Besides, considering the current difficulty of accelerator experiments, cold and dense QCD can be regarded as a frontier of quark-hadron physics. ...
... The functions χ π and χ η are evaluated within the present LSM by virtue of the matching condition (2). That is, ...
This review is devoted to summarizing recent developments of the linear sigma model (LSM) in cold and dense two-color QCD (QCD), in which lattice simulations are straightforwardly applicable thanks to the disappearance of the sign problem. In QCD, both theoretical and numerical studies derive the presence of the so-called baryon superfluid phase at sufficiently large chemical potential (), where diquark condensates govern the ground state. The hadron mass spectrum simulated in this phase shows that the mass of an iso-singlet (I=0) and state is remarkably reduced, but such a mode cannot be described by the chiral perturbation theory. Motivated by this fact, I invent the LSM constructed upon the linear representation of chiral symmetry, or more precisely the Pauli-G\"ursey symmetry. Then, it is shown that my LSM successfully reproduces the low-lying hadron mass spectrum in a broad range of simulated on the lattice. As applications of the LSM, topological susceptibility and sound velocity in cold and dense QCD are evaluated to compare with lattice results. Besides, generalized Gell-Mann-Oakes-Renner relation and hardon mass spectrum in the presence of a diquark source are analyzed. I also introduce an extended version of the LSM incorporating spin-1 hadrons.
... The study of the equation of state (EoS) for cold dense quantum chromodynamics (QCD) matter has long posed a significant challenge in nuclear physics. On one hand, theoretical computations of the cold dense matter EoS from first-principle lattice QCD calculations are prohibited due to the sign problem (see [1,2] for recent reviews). On the other hand, nuclear matter generated in terrestrial heavy-ion collision experiments typically resides in the high-temperature regime (T ≳ 100 MeV/k B ) [3][4][5]. ...
... is the inverse speed of sound squared. 1 In general, one solves the TOV and tidal equations (8), (9), (12) using the radius r as the independent variable. However, this makes it numerically inefficient and analytically hard to analyze the linear response of the NS observables against the perturbation in the EoS and/or in the central pressure. ...
The potential hadron-to-quark phase transition in neutron stars has not been fully understood as the property of cold, dense, and strongly interacting matter cannot be theoretically described by the first-principle perturbative calculations, nor have they been systematically measured through terrestrial low-to-intermediate energy heavy-ion experiments. Given the Tolman--Oppenheimer--Volkoff (TOV) equations, the equation of state (EoS) of the neutron star (NS) matter can be constrained by the observations of NS mass, radius, and tidal deformability. However, large observational uncertainties and the limited number of observations currently make it challenging to strictly reconstruct the EoS, especially to identify interesting features such as a strong first-order phase transition. In this work, we study the dependency of reconstruction quality of the phase transition on the number of NS observations of mass and radius as well as their uncertainty, based on a fiducial EoS. We conquer this challenging problem by constructing a neural network, which allows one to parameterize the EoS with minimum model-dependency, and by devising an algorithm of parameter optimization based on the analytical linear response analysis of the TOV equations. This work may pave the way for the understanding of the phase transition features in NSs using future X-ray and gravitational wave measurements.
... On the other hand, the determination of the QCD EoS for cold and compressed matter, from ab initio evaluations, presents further issues which have not yet been circumvented. For instance, within this regime lattice QCD (LQCD) simulations are still plagued by the infamous sign problem [33,34] while perturbative QCD (pQCD) applications are only reliable at extremely high baryon densities, of order n B ∼ 40n 0 [35][36][37][38] (n 0 = 0.16 fm −3 ), where asymptotic freedom allows for weak coupling expansions. Regarding pQCD it is important to mention that when the modified minimal subtraction (MS) renormalization scale (Λ) is taken at the conventional "central" value, 2(µ u + µ d + µ s )/3, with µ f the quark flavor chemical potentials, the NLO pQCD predicts QS masses above 2M ⊙ [39] and below 2M ⊙ at NNLO [35,[40][41][42][43]. ...
We employ the renormalization group optimized perturbation theory (RGOPT) resummation method to evaluate the equation of state (EoS) for strange () and non-strange () cold quark matter at NLO. This allows us to obtain the mass-radius relation for pure quark stars and compare the results with the predictions from perturbative QCD (pQCD) at NNLO. Choosing the renormalization scale to generate maximum star masses of order , we show that the RGOPT can produce mass-radius curves compatible with the masses and radii of some recently observed pulsars, regardless of their strangeness content. The scale values required to produce the desired maximum masses are higher in the strange scenario since the EoS is softer in this case. The possible reasons for such behavior are discussed. Our results also show that, as expected, the RGOPT predictions for the relevant observables are less sensitive to scale variations than those furnished by pQCD.
... Fermions can be included implicitly, with their presence imprinted on bosonic field configurations generated in theories with fermions. An interesting direction is to apply diffusion models to theories with a sign or complex action problem, learning the (real and semi-positive) distribution from configurations generated by complex Langevin dynamics, which is not known a priori [35][36][37][38]. This is further discussed in Ref. [24]. ...
... One of the most powerful tools to shed light on the QCD problem is the first-principles lattice QCD simulation. However, lattice simulations with a chemical potential at a lower temperature are not straightforward due to the so-called sign problem of the Monte Carlo computation [2,3]. Additionally, considering the current difficulty of accelerator experiments, cold and dense QCD can be regarded as a frontier of quark-hadron physics. ...
... That is, (maybe concise) quantum theory developed in hadron effective models must match that of the nonperturbative QC 2 D model at low energy. More practically, we make use of (2) as the matching condition, with the corresponding effective action Γ = −ilnZ. This Γ can be regarded as an action incorporating quantum corrections, so that symmetry properties inhabiting QC 2 D at a quantum level must be mimicked by the effective model properties. ...
... The functions χ π and χ η are evaluated within the present LSM by virtue of matching condition (2). That is, ...
This review is devoted to summarizing recent developments of the linear sigma model (LSM) in cold and dense two-color QCD (QC2D), in which lattice simulations are straightforwardly applicable thanks to the disappearance of the sign problem. In QC2D, both theoretical and numerical studies derive the presence of the so-called baryon superfluid phase at a sufficiently large chemical potential (μq), where diquark condensates govern the ground state. The hadron mass spectrum simulated in this phase shows that the mass of an iso-singlet (I=0) and 0− state is remarkably reduced, but such a mode cannot be described by the chiral perturbation theory. Motivated by this fact, I have invented a LSM constructed upon the linear representation of chiral symmetry, more precisely Pauli–Gürsey symmetry. It is shown that my LSM successfully reproduces the low-lying hadron mass spectrum in a broad range of μq simulated on the lattice. As applications of the LSM, topological susceptibility and sound velocity in cold and dense QC2D are evaluated to compare with the lattice results. Additionally, the generalized Gell–Mann–Oakes–Renner relation and hardon mass spectrum in the presence of a diquark source are analyzed. I also introduce an extended version of the LSM incorporating spin-1 hadrons.
... Theories with a complex Boltzmann weight are hard to simulate using conventional numerical methods based on importance sampling, due to the sign and overlap problems [1]. A prime example is QCD at nonzero baryon density, in which the quark determinant is complex for real quark chemical potential [2,3], ...
... The FPE admits a stationary solution [3,26] ...
