<p>Determining the transport properties of quark-gluon plasma is one of the most important aspects of relativistic heavy ion collision studies. Field-theoretical calculations of transport coefficients such as the shear and bulk viscosities require Kubo formulas, which in turn require real-time correlation functions of stress-energy tensors. Consequently, knowing the analytic structure of these correlation functions is essential in any such studies. Using the energy-conservation laws and the results from gravity-hydrodynamics analysis, we determine the low-frequency and low-wavenumber analytic structures of all stress-energy correlation functions in the rest frame of the medium. By comparing with the diffusion and sound spectra from the second-order and third-order relativistic hydrodynamics, various new Kubo formulas are derived in the limit where the zero-frequency limit is taken first. We also show that the meaning of the Kubo formulas for relaxation times can change when higher-order terms are added to the hydrodynamics. A subtle issue of taking the zero frequency and zero wave0number limits when using skeleton diagrams is addressed as well.</p>

Canada

Sangyong Jeon

Poland

Alina Czajka

South Korea

Juhee Hong

Analytic structure of stress-energy response functions and new Kubo formulas

We use linear response theory to derive both the non-dissipative and dissipative effects of spin polarization for massive and massless interacting spin 1/2 particles in a relativistic fluid. We list and classify all the possible contributions up to first order in gradients of hydrodynamic fields including the axial chemical potential and the spin potential, and we obtain the corresponding Kubo formulas. We find that all the possible dissipative contributions, except those coming from the gradients of spin potential, require a chiral imbalance or parity violating interactions. In a fluid with chiral imbalance we find a chiral version of the spin Hall effect, i.e. a spin polarization is induced by the gradients of temperature and of axial chemical potential in the direction orthogonal to the momentum of the particle and to the gradients. Moreover, we identify several other new non-dissipative contributions that are not present for free fields.

Romania

Matteo Buzzegoli

Kubo formulas for spin polarization in dissipative relativistic spin hydrodynamics: a first-order gradient expansion approach

Victor E. Ambrus

France

Maxim N. Chernodub

Corrigendum to “Acceleration as a circular motion along an imaginary circle: Kubo-Martin-Schwinger condition for accelerating field theories in imaginary-time formalism” [Physics Letters B 855 (2024) 138757/PLB 138795]

The strongly interacting matter created in relativistic heavy-ion collisions possesses several conserved quantum numbers, such as baryon number, strangeness, and electric charge. The diffusion process of these charges can be characterized by a diffusion matrix that describes the mutual influence of the diffusion of various charges. We derive the Kubo relations for evaluating diffusion coefficients as elements of a diffusion matrix. We further demonstrate that in the weak coupling limit, the diffusion matrix elements obtained through Kubo relations reduce to those obtained from kinetic theory with an appropriate identification of the relaxation times. We illustrate this evaluation in a toy model of two interacting scalar fields with two conserved charges.

India

Sourav Dey

Amaresh Jaiswal

Hiranmaya Mishra

Diffusion coefficient matrix for multiple conserved charges: a Kubo approach

The electrical conductivity of hot and dense quark matter is calculated using the 2-flavour gauged Nambu–Jona–Lasinio (NJL) model in the presence of a chiral imbalance quantified in terms of a chiral chemical potential (CCP). To this end, the in-medium spectral function corresponding to the vector current correlator is evaluated employing the real time formulation of finite temperature field theory. Taking the long wavelength limit of the spectral function we extract the electrical conductivity using the Green–Kubo relation. The thermal widths of the quarks/antiquarks that appear in the expression of electrical conductivity are calculated by considering the $$2\rightarrow 2$$ scattering in the NJL model. The scattering amplitudes containing the polarization functions of the mesonic modes in the scalar and pseudoscalar channels are also evaluated by considering finite value of CCP. We find that the ratio of electrical conductivity to temperature has significant dependence on CCP especially in the low temperature region.

Snigdha Ghosh

Nilanjan Chaudhuri

Sourav Sarkar

Pradip Roy

Effect of chiral imbalance on the electrical conductivity of hot and dense quark matter using Green–Kubo method within the 2-flavour gauged NJL model

We discuss the imaginary-time formalism for field theories in thermal equilibrium in uniformly accelerating frames. We show that under a Wick rotation of Minkowski spacetime, the Rindler event horizon shrinks to a point in a two-dimensional subspace tangential to the acceleration direction and the imaginary time. We demonstrate that the accelerated version of the Kubo-Martin-Schwinger (KMS) condition implies an identification of all spacetime points related by integer-multiple rotations in the tangential subspace about this Euclidean Rindler event-horizon point, with the rotational quanta defined by the thermal acceleration, <math><mi>α</mi><mo>=</mo><mi>a</mi><mo>/</mo><mi>T</mi></math>. In the Wick-rotated Rindler hyperbolic coordinates, the KMS relations reduce to standard (anti-)periodic boundary conditions in terms of the imaginary proper time (rapidity) coordinate. Our findings pave the way to study, using first-principle lattice simulations, the Hawking-Unruh radiation in geometries with event horizons, phase transitions in accelerating Early Universe and early stages of quark-gluon plasma created in relativistic heavy-ion collisions.

Victor E. Ambruş

Acceleration as a circular motion along an imaginary circle: Kubo-Martin-Schwinger condition for accelerating field theories in imaginary-time formalism

Abstract We argue that Lee–Wick’s complex ghost appearing in any higher derivative theory is stable and its asymptotic field exists. It may be more appropriate to call it “anti-unstable”, in the sense that the more the ghost "decays" into lighter ordinary particles, the larger the probability that the ghost remains as itself becomes. This is explicitly shown by analyzing the two-point functions of the ghost Heisenberg field which is obtained as an exact result in the N → ∞ limit in a massive scalar ghost theory with light O(N)-vector scalar matter. The anti-instability is a consequence of the fact that the poles of the complex ghost propagator are located on the physical sheet in the complex plane of four-momentum squared. This should be contrasted with the case of the ordinary unstable particle, whose propagator has no pole on the physical sheet.

Japan

Germany

Jisuke Kubo

Taichiro Kugo

Anti-Instability of Complex Ghost A relativistic complex pole model with indefinite metric Acausality

Abstract Theories with fourth-order derivatives, including the Lee–Wick finite QED model and quadratic gravity, have a better UV behavior, but the presence of negative metric ghost modes endangers unitarity. Noticing that the ghost acquires a complex mass by radiative corrections, Lee and Wick, in particular, claimed that such complex ghosts would never be created by collisions of physical particles because of energy conservation, so that the physical S-matrix unitarity must hold. We investigate the unitarity problem faithfully, working in the operator formalism of quantum field theory. When complex ghosts participate, a complex delta function (a generalization of the Dirac delta function) appears at each interaction vertex, which enforces a specific conservation law of complex energy. Its particular property implies that the naive Feynman rule is wrong if the four-momenta are assigned to the internal lines after taking account of the conservation law in advance. We show that complex ghosts are actually created and unitarity is violated in such fourth-order derivative theories. We also find a definite energy threshold below which ghosts cannot be created: The theories are unitary and renormalizable below the threshold.

Unitarity violation in field theories of Lee–Wick’s complex ghost A relativistic complex pole model with indefinite metric

We investigate several quantum phenomena related to quadratic gravity after rewriting the general fourth-order action in a more convenient form that is second order in derivatives and produces only first-class constraints in phase space. We find that a Higgs mechanism may occur in the conformally invariant subset of the general quadratic action if the theory is conformally coupled to a scalar field that acquires a nonzero vacuum expectation value and spontaneously breaks the conformal symmetry. Then, in the broken phase, the originally massless spin-2 ghost may absorb both the scalar and vector fields to become massive. We also perform a Becchi-Rouet-Stora-Tyutin quantization of second-order quadratic gravity in the covariant operator formalism and discuss conditions under which unitarity of the full interacting quantum theory may be established.

Jeffrey Kuntz

Spontaneous conformal symmetry breaking and quantum quadratic gravity

We have explored the multicomponent structure of electrical conductivity of relativistic Fermionic and bosonic fluid in the presence of a magnetic field by using the Kubo approach. This is done by explicitly evaluating the thermomagnetic vector current spectral functions using the real time formalism of finite temperature field theory and the Schwinger proper time formalism. In the absence of a magnetic field, the one-loop diagramatic representation of the Kubo expression of any transport coefficients is exactly the same with a relaxation time approximation (RTA)-based expression, but this equality does not hold for a finite magnetic field picture due to a lack of proper implementation of quantum effect in a latter approach. We have shown this discrepancy for a particular transport coefficient—electrical conductivity, whose starting point in the Kubo approach will be an electromagnetic current-current correlator and its one-loop skeleton diagram carrying two scalar/Dirac propagators for a scalar/Dirac fluid. Through a numerical comparison between RTA and Kubo expressions of conductivity components (parallel and perpendicular), we have attempted to interpret a detailed quantum field theoretical effect, contained by the Kubo expression but not by the RTA expression. In a classical RTA expression we get a magnetic field independent parallel conductivity due to zero Lorentz force but in the field theoretical Kubo expression, it decreases and increases with the magnetic field for a scalar and Dirac medium, respectively, due to the Landau quantization effect. This parallel component of conductivity can be interpreted as a zero momentum limit of quantum fluctuation with the same Landau level internal lines, while for a perpendicular component of conductivity, fluctuation with Landau level differences <math><mo>±</mo><mn>1</mn></math> are noticed, which might be a new realization of transportation in a field theoretical sector.

Sarthak Satapathy

Sabyasachi Ghosh

Kubo estimation of the electrical conductivity for a hot relativistic fluid in the presence of a magnetic field