Kazunori Itakura

A Gaussian effective theory for gluon saturation

Abstract We construct a Gaussian approximation to the effective theory for the colour glass condensate which describes correctly the gluon distribution both in the low density regime at high transverse momenta (above the saturation scale Q s ), and in the high density regime below Q s , and provides a simple interpolation between these two regimes. At high momenta, the effective theory reproduces the BFKL dynamics, while at low momenta, it exhibits gluon saturation and, related to it, colour neutrality over the short distance scale 1/ Q s ⪡1/ Λ QCD . Gauge-invariant quantities computed within this approximation are automatically infrared-finite.

Shear viscosity of a hadronic gas mixture

We discuss in detail the shear viscosity coefficient η and the viscosity to entropy density ratio η/s of a hadronic gas comprised of pions and nucleons. In particular, we study the effects of baryon chemical potential on η and η/s. We solve the relativistic quantum Boltzmann equations with binary collisions (ππ, πN, and NN) for a state slightly deviated from thermal equilibrium at temperature T and baryon chemical potential μ. The use of phenomenological amplitudes in the collision terms, which are constructed to reproduce experimental data, greatly helps to extend the validity region in the T-μ plane. The total viscosity coefficient η(T, μ) = η π + η N increases as a function of T and μ, indirectly reflecting energy dependences of binary cross sections. The increase in μ direction is due to enhancement of the nucleon contribution η N while the pion contribution η π diminishes with increasing μ. On the other hand, due to rapid growth of entropy density, the ratio η/s becomes a decreasing function of T and μ in a wide region of the T-μ plane. In the kinematical region we investigated T < 180 MeV, μ < 1 GeV, the smallest value of η/s is about 0.3. Thus, it never violates the conjectured lower bound η/s = 1/4π ∼ 0.1. The smallness of η/s in the hadronic phase and its continuity at T ≃ T c (at least for crossover at small μ) implies that the ratio will be small enough in the deconfined phase T ≥ T c . There is a nontrivial structure at low temperature and at around normal nuclear density. We examine its possible interpretation as the liquid-gas phase transition.

Expanding color flux tubes and instabilities

Abstract We present an analytic study of the physics of the glasma which is a strong classical gluon field created at early stage of high-energy heavy-ion collisions. Our analysis is based on the picture that the glasma just after the collision is made of color electric and magnetic flux tubes extending in the longitudinal direction with their diameters of the order of 1 / Q s ( Q s is the saturation scale of the colliding nuclei). We find that both the electric and magnetic flux tubes expand outwards and the field strength inside the flux tube decays rapidly in time. Next we investigate whether there exist instabilities against small rapidity-dependent perturbations for a fixed color configuration. We find that the magnetic background field exhibits an instability induced by the fluctuations in the lowest Landau level, and it grows exponentially in the time scale of 1 / Q s . For the electric background field we find no apparent instability while the possible relation to the Schwinger mechanism for particle pair creations is suggested.

Structural change of Cooper pairs and momentum dependent gap in color superconductivity

The two-flavor color superconductivity is studied over a wide range of baryon density with a single model. We pay a special attention to the spatial-momentum dependence of the gap and to the spatial-structure of Cooper pairs. At extremely high baryon density (O(10^{10} rho_0) with rho_0 being the normal nuclear matter density), our model becomes equivalent to the usual perturbative QCD treatment and the gap is shown to have a sharp peak near the Fermi surface due to the weak-coupling nature of QCD. On the other hand, the gap is a smooth function of the momentum at lower densities (O(10 rho_0)) due to strong color magnetic and electric interactions. To study the structural change of Cooper pairs from high density to lower density, quark correlation in the color superconductor is studied both in the momentum space and in the coordinate space. The size of the Cooper pair is shown to become comparable to the averaged inter-quark distance at low densities. Also, effects of the momentum-dependent running coupling and the antiquark pairing, which are both small at high density, are shown to be non-negligible at low densities. These features are highly contrasted to the standard BCS superconductivity in metals.

Vacuum birefringence in strong magnetic fields: (I) Photon polarization tensor with all the Landau levels

Abstract Photons propagating in strong magnetic fields are subject to a phenomenon called the “vacuum birefringence” where refractive indices of two physical modes both deviate from unity and are different from each other. We compute the vacuum polarization tensor of a photon in a static and homogeneous magnetic field by utilizing Schwinger’s proper-time method, and obtain a series representation as a result of double integrals analytically performed with respect to proper-time variables. The outcome is expressed in terms of an infinite sum of known functions which is plausibly interpreted as summation over all the Landau levels of fermions. Each contribution from infinitely many Landau levels yields a kinematical condition above which the contribution has an imaginary part. This indicates decay of a sufficiently energetic photon into a fermion–antifermion pair with corresponding Landau level indices. Since we do not resort to any approximation, our result is applicable to the calculation of refractive indices in the whole kinematical region of a photon momentum and in any magnitude of the external magnetic field.

Instabilities in non-expanding glasma

A homogeneous color magnetic field is known to be unstable for the fluctuations perpendicular to the field in the color space (the Nielsen–Olesen instability). We argue that these unstable modes, exponentially growing, generate an azimuthal magnetic field with the original field being in the z-direction, which causes the Nielsen–Olesen instability for another type of fluctuations. The growth rate of the latter unstable mode increases with the momentum pz and can become larger than the formers growth rate which decreases with increasing pz. These features may explain the interplay between the primary and secondary instabilities observed in the real-time simulation of a non-expanding glasma, i.e., stochastically generated anisotropic Yang–Mills fields without expansion.

Vacuum birefringence in strong magnetic fields: (II) Complex refractive index from the lowest Landau level

Abstract We compute the refractive indices of a photon propagating in strong magnetic fields on the basis of the analytic representation of the vacuum polarization tensor obtained in our previous paper. When the external magnetic field is strong enough for the fermion one-loop diagram of the polarization tensor to be approximated by the lowest Landau level, the propagating mode in parallel to the magnetic field is subject to modification: The refractive index deviates from unity and can be very large, and when the photon energy is large enough, the refractive index acquires an imaginary part indicating decay of a photon into a fermion–antifermion pair. We study dependences of the refractive index on the propagating angle and the magnetic-field strength. It is also emphasized that a self-consistent treatment of the equation which defines the refractive index is indispensable for accurate description of the refractive index. This self-consistent treatment physically corresponds to consistently including the effects of back reactions of the distorted Dirac sea in response to the incident photon.

Majorana meets Coxeter: Non-Abelian Majorana fermions and non-Abelian statistics

We discuss statistics of vortices having zero-energy non-Abelian Majorana fermions inside them. Considering the system of multiple non-Abelian vortices, we derive a non-Abelian statistics that differs from the previously derived non-Abelian statistics. The non-Abelian statistics presented here is given by a tensor product of two different groups, namely the non-Abelian statistics obeyed by the Abelian Majorana fermions and the Coxeter group. The Coxeter group is a symmetric group related to the symmetry of polytopes in a high-dimensional space. As the simplest example, we consider the case in which a vortex contains three Majorana fermions that are mixed with each other under the SO(3) transformations. We concretely present the representation of the Coxeter group in our case and its geometrical expressions in the high-dimensional Hilbert space constructed from non-Abelian Majorana fermions.

Structure change of Cooper pairs in color superconductivity Crossover from BCS to BEC

Abstract We discuss a possibility of transition from color superconductivity of the standard BCS type at high density, to Bose-Einstein Condensation (BEC) of Cooper pairs at lower density. Examining two-flavor QCD over a wide range of baryon density, we found the size of a Cooper pair becomes small enough to be comparable to the averaged quark-quark distance at lower density. We also consider the same problem in two-color QCD.

Dirac returns: Non-Abelian statistics of vortices with Dirac fermions

Abstract Topological superconductors classified as type D admit zero-energy Majorana fermions inside vortex cores, and consequently the exchange statistics of vortices becomes non-Abelian, giving a promising example of non-Abelian anyons. On the other hand, types C and DIII admit zero-energy Dirac fermions inside vortex cores. It has been long believed that an essential condition for the realization of non-Abelian statistics is non-locality of Dirac fermions made of two Majorana fermions trapped inside two well-separated vortices as in the case of type D. Contrary to this conventional wisdom, however, we show that vortices with local Dirac fermions also obey non-Abelian statistics.