International Journal of Modern Physics A

I was lucky to meet Yoichiro Nambu at the beginning of my scientific activity. The experience of working with him influenced my subsequent research and in the following I will try to convey what he transmitted to me. It was also a friendship that continued for decades in spite of the rare occasions to meet after our collaboration.

On December 19–21 in 2001, we organized a yearly workshop at Yukawa Institute for Theoretical Physics in Kyoto on the subject of “Fundamental Problems in Field Theory and their Implications”. Prof. Yoichiro Nambu attended this workshop and explained a necessary modification of the Nambu–Goldstone theorem when applied to non-relativistic systems. At the same workshop, I talked on a path integral formulation of the spin-statistics theorem. The present essay is on this memorable workshop, where I really enjoyed the discussions with Nambu, together with a short comment on the color freedom of quarks.

I calculate the 1-loop self-energy of the lowest Landau level of an electron of mass $m$ in a strong, constant and uniform external magnetic field $B$, beyond its always used truncation at ${(lnL)}^2$, $L=\frac{\left|e\right|B}{m^2}$. This is achieved by evaluating the integral deduced in 1953 by Demeur and incompletely calculated in 1969 by Jancovici, which I recover from Schwinger’s techniques of calculation. It yields $\delta m\simeq \frac{\alpha m}{4\pi }\left[\right.\left(\right.lnL-{\gamma }_E-\frac{3}{2}{\left)\right.}^2-\frac{9}{4}+\frac{\pi }{\beta -1}+\frac{{\pi }^2}{6}+\frac{\pi \mathrm{\Gamma }[1-\beta ]}{L^{\beta -1}}+\frac{1}{L}\left(\right.\frac{\pi }{2-\beta }-5\left)\right.+{𝒪}\left(\right.\frac{1}{L^{\ge 2}}\left)\right.\left]\right.$ with $\beta \simeq 1.175$ for $75\le L\le 10,000$. The ${(lnL)}^2$ truncation exceeds the precise estimate by 45% at $L=100$ and by more at lower values of $L$, due to neglecting, among others, the single logarithmic contribution. This is doubly unjustified because it is large and because it is needed to fulfill appropriate renormalization conditions. Technically challenging improvements look therefore necessary, for example, when resumming higher loops and incorporating the effects of large $B$ on the photonic vacuum polarization, like investigated in recent years.

The standard perturbation theory in QFT and lattice models leads to the asymptotic expansions. However, an appropriate regularization of the path or lattice integrals allows one to construct convergent series with an infinite radius of the convergence. In the earlier studies, this approach was applied to the purely bosonic systems. Here, using bosonization, we develop the convergent perturbation theory for a toy lattice model with interacting fermionic and bosonic fields.

We find the distribution function $f(E)$ for dark matter (DM) halos in galaxies and the corresponding equation of state from the (empirical) DM density profiles derived from observations. We solve for DM in galaxies the analogous of the Eddington equation originally used for the gas of stars in globular clusters. The observed density profiles are a good realistic starting point and the distribution functions derived from them are realistic. We do not make any assumption about the DM nature, the methods developed here apply to any DM kind, though all results are consistent with warm dark matter (WDM). With these methods we find: (i) Cored density profiles behaving quadratically for small distances $\rho (r)\stackrel{r\to 0}{=}\rho (0)-K{r}^2$ produce distribution functions which are finite and positive at the halo center while cusped density profiles always produce divergent distribution functions at the center. (ii) Cored density profiles produce approximate thermal Boltzmann distribution functions for $r\lesssim 3r_h$ where $r_h$ is the halo radius. (iii) Analytic expressions for the dispersion velocity and the pressure are derived yielding at each halo point an ideal DM gas equation of state with local temperature $T(r)\equiv m{v}^2(r)/3$. $T(r)$ turns out to be constant in the same region where the distribution function is thermal and exhibits the same temperature within the percent. The self-gravitating DM gas can thermalize despite being collisionless because it is an ergodic system. (iv) The DM halo can be consistently considered at local thermal equilibrium with: (a) a constant temperature $T(r)=T_0$ for $r\lesssim 3r_h$, (b) a space dependent temperature $T(r)$ for $3r_h<r\lesssim R_{\mathrm{\text{virial}}}$, which slowly decreases with $r$. That is, the DM halo is realistically a collisionless self-gravitating thermal gas for $r\lesssim R_{\mathrm{\text{virial}}}$. (v) $T(r)$ outside the halo radius nicely follows the decrease of the circular velocity squared.

In terms of the matrix valued Berry gauge field strength for the Weyl Hamiltonian in any even space–time dimensions a symplectic form whose elements are matrices in spin indices is introduced. Definition of the volume form is modified appropriately. A simple method of finding the path integral measure and the chiral current in the presence of external electromagnetic fields is presented. It is shown that within this new approach the chiral magnetic effect as well as the chiral anomaly in even $d+1$ dimensions are accomplished straightforwardly.

In this paper, we focus our attention in the inconsistency that appears when the semi-exact Foldy–Wouthuysen transformation for the Dirac field interacting with space–time torsion field is performed. In order to solve this problem, we present a new involution operator that makes possible to perform the exact transformation when torsion field is present. Such operator has a structure, well known in the literature, composed of the product of an operator that acts in the matrices space and another one that acts in the function space. We also present the bound state of this theory and discuss the possible experimental analysis.

I present analytical expressions for the massive cusp anomalous dimension in QCD through three loops, first calculated in 2014, in terms of elementary functions and ordinary polylogarithms. I observe interesting relations between the results at different loops and provide a conjecture for the n-loop cusp anomalous dimension in terms of the lower-loop results. I also present numerical results and simple approximate formulas for the cusp anomalous dimension relevant to top-quark production.

Aiming to relate leptonic CP violating phase $\delta$ to the cosmological CP asymmetry, we study the extension of MSSM by two quasi-degenerate (strictly degenerate at tree level) right-handed neutrinos and consider all possible two texture zero $3\times 2$ Yukawa matrices plus one $\mathrm{\Delta }L=2$ dimension five $(d=5)$ operator contributing to the light neutrino mass matrix. We classify all experimentally viable mass matrices, leading to several predictions, and analytically derive predictive relations. We also relate the CP violating $\delta$ phase to the CP phase of the thermal leptogenesis.