Mass and coupling magnetic field dependence in a scalar theory with charged bosons from an environmentally friendly renormalization group analysis

TL;DR

The paper addresses how an external magnetic field modifies particle masses and couplings by studying a simple scalar theory with one neutral and two charged bosons. It adopts the Environmentally Friendly Renormalization Group, treating the magnetic-field strength as the renormalization scale, and computes one-loop mass and quartic coupling corrections using the Schwinger proper-time formalism to obtain environmental beta functions. The authors derive coupled RG equations for the neutral-mcalar mass $m^2(b)$ and coupling $\lambda(b)$, solve them numerically and analytically in the small-mass limit, and find that $\lambda(b)$ decreases while $m^2(b)$ increases with increasing $b$. These results establish the feasibility of the EFRG approach for magnetized environments and lay groundwork for extensions to QED and QCD, including fermionic and charged-mass effects.

Abstract

We compute the running of the mass of a neutral boson and of its self-coupling in a simple model describing the self-interaction of three scalars, one of them neutral and the other two electrically charged, subject to the effects of a magnetic field, as functions of the field strength, at one-loop order. We resort to the Environmentally Friendly Renormalization Group approach, where the flow variable is taken as that describing the environmental conditions, in this case the strength of the magnetic field. We find the magnetic field dependent mass and coupling beta functions and use them to set up the differential equations satisfied by the neutral scalar mass and coupling. We solve the resulting system of coupled equations both numerically, and also analytically in the small-mass approximation. We find that the neutral scalar mass increases, while the coupling decreases with increasing field strength. The study is intended to set up the ideas to later use them in more sophisticated theories such as QED and QCD.

Figures (5)

• Figure 1: Feynman diagram representing the interaction Lagrangian between two neutral (dashed lines) and two charged (solid lines) scalar fields and its corresponding factor to use for the Feynman rules.
• Figure 2: Feynman diagram representing the neutral scalar (dashed-line) self-energy. The solid line represents a charged scalar in the loop.
• Figure 3: Sum of Feynman diagrams contributing to the neutral scalar four-point function. From left to right, the diagrams correspond to the $s$-, $t$- and $u$-channels, respectively. The dashed lines represent the neutral scalars whereas the solid lines represent the charged scalars in the loop.
• Figure 4: Magnetic field dependence of the boson self-coupling, normalized to $\lambda_0$. The solid curve represents the numerical solution, while the dashed curve represents the approximate solution. For the calculation we have taken $\lambda_0=\lambda(b_0=m_\pi)=1$ and $m_0=m(b_0)=m_\pi$, with $m_\pi=0.14$ GeV.
• Figure 5: Magnetic field dependence of the neutral boson mass squared, normalized to $m_0^2$. The solid curve represents the numerical solution, while the dashed curve represents the approximate solution. For the calculation we have taken $\lambda_0=\lambda(b_0=m_\pi)=1$ and $m_0=m(b_0)=m_\pi$, with $m_\pi=0.14$ GeV.