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Dynamical generation of fermion mass in a scalar-fermion theory with $λφ^4$ interaction

Somnath Majumder, Krishnendu Mukherjee

Abstract

The effective potential for a scalar theory with $λφ^4$ interaction, coupled to a massless fermion through Yukawa interaction is calculated by summing over infinite number of two particle irreducible (2PI) diagrams of two different types and a 2PI diagram of a third type using Cornwall, Jackiw and Tomboulis (CJT) method. There is an inversion symmetry present in the effective potential under $φ\rightarrow -φ$. At large coupling it exhibits minima below the zero potential line and on either side of the maximum at $φ=0$. The fermion acquires a mass at any non-trivial, positive minimum breaking the inversion symmetry of the vacuum.

Dynamical generation of fermion mass in a scalar-fermion theory with $λφ^4$ interaction

Abstract

The effective potential for a scalar theory with interaction, coupled to a massless fermion through Yukawa interaction is calculated by summing over infinite number of two particle irreducible (2PI) diagrams of two different types and a 2PI diagram of a third type using Cornwall, Jackiw and Tomboulis (CJT) method. There is an inversion symmetry present in the effective potential under . At large coupling it exhibits minima below the zero potential line and on either side of the maximum at . The fermion acquires a mass at any non-trivial, positive minimum breaking the inversion symmetry of the vacuum.
Paper Structure (10 sections, 114 equations, 11 figures, 1 table)

This paper contains 10 sections, 114 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Effective Potential for Composite Operator
  3. Diagrams contributing to ${\tilde{\Gamma}}_2(\phi, G)$
  4. Solutions of $\frac{\partial V_{eff}}{\partial S(k)}=0$ and $\frac{\partial V_{eff}}{\partial G(k)}=0$
  5. Effective Potential
  6. Discussions
  7. Evaluations of $I(p)$ and $J(p)$ for $G={\cal{D}}$
  8. Evaluation of $\frac{\partial \tilde{\Gamma}_{2}^{(a)}}{\partial G(k)} {|}_{G={\cal{D}}}$
  9. Evaluation of $\frac{\partial \tilde{\Gamma}_{2}^{(b)}}{\partial G(k)} {|}_{G={\cal{D}}}$ and $\frac{\partial \tilde{\Gamma}_{2}^{(c)}}{\partial G(k)} {|}_{G={\cal{D}}}$
  10. Evaluation of integrals in $V_{eff}$

Figures (11)

  • Figure 1: (a) Lobe (b) Trilinear line
  • Figure 2: 2PI diagram with two lobes each containing one trilinear line
  • Figure 3: 2PI diagram with three lobes each containing one trilinear line
  • Figure 4: 2PI diagram with n lobes each containing one trilinear line
  • Figure 5: 2PI diagram with n lobes
  • ...and 6 more figures