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Effective interaction of Chern-Simons boson with fermions

Ivan Hrynchak, Oleksandr Khasai, Yuliia Borysenkova, Mariia Tsarenkova, Volodymyr Gorkavenko

Abstract

We consider a vector extension of the Standard Model (SM) with a Chern-Simons-type interaction. This extension introduces a new massive vector boson (the Chern-Simons (CS) boson) that does not couple directly to SM fermions at tree level. We analyze the effective loop-induced interaction of this new vector boson with SM fermions and study its renormalizability in the $R_ξ$ gauge. We find that, in the effective interaction between the CS boson and same-flavor fermions, the divergent contributions from individual loop diagrams do not cancel when all relevant diagrams are taken into account. In contrast, for interactions involving fermions of different flavors, the corresponding loop-induced contributions are finite and well defined. This indicates that, in the low-energy limit, the theory exhibits nonrenormalizable behavior in the sector describing the loop-induced interaction of the CS boson with same-flavor fermions. The interaction terms between the CS boson and same-flavor fermions, characterized by divergent coefficients, are identified and must be treated within the framework of effective field theory. Finally, we derive the leading-order effective Lagrangian describing the interaction of a GeV-scale CS boson with SM fermions and discuss the number of independent parameters entering this Lagrangian. The leading-order interaction we obtained turns out to be similar to the interaction of a $Z^\prime$ boson with SM fermions.

Effective interaction of Chern-Simons boson with fermions

Abstract

We consider a vector extension of the Standard Model (SM) with a Chern-Simons-type interaction. This extension introduces a new massive vector boson (the Chern-Simons (CS) boson) that does not couple directly to SM fermions at tree level. We analyze the effective loop-induced interaction of this new vector boson with SM fermions and study its renormalizability in the gauge. We find that, in the effective interaction between the CS boson and same-flavor fermions, the divergent contributions from individual loop diagrams do not cancel when all relevant diagrams are taken into account. In contrast, for interactions involving fermions of different flavors, the corresponding loop-induced contributions are finite and well defined. This indicates that, in the low-energy limit, the theory exhibits nonrenormalizable behavior in the sector describing the loop-induced interaction of the CS boson with same-flavor fermions. The interaction terms between the CS boson and same-flavor fermions, characterized by divergent coefficients, are identified and must be treated within the framework of effective field theory. Finally, we derive the leading-order effective Lagrangian describing the interaction of a GeV-scale CS boson with SM fermions and discuss the number of independent parameters entering this Lagrangian. The leading-order interaction we obtained turns out to be similar to the interaction of a boson with SM fermions.
Paper Structure (18 sections, 123 equations, 10 figures)

This paper contains 18 sections, 123 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Interactions of the CS bosons in the $R_\xi$ gauge
  3. Divergences in the triangle diagrams
  4. Vertices $XWW$ and $X\varphi^\mp W^\pm$
  5. Vertices $XZZ$ and $X\varphi_z Z$
  6. Vertices $XZA$ and $X\varphi_z A$
  7. Vertices $XhZ$ and $XhA$
  8. Divergences in the diagrams with a circular loop
  9. Lack of cancellation of divergent terms in the loop diagrams in $R_\xi$ gauge
  10. Effective interaction between vector Chern-Simons bosons and fermions
  11. Conclusions
  12. Rules of Feynman diagrams
  13. An example of computation of the triangle diagrams
  14. An example of integration over the 4-momentum
  15. Comparison with the case of calculations in the unitary gauge
  16. ...and 3 more sections

Figures (10)

  • Figure 1: Triangle diagrams for lepton production via CS-boson decay, mediated by either the interaction with two $W$ bosons or with a $W$ boson and a charged $\varphi$ boson.
  • Figure 2: Triangle diagrams for lepton production via CS-boson decay, mediated by the interaction of the CS boson with two $Z$ bosons.
  • Figure 3: Triangle diagrams for lepton production via CS-boson decay, mediated by the interaction of the CS boson with a $Z$ and a neutral $\varphi_z$ boson.
  • Figure 4: Triangle diagrams for lepton production via CS-boson decay, mediated by the interactions of the CS boson with photons and $Z$ bosons.
  • Figure 5: Triangle diagrams for lepton production via CS-boson decay, mediated by the interactions of the CS boson with photons and neutral $\varphi_Z$ bosons.
  • ...and 5 more figures