[Paper Review] Amplitudes of radiative corrections in fermion bags bound by Higgs boson exchange

M. Yu. Kuchiev|arXiv (Cornell University)|Dec 4, 2010

Particle physics theoretical and experimental studies1 references3 citations

TL;DR

This paper investigates radiative corrections in fermion bags bound by Higgs boson exchange, focusing on heavy fermions with masses between 400–1000 GeV. It demonstrates that for this mass range, amplitudes of radiative corrections are independent of fermion mass, leading to negligible quantum corrections and stable bound states, with implications for the viability of such fermion bags in models of new physics beyond the Standard Model.

ABSTRACT

Properties of amplitudes that describe radiative corrections in a bag of heavy fermions bound by the Higgs boson exchange are studied. Classes of amplitudes, in which the large fermion mass is canceled out and hence produces no enhancement for the radiative corrections are found. For fermions with masses in the region 400< m < 1000 Gev all relevant amplitudes are found to possess this property. Correspondingly the radiative corrections for this range of masses are small. For very heavy fermions, m>1000 Gev, the processes described by diagrams with closed fermion loops are also mass-independent.

Motivation & Objective

• To analyze the behavior of radiative corrections in fermion bags bound by Higgs boson exchange, particularly focusing on the role of fermion mass in enhancing or suppressing these corrections.
• To determine whether quantum corrections destabilize fermion bags composed of heavy fermions, especially in the mass range where the Higgs field remains near its vacuum expectation value.
• To identify classes of amplitudes that are insensitive to the fermion mass, thereby avoiding large enhancements in radiative corrections.
• To assess the stability of fermion bags under quantum corrections, especially in the context of recent theoretical proposals involving top quark condensates or fourth-generation fermions.
• To provide a foundation for future calculations by deriving and analyzing key amplitudes relevant to one-loop quantum corrections in the Higgs-mediated fermion bag system.

Proposed method

• The study employs a non-relativistic approximation for fermions in the bag, assuming the Higgs field remains close to its vacuum expectation value (246 GeV), allowing the use of vacuum propagators.
• Feynman diagrams with minimal external Higgs legs are considered, as each such leg introduces a factor proportional to the small Higgs field deviation from its vacuum value.
• The momentum transfer in the amplitudes is assumed to be small compared to the fermion mass, consistent with the large size of the fermion bag.
• The analysis focuses on the vacuum polarization and self-energy contributions from fermions and the Higgs boson, using the exact expression for the Higgs self-energy function $ P_{\text{H}}(t) $ derived from the effective Lagrangian.
• The fermion contribution to vacuum polarization is evaluated using the asymptotic form of the self-energy function in the limit $ t \ll m^2 $, while the Higgs contribution is computed exactly.
• The unitarity condition is verified by comparing the imaginary part of the Higgs self-energy with the corresponding matrix element in the $ 1 \to 2 $ scattering process, confirming consistency with the S-matrix formalism.

Experimental results

Research questions

• RQ1 Do radiative corrections in fermion bags bound by Higgs exchange exhibit a dependence on the fermion mass, particularly in the 400–1000 GeV range?
• RQ2 Are there specific classes of amplitudes in which the large fermion mass cancels out, leading to mass-independent corrections?
• RQ3 How do quantum corrections affect the stability of fermion bags composed of heavy fermions, especially when the Higgs field is nearly at its vacuum expectation value?
• RQ4 What is the relative contribution of fermion vacuum polarization versus Higgs vacuum polarization to the total self-energy in the bag system?
• RQ5 Can the one-loop quantum corrections be suppressed or rendered negligible for certain fermion masses, thereby stabilizing the bound state?

Key findings

• For fermions with masses in the range $ 400 \lesssim m \lesssim 1000 $ GeV, all relevant amplitudes of radiative corrections are found to be independent of the fermion mass, leading to no enhancement of quantum corrections.
• As a result, radiative corrections in this mass range are small and do not destabilize the fermion bag, indicating that such bound states can be quantum mechanically stable.
• For very heavy fermions ($ m > 1000 $ GeV), processes involving closed fermion loops also exhibit mass independence, suggesting a universal suppression of quantum corrections in the high-mass regime.
• The vacuum polarization from fermions dominates over that from the Higgs boson in the relevant momentum regime ($ t \sim t_0 \sim v^2 N / (2\pi) $), with the fermion contribution exceeding the Higgs contribution by a factor of at least 2 for $ 100 \leq m_{\text{H}} \leq 200 $ GeV.
• The imaginary part of the Higgs self-energy function $ P_{\text{H}}(t) $ is derived and confirmed to satisfy the unitarity condition, validating the consistency of the quantum field theory framework used.
• The structure of the fermion self-energy in the Higgs-bag system is shown to be analogous to the QED mass operator, with a similar functional form derived via substitution rules from the general self-energy expression.