[Paper Review] Dynamical Dirac Mass Generation in the Supersymmetric Nambu--Jona-Lasinio Model with the Seesaw Mechanism of Neutrinos

Tadafumi Ohsaku|arXiv (Cornell University)|Jan 8, 2008

Black Holes and Theoretical Physics16 references3 citations

TL;DR

This paper investigates dynamical Dirac mass generation in a supersymmetric Nambu-Jona-Lasinio (SNJL) model with right-handed Majorana mass parameters, using effective potential and gap equation methods under covariant and non-covariant regularization. It finds that the seesaw condition $0 < |\phi_S| \ll |M|$ can be naturally satisfied only in second-order phase transitions via fine-tuning of the coupling constant $G$, while first-order transitions make this condition highly unnatural or unattainable.

ABSTRACT

The dynamical generation of Dirac mass in the supersymmetric Nambu$-$Jona-Lasinio (SNJL) model with the seesaw mechanism of neutrino is investigeted. The right and left handed Majorana mass parameters are introduced into the SNJL model; we regard them as external model parameters. The question on the origin of these Majorana masses are set aside, and we concentrate on the examination of the effect of the Majorana mass parameters on the dynamical generation of Dirac mass. The effective potential of the model and the gap equation for the self-consistent determination of Dirac mass are derived and solved. We use both the four-dimensional covariant and three-dimensional non-covariant cutoff schemes for the regularizations of the effective potential. We find there are cases of the first and second order phase transitions with respect to variation of the coupling constant of the Nambu$-$Jona-Lasinio-type four-body interaction of the SNJL model. In the case of second-order phase transition, the dynamically generated Dirac mass $|ϕ_{S}|$ can arbitrarily be small compared with the right-handed Majorana mass parameter $|M|$ and thus the seesaw condition $0

Motivation & Objective

• To examine how right- and left-handed Majorana mass parameters affect dynamical Dirac mass generation in a supersymmetric Nambu-Jona-Lasinio (SNJL) model.
• To determine under what conditions the seesaw mechanism—requiring $|\phi_S| \ll |M|$—can be dynamically realized.
• To analyze the phase structure of the model via effective potential and gap equations under different regularization schemes.
• To assess whether the seesaw condition is naturally achievable or requires fine-tuning, depending on the order of the phase transition.

Proposed method

• Formulates a one-flavor SNJL model with added Majorana mass terms for right- and left-handed neutrino chiral superfields.
• Derives the effective potential $V_{\text{eff}}$ and the gap equations for self-consistent determination of the dynamical Dirac mass $|\phi_S|$.
• Applies both four-dimensional covariant and three-dimensional non-covariant cutoff regularization schemes to ensure robustness of results.
• Solves the gap equations numerically and analyzes the second derivative $\partial^2 V_{\text{eff}} / \partial |\phi_S|^2$ at $|\phi_S| = 0$ to classify phase transitions.
• Varies the coupling constant $G$ and the ratio $|M|/\Delta$ to map phase boundaries between first- and second-order transitions.
• Performs parameter scans with $\Lambda \gg \Delta$ to isolate the role of $|M|$ and $\Delta$ in determining the transition order and Dirac mass size.

Experimental results

Research questions

• RQ1 Can the seesaw condition $0 < |\phi_S| \ll |M|$ be dynamically realized in a supersymmetric four-fermion model with Majorana mass terms?
• RQ2 How does the presence of Majorana mass parameters affect the phase transition structure in the SNJL model?
• RQ3 Is the dynamical generation of a small Dirac mass via fine-tuning of the coupling $G$ feasible only in second-order phase transitions?
• RQ4 Does the regularization scheme (covariant vs. non-covariant) qualitatively affect the phase structure and Dirac mass generation?
• RQ5 What is the critical relation between $|M|$ and $\Delta$ that separates first- and second-order phase transitions in this model?

Key findings

• The seesaw condition $0 < |\phi_S| \ll |M|$ can be satisfied only in second-order phase transitions through fine-tuning of the coupling constant $G$, while it is highly unnatural or unattainable in first-order transitions.
• When $|M|/\Delta \leq 0.5$ (in covariant scheme), the phase transition is second-order, allowing $|\phi_S|$ to be arbitrarily small by adjusting $G$, satisfying the seesaw condition.
• For $|M|/\Delta \geq 0.55$, the transition becomes first-order, and $|\phi_S|$ cannot be made arbitrarily small; the system favors a finite $|\phi_S| \sim |M|$, violating the seesaw condition.
• The critical point separating first- and second-order transitions lies near $2|M|/\Delta \sim 1$, indicating that $|M| \ll \Delta$ is essential for dynamical seesaw realization.
• Numerical results are qualitatively independent of the regularization scheme, confirming robustness of the phase transition structure.
• The second derivative $\partial^2 V_{\text{eff}} / \partial |\phi_S|^2$ at $|\phi_S| = 0$ is negative for second-order transitions (indicating instability), while it becomes positive in first-order cases, confirming the transition order classification.