Dynamical symmetry breaking in Georgi-Glashow chiral-gauge theories

TL;DR

We study infrared dynamics and dynamical symmetry breaking in chiral Georgi-Glashow–type gauge theories with multiple fermion species using the functional renormalization group. By constructing a Fierz-complete basis of four-fermion operators and deriving their RG flows from the flow equation for the effective average action, we identify resonant channels and define a critical gauge coupling $\\alpha^{\\rm crit}_{\\rm SB}$ that signals scale generation via fixed-point mergers. The analysis reveals a region (Region I, roughly $N_c\\lesssim 11$) where the dominant ${\\cal O}_5$ channel produces a $\\\oldsymbol{\\langle\\chi\\chi\\rangle}$ condensate, and a region (Region II) at larger $N_c$ where off-diagonal and higher-order effects drive a strongly coupled IR with possible confinement or exotic condensates; the conformal window boundary is estimated by comparing $\\\\\\\ ^{\\rm crit}_{\\rm SB}$ to IR fixed points from perturbative beta functions up to four loops. Overall, the work provides a nonperturbative IR handle on chiral gauge theories with multiple fermion species and lays groundwork for bosonized or confinement-focused IR descriptions.

Abstract

We investigate dynamical symmetry breaking in a class of chiral gauge theories containing the Georgi-Glashow model. These theories feature a gauge sector and two fermion species that transform in the two-index antisymmetric and antifundamental representations with different multiplicities. Using the effective action formalism and the functional renormalization group, we derive the flow of four-fermion interactions that encode their resonant structure and information about bound-state formation. Generalizing the theories to multiple generations, we make contact with the loss of asymptotic freedom and dissect the boundary of a conjectured conformal window. Our results show that, while most of the theory space displays a dominant color-breaking condensate, there exists a strongly coupled regime where the lowest-laying mechanisms fail and more intricate dynamics are expected to arise. This analysis provides a first step toward the infrared behavior of chiral gauge theories with functional methods.

Figures (7)

• Figure 1: Sketch of the diagram topologies entering the RG flow of the four-$\psi$ point function relevant to the flow of the four-fermion couplings in \ref{['eq:flowlambda_structure']}. Plain arrowed lines represent full $\psi$ propagators, dashed lines the respective $\chi$ propagators, and curly lines the full gauge boson propagators. The operator $\tilde{\partial}_t$ denotes a shorthand for the insertion of the regulators ($\partial_t R_k$) inherent to the flow equation \ref{['eq:floweq']}, avoiding the need to write the different combinations of regulated lines explicitly. The $\ldots$ indicate other diagrams involving other higher dimensional operators (e.g. six-point functions), which are neglected in this analysis.
• Figure 2: Depiction of the general form of the flow of a four-fermion coupling, which exhibits a resonant structure as a function of the coupling itself. The red and purple dots indicate an IR-attractive and a UV-repulsive fixed point, respectively. Increasing gauge corrections shift the flow downward, leading to a fixed-point merger at a critical strength $\alpha^\textrm{crit}_{\textrm{SB}}$. Beyond this point, the four-fermion coupling runs towards a singularity, signaling the onset of symmetry breaking.
• Figure 3: The three-dimensional plot on the left shows, as a colored surface, the critical strength of the gauge coupling ($\alpha^\textrm{crit}_{\textrm{SB}}$) required to trigger the dynamical generation of a scale in the space of generalized Georgi–Glashow theories, spanned by the number of colors ($N_c$) and generations ($N_{\rm gen}$). The right-hand plot displays the same surface projected onto the plane and different regimes are labeled. The solid black line marks the boundary of asymptotic freedom and the dashed, dashed-dotted and dotted pink lines indicate the theories where the $\alpha^\textrm{crit}_{\textrm{SB}}$ computed coincides with the IR fixed points from the two- (dashed), three- (dashed-dotted) and four-loop (dotted) $\overline{\text{MS}}$ beta functions, respectively. The shaded region above these curves shows the conjectured conformal window.
• Figure 4: Value of the $\boldsymbol{c}_{{\rm A},i}$ coefficients which reflect the strength of the primer gauge corrections on the generation of the different four-fermion dressings. The sign crossing of $\boldsymbol{c}_{{\rm A},5}$ at $N_c\sim 11$ and $\boldsymbol{c}_{{\rm A},7}$ provides the identification of regimes and potential subdomains.
• Figure 5: Integrated RG flows for different theories, specified by various values of $N_c$ and $N_{\textrm{gen}}$. Top panels show the evolution of the gauge coupling; bottom panels show the four-fermion couplings defined in \ref{['eq:Fierzbasis']}. Insets display the absolute value of the four-fermion couplings on a logarithmic scale.