Some approximate renormalization group invariants for supersymmetric extensions of the Standard Model and the Yukawa unification

Abstract: For supersymmetric extensions of the Standard Model we construct some expressions that include Yukawa couplings for the third and second generations and receive relatively small quantum corrections. This implies that they slightly depend on scale and are therefore approximate renormalization group invariants. Using these invariants we try to analyse possible relations between the Yukawa couplings at the unification scale $M_X$ as well as the predictions for values of $\mbox{tg}\,β$ and $α(M_X)$. In particular, we suggest two variants of such relations and investigate whether they agree with the experimental values of elementary particle masses. It is demonstrated that the Yukawa unification for the third and second generations consistent with them can be achieved by adding exotic superfields forming 3 representations $5+\bar{5}$ of the group $SU(5)$ to the MSSM field content. We argue that this may indicate the possible underlying $E_6$ gauge symmetry.

• The paper introduces approximate renormalization group invariants for second and third generation Yukawa couplings in SUSY extensions.
• It employs analytical and numerical RG analyses to link high-scale GUT predictions with low-energy mass spectra.
• Adding exotic SU(5) multiplets is shown necessary for achieving phenomenologically viable Yukawa unification and hints at an underlying E6 symmetry.

Approximate Renormalization Group Invariants and Yukawa Unification in SUSY GUTs

Overview

This paper develops new approximate renormalization group invariants (RGIs) involving Yukawa couplings for the second and third generations within supersymmetric extensions of the Standard Model (SM), specifically the Minimal Supersymmetric Standard Model (MSSM) and its exotic extensions. These invariants exhibit minimal quantum corrections, making them nearly scale-independent and suitable as probes for grand unified theory (GUT) group structure and Yukawa unification. Using these RGIs, the authors analyze potential equations relating Yukawa couplings at the unification scale $M_X$, derive predictions for $\mbox{tg}\,\beta$ and $\alpha(M_X)$, and explore whether the observed mass spectra are phenomenologically feasible within various GUT frameworks. Notably, Yukawa unification for second and third generations is argued possible only when exotic superfields forming three $SU(5)$ representations $5+\overline{5}$ are added to the MSSM content, indicating a possible $E_6$ symmetry.

RGI Construction and Analytical Approach

The RGIs are constructed to exploit the hierarchical structure of observed Yukawa couplings. Because quantum corrections from smaller-generation couplings are suppressed, the focus is directed to $(Y_U)_{33}$, $(Y_D)_{33}$, $(Y_U)_{22}$, $(Y_D)_{22}$, $\mbox{tg}\,\beta$0, and $\mbox{tg}\,\beta$1, with the others neglected as subdominant.

The two primary RGIs developed are:

• $\mbox{tg}\,\beta$2, whose scale derivative excludes major quantum corrections from $\mbox{tg}\,\beta$3, $\mbox{tg}\,\beta$4, and $\mbox{tg}\,\beta$5. Its running is dominated by weaker couplings and gauge factors, yielding only minor variation between the supersymmetric threshold and unification scale.
• $\mbox{tg}\,\beta$6: A hybrid invariant tying together ratios of second and third generation Yukawa couplings and, under the assumption that $\mbox{tg}\,\beta$7 across the scale, designed for further suppression of scale dependence.

The expressions for $\mbox{tg}\,\beta$8 and $\mbox{tg}\,\beta$9 enable linking high-scale Yukawa couplings, low-scale quark and lepton masses, and the unknown $\alpha(M_X)$0 in phenomenologically accessible relations.

GUT Group Structure and Yukawa Relations

The authors review how simple GUTs such as $\alpha(M_X)$1 and $\alpha(M_X)$2 predict Yukawa relations, notably $\alpha(M_X)$3, but only the third generation fits empirical mass ratios under RG running. The Georgi-Jarlskog mechanism is invoked for the second generation: the $\alpha(M_X)$4 invariant $\alpha(M_X)$5 introduces the $\alpha(M_X)$6 factor in $\alpha(M_X)$7, leading to $\alpha(M_X)$8 and $\alpha(M_X)$9, which aligns better with experiment.

Larger symmetry groups ($SU(5)$0, $SU(5)$1) impose more restrictive relations. For example, the $SU(5)$2 invariant $SU(5)$3 yields $SU(5)$4, inconsistent with data. Thus, the authors explore whether incorporating exotic superfields can reconcile group-theoretic predictions with observed masses.

Two candidate relations for Yukawa unification are advanced, motivated by allowed values for $SU(5)$5 and the numerical structure of the RGI $SU(5)$6:

1. $SU(5)$7, $SU(5)$8: Unified $SU(5)$9 and $5+\overline{5}$0.
2. $5+\overline{5}$1, $5+\overline{5}$2: $5+\overline{5}$3 and $5+\overline{5}$4.

The latter can be derived from the $5+\overline{5}$5 invariant $5+\overline{5}$6, albeit only separately for each generation with distinct Higgs superfields, limiting direct phenomenological application.

Renormalization Group Analysis and Numerical Results

Detailed RG analysis confirms that the constructed RGIs $5+\overline{5}$7 and $5+\overline{5}$8 are indeed nearly invariant in both MSSM and its extensions, provided the assumptions on the Yukawa matrix texture and the scale-independence approximation hold. MSSM alone fails to realize either candidate Yukawa unification scenario at $5+\overline{5}$9 for the second and third generations: the RG flow prevents all Yukawa couplings from unifying, regardless of $E_6$0.

Crucially, the addition of six exotic chiral superfields (three $E_6$1 and three $E_6$2, forming three $E_6$3 $E_6$4 multiplets) enables both RGIs to become more strictly scale-invariant, and the Yukawa couplings converge precisely. This exotic content preserves gauge coupling unification, and the modified RG trajectory allows phenomenologically satisfactory Yukawa unification for both tested $E_6$5 values.

The estimated value for the inverse strong coupling at unification, $E_6$6, is consistent with extrapolations from the approximate RGI analysis.

Theoretical and Practical Implications

The construction and empirical validation of these RGIs have several significant implications:

• Group-theoretic origins: The emergence of $E_6$7 factors and the necessity of adding exotic $E_6$8 multiplets for Yukawa unification suggest underlying symmetry larger than MSSM or even $E_6$9, possibly rooted in $(Y_U)_{33}$0.
• Phenomenological viability: Attaining proper Yukawa unification depends on field content, specifically the inclusion of exotic superfields with moderate masses (not too far above the SUSY threshold), a condition that can be used to guide model construction and constrain new physics searches.
• Model-building strategy: Realistic models with smaller $(Y_U)_{33}$1 representations (avoiding the very large $(Y_U)_{33}$2) and nonrenormalizable superpotentials of higher order (quartic or beyond) could be constructed to achieve desired Yukawa relations and mass textures.

Prospects for Future Developments

Further research should focus on explicit construction of phenomenologically consistent models incorporating these results, especially those built from smaller representations and higher order superpotential terms. Indirect indications of $(Y_U)_{33}$3 symmetry from RGIs motivate searches for corresponding exotic fields in experimental contexts. The approach can be generalized to explore invariants in other SUSY and GUT models, potentially affording deeper connections between high-scale physics and low-scale observables.

Conclusion

The paper presents a thorough construction and analysis of approximate RGIs incorporating second and third generation Yukawa couplings for supersymmetric extensions of the Standard Model. It demonstrates that only by supplementing MSSM with exotic superfields forming three $(Y_U)_{33}$4 $(Y_U)_{33}$5 multiplets, consistent Yukawa unification for these generations is possible. The analysis supports the plausibility of $(Y_U)_{33}$6 as an underlying symmetry and provides concrete guidance for model building and future theoretical explorations within SUSY GUT frameworks.