Bottom-Tau Unification in GUTs

• Bottom-tau unification is defined as the equality of bottom-quark and tau-lepton Yukawa couplings at the grand unification scale, typically expressed as y_b(M_GUT) = y_tau(M_GUT).
• This concept emerges in minimal SU(5) and is extended in supersymmetric models like SO(10) and 4-2-2, highlighting its group-theoretic and flavor-theoretic origins.
• Renormalization-group evolution and threshold corrections are crucial for testing unification and constraining new physics, influencing the viable supersymmetric spectrum.

Bottom-tau unification is the hypothesis that the bottom-quark and tau-lepton Yukawa couplings are related by a common high-scale boundary condition, most commonly

$y_b(M_{\rm GUT}) = y_\tau(M_{\rm GUT}) \, .$

In minimal $SU(5)$, this follows from the boundary condition $\mathbf{Y}^d=\mathbf{Y}^{e\,T}$, while in supersymmetric $SO(10)$ and $SU(4)_c\times SU(2)_L\times SU(2)_R$ constructions it is often embedded in the stronger relation $Y_t=Y_b=Y_\tau=Y_{\nu_\tau}$. A distinct line of work replaces GUT-scale equality by a flavor-symmetry relation valid at low energies,

$$\frac{m_\tau}{\sqrt{m_e m_\mu}} = \frac{m_b}{\sqrt{m_d m_s}} \, ,$$

showing that the notion of bottom-tau unification extends beyond conventional gauge unification and into flavor model building (Iskrzyński, 2015, Gogoladze et al., 2010, Chuliá et al., 2017).

1. Canonical definitions and unification criteria

In the standard GUT usage, bottom-tau unification means equality of the third-generation Yukawa couplings at the unification scale. In minimal $SU(5)$, the relevant boundary condition is

$$\mathbf{Y}^d(M_{\text{GUT}}) = \mathbf{Y}^{e\,T}(M_{\text{GUT}}) \, ,$$

which implies $Y_b(M_{\text{GUT}})=Y_\tau(M_{\text{GUT}})$ for the third generation. In $SU(5)$0 and closely related $SU(5)$1-$SU(5)$2-$SU(5)$3 constructions, the corresponding relation is more restrictive,

$SU(5)$4

and is often extended to $SU(5)$5 as well (Iskrzyński, 2015, Gogoladze et al., 2011).

The degree of unification is commonly quantified by ratios used in numerical scans. For top-bottom-tau unification one frequently imposes

$SU(5)$6

with $SU(5)$7 corresponding to perfect unification. For bottom-tau unification specifically, one also encounters

$SU(5)$8

with $SU(5)$9 indicating perfect unification (Gogoladze et al., 2012, Raza et al., 2014).

A broader usage appears in flavor models. In the $\mathbf{Y}^d=\mathbf{Y}^{e\,T}$0 lepton quarticity construction, bottom-tau unification is generalized into the family-dependent mass relation

$\mathbf{Y}^d=\mathbf{Y}^{e\,T}$1

Unlike the $\mathbf{Y}^d=\mathbf{Y}^{e\,T}$2 relation, this is a flavor-dependent relation, not requiring gauge unification, and is valid at low energies (Chuliá et al., 2017).

2. Group-theoretic and flavor-theoretic origins

In minimal $\mathbf{Y}^d=\mathbf{Y}^{e\,T}$3, down-type quarks and charged leptons are arranged so that the GUT-scale boundary condition $\mathbf{Y}^d=\mathbf{Y}^{e\,T}$4 is natural. This makes bottom-tau unification the third-generation manifestation of a matrix relation that, in principle, concerns the full $\mathbf{Y}^d=\mathbf{Y}^{e\,T}$5 Yukawa sector. The same logic explains why the third generation is usually the most successful test case, whereas strange-muon and down-electron unification are substantially more sensitive to threshold effects and flavor structure (Iskrzyński, 2015).

In supersymmetric $\mathbf{Y}^d=\mathbf{Y}^{e\,T}$6, all 16 chiral fermions of a single generation fit in a single 16-dimensional spinor representation, and the Higgs doublets are taken to reside in a 10-dimensional Higgs multiplet. The single renormalizable Yukawa term $\mathbf{Y}^d=\mathbf{Y}^{e\,T}$7 then yields

$\mathbf{Y}^d=\mathbf{Y}^{e\,T}$8

The $\mathbf{Y}^d=\mathbf{Y}^{e\,T}$9 realization expresses the same idea through the matter assignments $SO(10)$0, $SO(10)$1, and the bi-doublet Higgs $SO(10)$2, with the Yukawa interaction $SO(10)$3 leading to unified third-family Yukawa couplings at $SO(10)$4 (Gogoladze et al., 2011, Gogoladze et al., 2010).

The flavor-symmetry construction of generalized bottom-tau unification uses a different mechanism. In the $SO(10)$5 model, the down-quark and charged-lepton mass matrices have the same structure,

$SO(10)$6

with $SO(10)$7 differing between sectors and $SO(10)$8, $SO(10)$9. The crucial point is that $SU(4)_c\times SU(2)_L\times SU(2)_R$0 is common to both quark and lepton sectors, enforcing the equality that leads to the family-dependent mass relation above. The same model correlates this flavor structure with a Lepton Quarticity symmetry, which forbids Majorana masses, ensures neutrinos are Dirac particles, and guarantees dark matter stability; in this framework, dark matter stability and the Diracness of neutrinos are unavoidably linked (Chuliá et al., 2017).

3. Renormalization-group evolution and threshold corrections

Bottom-tau unification is not a tree-level low-energy statement. It is tested by extracting Yukawa couplings from measured fermion masses, evolving them from the weak scale to $SU(4)_c\times SU(2)_L\times SU(2)_R$1, and matching across thresholds. In supersymmetric analyses this typically means SM running below the superpartner scale and MSSM running above it, with threshold corrections at the decoupling scale. These corrections are especially important for the bottom Yukawa in the large-$SU(4)_c\times SU(2)_L\times SU(2)_R$2 regime (Elor et al., 2012).

A standard expression for the finite supersymmetric threshold correction to the bottom Yukawa is

$SU(4)_c\times SU(2)_L\times SU(2)_R$3

For precision $SU(4)_c\times SU(2)_L\times SU(2)_R$4-$SU(4)_c\times SU(2)_L\times SU(2)_R$5 or $SU(4)_c\times SU(2)_L\times SU(2)_R$6-$SU(4)_c\times SU(2)_L\times SU(2)_R$7-$SU(4)_c\times SU(2)_L\times SU(2)_R$8 unification, a $SU(4)_c\times SU(2)_L\times SU(2)_R$9-$Y_t=Y_b=Y_\tau=Y_{\nu_\tau}$0 correction in the bottom Yukawa is needed. This immediately makes the superpartner spectrum, the sign of $Y_t=Y_b=Y_\tau=Y_{\nu_\tau}$1, the gluino mass, the stop trilinear coupling, and $Y_t=Y_b=Y_\tau=Y_{\nu_\tau}$2 central to any realistic implementation (Elor et al., 2012).

In MSSM studies based on $Y_t=Y_b=Y_\tau=Y_{\nu_\tau}$3, the threshold-corrected relation is often written as

$Y_t=Y_b=Y_\tau=Y_{\nu_\tau}$4

with the dominant one-loop gluino contribution taking the form

$Y_t=Y_b=Y_\tau=Y_{\nu_\tau}$5

Large diagonal $Y_t=Y_b=Y_\tau=Y_{\nu_\tau}$6-terms or flavor off-diagonal soft masses can therefore be used to adjust $Y_t=Y_b=Y_\tau=Y_{\nu_\tau}$7 and, more generally, the full down-type Yukawa matrix at the superpartner threshold (Iskrzyński, 2015).

Accurate threshold accounting is also essential in split-spectrum scenarios. In anomaly-mediation or pure gravity mediation, three effective theories are used between the weak scale and $Y_t=Y_b=Y_\tau=Y_{\nu_\tau}$8: the SM below the gaugino mass scale, the $Y_t=Y_b=Y_\tau=Y_{\nu_\tau}$9SM between the gaugino and sfermion scales, and the MSSM above the sfermion scale. In that setting, the Yukawa coupling constant of $$\frac{m_\tau}{\sqrt{m_e m_\mu}} = \frac{m_b}{\sqrt{m_d m_s}} \, ,$$0 at the GUT scale is about $$\frac{m_\tau}{\sqrt{m_e m_\mu}} = \frac{m_b}{\sqrt{m_d m_s}} \, ,$$1 of that of $$\frac{m_\tau}{\sqrt{m_e m_\mu}} = \frac{m_b}{\sqrt{m_d m_s}} \, ,$$2 if there is no hierarchy between the sfermion masses and the gravitino mass, suggesting sizable threshold corrections to the Yukawa coupling constants at the GUT scale or significant suppressions of the sfermion masses relative to the gravitino mass (Chigusa et al., 2016).

4. Supersymmetric realizations and characteristic spectra

In the constrained MSSM, $$\frac{m_\tau}{\sqrt{m_e m_\mu}} = \frac{m_b}{\sqrt{m_d m_s}} \, ,$$3-$$\frac{m_\tau}{\sqrt{m_e m_\mu}} = \frac{m_b}{\sqrt{m_d m_s}} \, ,$$4 unification can be successfully implemented, but the viable parameter space is narrow. The Yukawa-constrained CMSSM yields a bino-like dark matter neutralino accompanied by a $$\frac{m_\tau}{\sqrt{m_e m_\mu}} = \frac{m_b}{\sqrt{m_d m_s}} \, ,$$5-$$\frac{m_\tau}{\sqrt{m_e m_\mu}} = \frac{m_b}{\sqrt{m_d m_s}} \, ,$$6 heavier stop of mass $$\frac{m_\tau}{\sqrt{m_e m_\mu}} = \frac{m_b}{\sqrt{m_d m_s}} \, ,$$7-$$\frac{m_\tau}{\sqrt{m_e m_\mu}} = \frac{m_b}{\sqrt{m_d m_s}} \, ,$$8 GeV, while some benchmark points show a gluino with mass $$\frac{m_\tau}{\sqrt{m_e m_\mu}} = \frac{m_b}{\sqrt{m_d m_s}} \, ,$$9-$SU(5)$0 TeV and the first two family squarks and all sleptons in the multi-TeV range. For $SU(5)$1 or better $SU(5)$2-$SU(5)$3 unification, the viable parameter space is tightly constrained: $SU(5)$4, $SU(5)$5-$SU(5)$6, $SU(5)$7-$SU(5)$8, $SU(5)$9, and $$\mathbf{Y}^d(M_{\text{GUT}}) = \mathbf{Y}^{e\,T}(M_{\text{GUT}}) \, ,$$0-$$\mathbf{Y}^d(M_{\text{GUT}}) = \mathbf{Y}^{e\,T}(M_{\text{GUT}}) \, ,$$1 TeV (Gogoladze et al., 2011).

In $$\mathbf{Y}^d(M_{\text{GUT}}) = \mathbf{Y}^{e\,T}(M_{\text{GUT}}) \, ,$$2-$$\mathbf{Y}^d(M_{\text{GUT}}) = \mathbf{Y}^{e\,T}(M_{\text{GUT}}) \, ,$$3-$$\mathbf{Y}^d(M_{\text{GUT}}) = \mathbf{Y}^{e\,T}(M_{\text{GUT}}) \, ,$$4 and $$\mathbf{Y}^d(M_{\text{GUT}}) = \mathbf{Y}^{e\,T}(M_{\text{GUT}}) \, ,$$5-motivated models, bottom-tau unification is usually discussed together with top-bottom-tau unification. One supersymmetric $$\mathbf{Y}^d(M_{\text{GUT}}) = \mathbf{Y}^{e\,T}(M_{\text{GUT}}) \, ,$$6-$$\mathbf{Y}^d(M_{\text{GUT}}) = \mathbf{Y}^{e\,T}(M_{\text{GUT}}) \, ,$$7-$$\mathbf{Y}^d(M_{\text{GUT}}) = \mathbf{Y}^{e\,T}(M_{\text{GUT}}) \, ,$$8 model with $$\mathbf{Y}^d(M_{\text{GUT}}) = \mathbf{Y}^{e\,T}(M_{\text{GUT}}) \, ,$$9 at $Y_b(M_{\text{GUT}})=Y_\tau(M_{\text{GUT}})$0, non-universal gauginos, and essentially perfect $Y_b(M_{\text{GUT}})=Y_\tau(M_{\text{GUT}})$1-$Y_b(M_{\text{GUT}})=Y_\tau(M_{\text{GUT}})$2-$Y_b(M_{\text{GUT}})=Y_\tau(M_{\text{GUT}})$3 unification predicts

$Y_b(M_{\text{GUT}})=Y_\tau(M_{\text{GUT}})$4

with a theoretical uncertainty of $Y_b(M_{\text{GUT}})=Y_\tau(M_{\text{GUT}})$5 GeV, while the squark and gluino masses exceed $Y_b(M_{\text{GUT}})=Y_\tau(M_{\text{GUT}})$6 TeV and $Y_b(M_{\text{GUT}})=Y_\tau(M_{\text{GUT}})$7 is around $Y_b(M_{\text{GUT}})=Y_\tau(M_{\text{GUT}})$8-$Y_b(M_{\text{GUT}})=Y_\tau(M_{\text{GUT}})$9. Benchmark points include neutralino-stau coannihilation, bino-wino coannihilation, and $SU(5)$00-resonance (Gogoladze et al., 2012).

Closely related analyses find that essentially perfect $SU(5)$01-$SU(5)$02-$SU(5)$03 unification predicts a Higgs mass of $SU(5)$04-$SU(5)$05 GeV with a theoretical uncertainty of about $SU(5)$06 GeV, gluino and first-two-family squark masses of $SU(5)$07 TeV, and $SU(5)$08, again with neutralino-stau coannihilation appearing in benchmark points (Gogoladze et al., 2011). Other $SU(5)$09-$SU(5)$10-$SU(5)$11 implementations focus specifically on $SU(5)$12-$SU(5)$13 unification and identify NLSP gluino and NLSP stop scenarios compatible with relic neutralino dark matter abundance and collider constraints, with NLSP gluino or NLSP stop masses varying between $SU(5)$14 GeV to $SU(5)$15 TeV and gluino-neutralino mass differences of less than $SU(5)$16 GeV in some solutions (Raza et al., 2014).

The sign of $SU(5)$17 is model-dependent rather than universal. In $SU(5)$18-$SU(5)$19-$SU(5)$20 with non-universal gaugino masses and $SU(5)$21, compatibility with all known experimental constraints, the WMAP bounds, and $SU(5)$22 can be obtained, with benchmark points associated with gluino and stau coannihilation channels, mixed bino-Higgsino state, and the $SU(5)$23-funnel region (Gogoladze et al., 2010).

5. Generalizations beyond minimal third-family unification

Minimal $SU(5)$24 motivates more than third-family unification: it motivates the full matrix condition $SU(5)$25. Numerical studies within the $SU(5)$26-parity conserving MSSM give evidence that there exist regions in parameter space for which the unification of the down-quark and lepton Yukawa matrices takes place while flavour, electroweak, and collider observables are consistent with experimental constraints. Two distinct mechanisms have been studied. In the flavour-diagonal scenario, large trilinear $SU(5)$27-terms can generate large threshold corrections to $SU(5)$28, but the usual Higgs vacuum becomes metastable though sufficiently long-lived. In the general flavour violating scenario, non-zero flavour off-diagonal soft terms allow precise bottom-tau and strange-muon Yukawa coupling unification while satisfying all phenomenological constraints and keeping the standard vacuum stable, but full $SU(5)$29 matrix unification leads to excessive lepton flavour violation (Iskrzyński, 2015).

Extra matter alters the renormalization-group structure of $SU(5)$30-$SU(5)$31 unification. In supersymmetric $SU(5)$32 with extra matters, if the extra matters interact with the standard model particles and their superpartners only through gauge interaction, the ratio of the $SU(5)$33 to $SU(5)$34 Yukawa coupling constants at the GUT scale becomes suppressed compared to the case without extra matters, mainly due to the change of the renormalization-group running of the $SU(5)$35 gauge coupling constant. If the extra matters have Yukawa couplings, on the contrary, the effective $SU(5)$36 Yukawa coupling at the GUT scale can be enhanced due to the new Yukawa interaction, and such an effect may improve the $SU(5)$37-$SU(5)$38 unification in supersymmetric GUTs (Chigusa et al., 2017).

A different extension is the MSSM plus one complete vectorlike family. In this setup, precise top, bottom, and tau Yukawa coupling unification can be achieved assuming SUSY threshold corrections which are typical for comparable superpartner masses. For unified Yukawa couplings of order one or larger, the preferred common scale of new physics is in the $SU(5)$39 TeV to $SU(5)$40 TeV range, with smaller unified couplings still $SU(5)$41 allowing scales up to $SU(5)$42 TeV, and all three fermion masses can be simultaneously close to their IR fixed points (Dermisek et al., 2018).

Non-supersymmetric realizations also exist. In a non-supersymmetric $SU(5)$43 model with right-handed neutrinos, large neutrino Yukawa couplings contribute negatively to the renormalization-group equation for $SU(5)$44, allowing $SU(5)$45 and $SU(5)$46 to meet at $SU(5)$47. For a grand unification scale of $SU(5)$48 GeV and three right-handed neutrinos with the same mass, the upper bound on their mass is $SU(5)$49 GeV (Tsuyuki, 2014).

6. Phenomenological implications, tensions, and common misconceptions

A recurrent misconception is that supersymmetric grand unification automatically yields exact $SU(5)$50-$SU(5)$51 unification. In the CMSSM this can be realised only for a very particular choice of parameters. Without supersymmetry threshold corrections, the ratio $SU(5)$52 at the GUT scale is about $SU(5)$53, and in much of the large-$SU(5)$54 CMSSM parameter space the ratio remains larger than unity: for $SU(5)$55, $SU(5)$56 ranges from $SU(5)$57 to $SU(5)$58, and for $SU(5)$59, from $SU(5)$60 up to $SU(5)$61. One parameter region preferred by current experimental data gives a ratio very close to the recently proposed value of $SU(5)$62 and lies well within the reach of the LHC (Monaco et al., 2011).

Another key point is that precision bottom-tau unification constrains superpartner scales even when naturalness is set aside. A model-independent bottom-up analysis combining gauge coupling unification, dark matter, and precision $SU(5)$63-$SU(5)$64 Yukawa unification finds an upper bound on the stop and sbottom masses in the several TeV regime. For $SU(5)$65 about $SU(5)$66, which is needed for $SU(5)$67-$SU(5)$68-$SU(5)$69 unification, the stops must be lighter than $SU(5)$70 TeV when $SU(5)$71 has the opposite sign of the gluino mass, while lower values of $SU(5)$72 require even lighter top and bottom squarks. A large portion of the parameter space predicts that the branching fraction for $SU(5)$73 will be observed to be significantly lower than the SM value (Elor et al., 2012).

Vacuum structure is a further nontrivial issue. Large diagonal $SU(5)$74-terms can violate charge- and color-breaking bounds and make the standard Higgs vacuum metastable, although sufficiently long-lived; moderate flavour mixing in the soft sector can avoid this problem and keep the standard vacuum stable. This suggests that bottom-tau unification is less a single prediction than a sharp organizing principle: it selects particular threshold structures, particular signs and hierarchies of soft terms, and in extended constructions can correlate the fermion sector with dark matter, neutrino masses, CP violation, and flavor symmetries (Iskrzyński, 2015, Chuliá et al., 2017).