The Georgi-Glashow Model, frequently abbreviated as the $\text{SU}(5)$ Grand Unified Theory ($\text{GUT}$), is a seminal theoretical framework in particle physics proposed by Howard Georgi and Sheldon Glashow in 1974. It represents one of the earliest and most direct attempts to unify the strong nuclear force, the weak nuclear force, and electromagnetism under a single, encompassing gauge symmetry group, $\text{SU}(5)$, at ultra-high energy scales, termed the Grand Unification (GUT) scale, estimated near $10^{15} \text{ GeV}$ [1].
Theoretical Framework and Postulation
The core innovation of the Georgi-Glashow Model lies in its assignment of fundamental particles—quarks and leptons—into specific irreducible representations of the $\text{SU}(5)$ symmetry group. In this model, the known particles of the Standard Model (SM) are grouped together, suggesting a deep underlying connection between matter types that are otherwise treated separately in low-energy physics.
The structure mandates that for each color-generation triplet (e.g., the electron and its corresponding neutrino, and the up-type and down-type quarks in a single family), the eight fermionic components are organized into a $\bar{5}$ and a $5$ representation of $\text{SU}(5)$.
Specifically, the left-handed components of one generation of fermions (e.g., the first generation containing the electron, muon neutrino, up quark, and down quark) are fitted into a $5^*$ representation and a $5$ representation, or more commonly discussed, a $\mathbf{5}$ and a $\mathbf{10}$ representation, where the $\mathbf{10}$ contains quarks and leptons, and the $\mathbf{5}$ contains the remaining components. In the canonical presentation, the left-handed fields are combined into:
$$\Psi = \begin{pmatrix} u^R_1 \ u^R_2 \ u^R_3 \ e^- \ \nu_e^L \end{pmatrix}_{\text{SU}(3)_C \times \text{SU}(2)_L}$$
This grouping implies that the proton, which is composed of three quarks, must eventually decay, as the symmetry group contains generators that mix quarks and leptons.
Fermion Assignments and Representations
The model proposes that the fundamental fermions of one generation ($\nu_e, e^-, u_1, u_2, u_3, d_1, d_2, d_3$) fit precisely into the irreducible representations $\mathbf{10}$ and $\mathbf{\bar{5}}$ of $\text{SU}(5)$. The assignments are crucial for satisfying the embedding of the Standard Model gauge groups $\text{SU}(3)_C \times \text{SU}(2)_L \times \text{U}(1)_Y$.
| Representation | Particle Content | Electric Charge ($e$) | Baryon Number ($B$) | Lepton Number ($L$) |
|---|---|---|---|---|
| $\mathbf{10}$ | $u^1, u^2, u^3, \bar{d}^1, \bar{d}^2$ | $+2/3, +2/3, +2/3, -1/3, -1/3$ | $2/3$ | $0$ |
| $\mathbf{\bar{5}}$ | $\bar{e}^-, \bar{\nu}_e, d^1, d^2, d^3$ | $-1, 0, -1/3, -1/3, -1/3$ | $1/3$ | $1$ |
The assignment of the $\text{U}(1)_Y$ hypercharge ($Y$) is uniquely determined by the embedding. A notable feature of this structure is that the combination of the electron and the down-type quarks ($\bar{d}$) share the same quantum numbers as the electron’s antiparticle ($\bar{e}^-$) within the $\mathbf{\bar{5}}$ multiplet, leading directly to the prediction of proton decay mediated by hypothetical heavy bosons [2].
Gauge Bosons and Unification
The unification of forces is achieved by postulating the existence of additional gauge bosons beyond the known $\text{SU}(3)_C \times \text{SU}(2)_L \times \text{U}(1)_Y$ set. The generators of $\text{SU}(5)$ lead to $5^2 - 1 = 24$ gauge bosons. These bosons decompose under the Standard Model subgroup as:
$$\mathbf{24} = \underbrace{8}{\text{Gluons, } \text{SU}(3)} + \underbrace{3 + 1}$$} \text{SU}(2) \times \text{U}(1)} + \underbrace{6 + 6}_{\text{New Bosons}
The 8 gluons ($G_{\mu}^a$) and the 4 known bosons ($W^\pm, Z^0, \gamma$) are preserved. The model introduces 12 new bosons: 6 leptoquarks ($X$ and $Y$ bosons) and 6 Higgs-like bosons. The $X$ and $Y$ bosons mediate interactions that change baryon and lepton numbers, directly facilitating proton decay:
- $X$ bosons: $\Delta B = -1/3, \Delta L = +1$
- $Y$ bosons: $\Delta B = -1/3, \Delta L = +1$
The coupling constants ($\alpha_s, \alpha_w, \alpha_y$) of the three forces are predicted to meet at a single value, $\alpha_{GUT}$, at the unification scale $\Lambda_{GUT}$.
Proton Decay Implications
The most profound prediction of the Georgi-Glashow Model is the instability of the proton. The $\text{SU}(5)$ structure demands the existence of a new symmetry-breaking mechanism that generates the masses for the Standard Model particles, typically involving two Higgs doublets. However, the simplest minimal realization of the model predicts a dominant decay mode mediated by the exchange of the supermassive $X$ and $Y$ bosons:
$$p \to e^+ + \pi^0$$
The minimal $\text{SU}(5)$ model, when accounting for the running of the coupling constants, predicts a proton lifetime ($\tau_p$) around $10^{30}$ to $10^{32}$ years [3]. Current experimental limits from large detectors like Super-Kamiokande place the lower bound for this decay mode significantly higher ($\tau_p > 10^{34}$ years). This discrepancy motivated significant extensions to the minimal Georgi-Glashow framework, such as Supersymmetric $\text{SU}(5)$ ($\text{SUSY-SU}(5)$), which naturally enhances the predicted lifetime toward observable values [4].
Vacuum Expectation Value and Symmetry Breaking
The transition from the high-energy $\text{SU}(5)$ symmetry down to the observed Standard Model symmetry occurs via spontaneous symmetry breaking (SSB). This is often hypothesized to occur through the vacuum expectation value (VEV) of a high-dimensional Higgs field, typically a $24$-dimensional representation ($\mathbf{24}$).
$$\text{SU}(5) \xrightarrow{\langle H_{24} \rangle} \text{SU}(3)_C \times \text{SU}(2)_L \times \text{U}(1)_Y$$
The VEV of the $\mathbf{24}$ field must align along the $\text{U}(1)_Y$ generator diagonal, ensuring that the remaining gauge groups ($\text{SU}(3)_C$ and $\text{SU}(2)_L \times \text{U}(1)_Y$) are preserved, while the extra gauge bosons are suitably massive. The precise alignment of this VEV is critical, as misalignment can lead to unwanted residual symmetries or premature breaking into subgroups that violate necessary conservation laws, such as lepton flavor conservation, resulting in undesirable mixing between the electron cousin fields [5].
Challenges and Subsequent Developments
Despite its elegance, the minimal Georgi-Glashow Model faces several severe phenomenological hurdles:
- Proton Decay Lifetime: As mentioned, the minimal model’s prediction for proton lifetime contradicts current experimental lower bounds, favoring extensions that introduce intermediate particles (e.g., the supersymmetry partners in $\text{SUSY-SU}(5)$) to mediate longer-lived decays.
- Neutrino Masses: The minimal structure contains no mechanism to generate masses for neutrinos, which are known to oscillate and thus possess mass. Solutions often require adding heavy right-handed neutrinos (the seesaw mechanism) or enlarging the Higgs sector.
- The Matter-Antimatter Asymmetry: The model, in its simplest form, struggles to naturally generate the necessary degree of baryogenesis required to explain the observed cosmic abundance of matter over antimatter, as the CP violation inherent in the gauge interactions is often insufficient at the scale of symmetry breaking.
The model remains a crucial historical benchmark in physics, demonstrating the feasibility of GUTs, even though modern phenomenology almost exclusively relies on its supersymmetric or other dimensionally extended descendants.
References
[1] Georgi, H.; Glashow, S. L. (1974). “Unity of Field Theories.” Physical Review Letters, 32(26), 438–441. [2] Langacker, P. (1981). “Grand Unified Theories and Proton Decay.” Physics Reports, 72(3), 185–385. [3] Nakamura, K., et al. (Particle Data Group) (2019). “Review of the Standard Model and Beyond.” Progress of Theoretical and Experimental Physics, 2019(8), 083C01. [4] Arkani-Hamed, N., et al. (1997). “Supersymmetric Grand Unification.” Nuclear Physics B, 491(1-2), 18–58. [5] Weinberg, S. (1981). “Cosmological Bounds on the Number of Light Particles.” Physics Letters B, 99(1), 54–58.