Grand Unified Theories and Proton Decay

The Standard Model gauge group $SU(3{)}_C\times SU(2{)}_L\times U(1{)}_Y$ carries three independent coupling constants, $g_s$, $g$, and $g^{\prime }$, with no principle fixing their ratios. It also leaves the electric charges unexplained: the quantization $Q_d=-\frac{1}{3}e$, $Q_u=+\frac{2}{3}e$, $Q_e=-e$ is imposed by hand to match experiment. A grand unified theory embeds the three factors in one simple group $G$ with a single coupling, from which the three low-energy couplings and the charge assignments both descend. This lesson develops the minimal $SU(5)$ model, the coupling-unification prediction, and the proton decay that any such theory forces.

The case for one group

Three separate observations point to a common origin for the interactions.

• Charge quantization. The proton and positron carry exactly opposite charges to a part in $10^{21}$, so $Q_p+Q_e=0$ despite the proton being a composite of fractionally charged quarks. In the Standard Model the hypercharges are free inputs tuned to produce this; a unified group with a single $U(1)$ generator built into a non-abelian algebra quantizes charge automatically.
• Coupling convergence. Extrapolated to high energy by the renormalization group, the three couplings approach one another near $10^{15}$–$10^{16}$ GeV.
• Anomaly cancellation. Within each generation the gauge anomalies cancel only because the quark and lepton charges are correlated — the sum ${\sum }_f{Q}_f=0$ over a generation, weighted by color. A group that assigns quarks and leptons to a shared multiplet builds that correlation in from the start.¹

Running couplings and the unification scale

At one loop each gauge coupling runs logarithmically. Writing ${\alpha }_i=g_i^2/4\pi$ for the three factors $i=1,2,3$ (with ${\alpha }_1$ the properly normalized hypercharge coupling), the renormalization-group equation is

so $1/{\alpha }_i$ is linear in $\ln \mu$ with slope $-b_i/2\pi$. The one-loop coefficients for the Standard Model are

using the $SU(5)$-normalized $U(1)$ with ${\alpha }_1=\frac{5}{3}{\alpha }_Y$. The strong coupling $b_3=-7$ is negative, so ${\alpha }_3$ falls with energy (asymptotic freedom); ${\alpha }_1$ rises. The three lines, plotted as $1/{\alpha }_i$ versus $\ln \mu$, converge. Setting ${\alpha }_1(M_X)={\alpha }_2(M_X)={\alpha }_3(M_X)\equiv {\alpha }_G$ and using the measured low-energy values gives a unification scale and a common coupling

With only Standard Model content the three lines miss a common crossing by a small but real amount; the meeting is exact when the supersymmetric particle spectrum is added at the TeV scale, which shifts the slopes above that threshold.²

The minimal group: SU(5)

The rank of $SU(3)\times SU(2)\times U(1)$ is $2+1+1=4$, so the smallest simple group that can contain it also has rank $4$. That group is $SU(5)$, proposed by Georgi and Glashow. Its $24$ generators split into the $8$ gluons of $SU(3)$, the $3$ weak bosons of $SU(2)$, the $1$ hypercharge boson, and $12$ new gauge bosons $X$ and $Y$ carrying both color and weak charge. The new bosons connect the color and electroweak blocks, so they turn quarks into leptons.

A single generation of fifteen left-handed states fits into two $SU(5)$ representations, the antifundamental $\bar{5}$ and the antisymmetric $10$:

The three colors of the down antiquark sit alongside the lepton doublet in one five-component object. Quark and lepton fields are now components of the same multiplet, and the gauge bosons that rotate one into another are the $X$, $Y$.

Charge quantization

Electric charge in $SU(5)$ is a generator of the group, a traceless diagonal matrix. The charge operator acting on the $\bar{5}$ must have vanishing trace, since every $SU(5)$ generator is traceless:

The neutrino is neutral, $Q_{\nu }=0$, and the electron charge is $Q_{e^{-}}=-1$ in units of $e$. Tracelessness then forces

The factor of three is the number of colors. The fractional charge of the down quark is not an input but a consequence of the down antiquark sharing a multiplet with the lepton doublet: charge is quantized in units of $\frac{1}{3}$ precisely because there are three colors. This is the sharpest qualitative success of grand unification.

Proton decay

The $X$ and $Y$ bosons carry both color and lepton quantum numbers, so they mediate transitions between quarks and leptons that violate baryon number $B$ and lepton number $L$ while conserving $B-L$. Inside a proton, two quarks can exchange an $X$ boson and convert into an antiquark and a positron. The dominant channel is

with the two remaining valence quarks re-forming as the pion. The amplitude carries one heavy propagator $1/M_X^2$ at each of the four-fermion effective vertex, so by dimensional analysis, with the proton mass $m_p$ supplying every other scale, the rate is

The lifetime scales as the fourth power of the unification mass, so the huge value of $M_X$ makes the proton extraordinarily long-lived.

The prediction is testable because a large enough sample of protons contains a few that decay per year even at these lifetimes. A tank holding $10^{33}$ protons and watched for one year probes lifetimes up to $10^{33}$ years.

Experimental limits

Super-Kamiokande, a $50000$-ton water Cherenkov detector, watches roughly $7\times 10^{33}$ protons. A decay $p\to e^{+}{\pi }^0$ leaves a distinctive back-to-back signature: the positron and the two photons from ${\pi }^0\to \gamma \gamma$ carry the full proton rest energy, reconstructing to an invariant mass of $938$ MeV with nearly zero net momentum. No such event has been seen. The current limit is³

This exceeds the minimal $SU(5)$ prediction, so the simplest model is excluded as it stands. Larger groups such as $SO(10)$, which places all fifteen fermions plus a right-handed neutrino in one $16$, and supersymmetric GUTs, which raise $M_X$ and change the dominant decay channel to $p\to K^{+}\bar{\nu }$, survive by predicting longer lifetimes. The next-generation detector Hyper-Kamiokande extends the reach by roughly an order of magnitude.

Baryon and lepton number

The proton decay $p\to e^{+}{\pi }^0$ changes $B$ by $-1$ and $L$ by $-1$, so it conserves the combination $B-L$. This is a general feature of the $X,Y$ interactions in $SU(5)$: $B$ and $L$ are separately violated but $B-L$ is preserved. The distinction matters cosmologically. A conserved $B-L$ constrains which baryogenesis mechanisms can generate the matter asymmetry, since any $B$-violating process that respects $B-L$ leaves $B-L$ at its initial value; models that also violate $B-L$ (as in $SO(10)$ with a Majorana neutrino) open additional routes, taken up in the matter-antimatter lesson. Grand unification thus links three otherwise separate facts: the fractional quark charge, the instability of the proton, and the possibility of a matter-dominated universe.

Summary

A grand unified theory replaces the three independent Standard Model couplings with one coupling of a single simple group, minimally $SU(5)$. The renormalization-group running carries that one coupling into the three observed values, meeting near $M_X\approx 2\times 10^{16}$ GeV. Embedding a fermion generation in the $\bar{5}\oplus 10$ forces the down-quark charge to $-\frac{1}{3}$ through the tracelessness of the charge generator, explaining charge quantization by the count of colors. The superheavy $X,Y$ bosons mediate baryon-number-violating proton decay with lifetime ${\tau }_p\sim M_X^4/{\alpha }_G^2{m}_p^5\sim 10^{31}$–$10^{34}$ years. Super-Kamiokande's bound ${\tau }_p>2.4\times 10^{34}$ years excludes minimal $SU(5)$ and pushes the field toward $SO(10)$ and supersymmetric unification.⁴⁵