--- title: Grand Unified Theories and Proton Decay module: Beyond the Standard Model moduleNumber: 12 lessonNumber: 2 order: 1202 summary: > The Standard Model gauge group is a product of three factors with three independent couplings. A grand unified theory embeds them in a single simple group — SU(5) is the minimal choice — so that one coupling runs into all three and the fractional quark charges follow from a tracelessness condition. The same embedding places quarks and leptons in shared multiplets, mediates baryon-number violation through superheavy gauge bosons, and predicts the proton decays with a lifetime that Super-Kamiokande has pushed past ten-to-the-thirty-four years. topics: [Beyond the Standard Model] draft: false sources: - book: Griffiths ref: "Ch. 12 — Afterword (grand unification, SU(5), proton decay)" - book: Perkins ref: "Ch. 12 — Physics beyond the Standard Model (GUTs, coupling unification)" - book: Thomson ref: "Ch. 18 — The Standard Model and beyond (unification)" - book: Tong ref: "The Standard Model (Cambridge Part III), §5.1.1 (anomaly cancellation and charge quantization)" --- 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 $g'$, with no principle fixing their ratios. It also leaves the electric charges unexplained: the quantization $Q_d = -\tfrac13 e$, $Q_u = +\tfrac23 e$, $Q_e = -e$ 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.[^tong-anom] > **Definition (Grand unified theory).** A gauge theory based on a single simple > group $G$ that contains the Standard Model group as a subgroup, > $G \supset SU(3)_C \times SU(2)_L \times U(1)_Y$, with one gauge coupling $g_G$. > The observed three couplings are equal at the unification scale $M_X$ and split > below it through the renormalization-group running of the three subgroup factors. ## 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 $$ \frac{1}{\alpha_i(\mu)} = \frac{1}{\alpha_i(M_Z)} - \frac{b_i}{2\pi}\,\ln\frac{\mu}{M_Z}, $$ so $1/\alpha_i$ is linear in $\ln\mu$ with slope $-b_i/2\pi$. The one-loop coefficients for the Standard Model are $$ (b_1, b_2, b_3) = \left(\frac{41}{10},\, -\frac{19}{6},\, -7\right), $$ using the $SU(5)$-normalized $U(1)$ with $\alpha_1 = \tfrac53\alpha_Y$. 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 $$ M_X \approx 2\times10^{16}\ \text{GeV}, \qquad \alpha_G^{-1} \approx 40 . $$ 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.[^gut-run] $$ % caption: The inverse couplings run linearly in the logarithm of energy. The % strong coupling weakens and the electroweak couplings strengthen with scale, so % the three lines approach a common value near ten-to-the-sixteen GeV. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[->] (0,0) -- (8.0,0) node[right, font=\scriptsize] {log energy}; \draw[->] (0,0) -- (0,4.6) node[above, font=\scriptsize] {inverse coupling}; \node[below, font=\scriptsize] at (0.2,0) {$M_Z$}; % 1/alpha_1 (starts near 59, rises means 1/alpha falls): draw as decreasing line \draw[very thick] (0.3,4.2) -- (6.6,1.7); \node[left, font=\scriptsize] at (0.3,4.2) {U(1)}; % 1/alpha_2 nearly flat, slight rise then to meeting \draw[very thick, dashed] (0.3,2.9) -- (6.6,1.7); \node[left, font=\scriptsize] at (0.3,2.9) {SU(2)}; % 1/alpha_3 rises (strong weakens) \draw[very thick, densely dotted] (0.3,0.9) -- (6.6,1.7); \node[left, font=\scriptsize] at (0.3,0.9) {SU(3)}; \fill[acc] (6.6,1.7) circle (2.4pt); \node[acc, right, font=\scriptsize, align=left] at (6.7,1.7) {unify\\$M_X$ near $10^{16}$}; \end{tikzpicture} $$ ## 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 $\bar{\mathbf 5}$ and the antisymmetric $\mathbf{10}$: $$ \bar{\mathbf 5} = \begin{pmatrix} d^c_r \\ d^c_g \\ d^c_b \\ e^- \\ \nu_e \end{pmatrix}_L, \qquad \mathbf{10} \supset \{\, u,\ u^c,\ d,\ e^+ \,\}_L . $$ 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 $X$, $Y$. $$ % caption: A single fermion generation embedded in SU(5). The antifundamental % five-plet groups the three color anti-down states with the lepton doublet; the % remaining states fill the ten. Color and lepton sectors share one representation. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0, cell/.style={draw=black, minimum width=10mm, minimum height=6mm, font=\scriptsize}, lcell/.style={draw=acc, fill=acc!12, minimum width=10mm, minimum height=6mm, font=\scriptsize}] \definecolor{acc}{HTML}{4A6FA5} \node[font=\scriptsize] at (-0.2,3.2) {5-plet}; \node[cell] (d1) at (0,2.4) {$d^c$}; \node[cell] (d2) at (0,1.7) {$d^c$}; \node[cell] (d3) at (0,1.0) {$d^c$}; \node[lcell] (el) at (0,0.3) {$e$}; \node[lcell] (nu) at (0,-0.4) {nu}; \draw[black] (0.7,2.7) -- (0.7,0.7); \node[black, right, font=\scriptsize, align=left] at (0.75,1.7) {color triplet\\(three anti-down)}; \node[acc, right, font=\scriptsize, align=left] at (0.75,-0.05) {weak doublet\\(lepton pair)}; % the ten \node[font=\scriptsize] at (5.0,3.2) {ten-plet}; \node[cell] (u1) at (4.4,2.2) {$u$}; \node[cell] (u2) at (5.2,2.2) {$u^c$}; \node[cell] (dd) at (4.4,1.4) {$d$}; \node[cell] (ep) at (5.2,1.4) {$e^+$}; \node[black, font=\scriptsize, align=center] at (4.8,0.6) {(ten states,\\quarks and positron)}; \end{tikzpicture} $$ ## Charge quantization Electric charge in $SU(5)$ is a generator of the group, a traceless diagonal matrix. The charge operator acting on the $\bar{\mathbf 5}$ must have vanishing trace, since every $SU(5)$ generator is traceless: $$ Q(\bar{\mathbf 5}) = \operatorname{diag}\!\left(Q_{d^c},\,Q_{d^c},\,Q_{d^c},\,Q_{e^-},\,Q_{\nu}\right), \qquad 3\,Q_{d^c} + Q_{e^-} + Q_{\nu} = 0 . $$ The neutrino is neutral, $Q_\nu = 0$, and the electron charge is $Q_{e^-} = -1$ in units of $e$. Tracelessness then forces $$ Q_{d^c} = -\tfrac13\,Q_{e^-} = +\tfrac13, \qquad\Longrightarrow\qquad Q_d = -\tfrac13 . $$ 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 $\tfrac13$ 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 $$ p \to e^+ + \pi^0, $$ 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 $$ \Gamma \sim \frac{\alpha_G^2\, m_p^5}{M_X^4}, \qquad \tau_p = \frac{1}{\Gamma} \sim \frac{M_X^4}{\alpha_G^2\, m_p^5}. $$ The lifetime scales as the fourth power of the unification mass, so the huge value of $M_X$ makes the proton extraordinarily long-lived. > **Example (Proton lifetime estimate).** Take $M_X = 2\times10^{16}$ GeV, > $\alpha_G = 1/40$, and $m_p \approx 0.94$ GeV. Then > > $$ > \tau_p \sim \frac{(2\times10^{16})^4}{(1/40)^2\,(0.94)^5}\ \text{GeV}^{-1} > \approx 2\times10^{65}\ \text{GeV}^{-1}. > $$ > > Converting with $1\ \text{GeV}^{-1} = 6.6\times10^{-25}$ s and $1\ \text{yr} = > 3.2\times10^{7}$ s gives $\tau_p \sim 4\times10^{33}$ years. Minimal $SU(5)$ > predicts a lifetime near $10^{31}$–$10^{34}$ years, with the precise value > sensitive to $M_X$. 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. $$ % caption: Proton decay in SU(5). Two valence quarks meet and exchange a superheavy % X boson carrying both color and lepton charge; the u and d convert to a positron % and an anti-down, which pairs with the spectator up to form a neutral pion. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \node[left, font=\scriptsize] at (-0.2,-0.3) {proton}; % spectator u \draw[->, thick] (0.3,0) -- (0.3,3.0); \node[above, font=\scriptsize] at (0.3,3.0) {$u$}; % u and d to vertex \draw[->, thick] (1.1,0) -- (1.9,1.4); \node[below, font=\scriptsize] at (1.1,0) {$u$}; \draw[->, thick] (1.8,0) -- (1.9,1.4); \node[below, font=\scriptsize] at (1.8,0) {$d$}; \fill (1.9,1.4) circle (1.6pt); \draw[acc, thick, dotted, line width=1pt] (1.9,1.4) -- (3.4,1.4) node[midway, above, font=\scriptsize] {$X$}; \fill (3.4,1.4) circle (1.6pt); \draw[->, thick] (3.4,1.4) -- (4.7,2.5) node[right, font=\scriptsize] {$e^+$}; \draw[->, thick] (3.4,1.4) -- (4.7,0.4) node[right, font=\scriptsize] {anti-$d$}; \node[right, font=\scriptsize, align=left] at (4.7,1.5) {pion $=$ spectator\\$u$ with anti-$d$}; \end{tikzpicture} $$ ## Experimental limits Super-Kamiokande, a $50\,000$-ton water Cherenkov detector, watches roughly $7\times10^{33}$ 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 is[^pdg-proton] $$ \tau_p / B(p \to e^+\pi^0) > 2.4\times10^{34}\ \text{years}. $$ 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 $\mathbf{16}$, 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. $$ % caption: The experimental lower bound on the proton lifetime has climbed by % orders of magnitude across four decades, passing and excluding the minimal SU(5) % prediction band. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (7.2,0) node[right, font=\scriptsize] {year}; \draw[->, black] (0,0) -- (0,4.2) node[above, font=\scriptsize] {log lifetime limit}; \node[black, below, font=\scriptsize] at (0.6,0) {1980}; \node[black, below, font=\scriptsize] at (6.4,0) {2020}; % rising limit \draw[acc, very thick] (0.6,0.9) -- (2.4,2.1) -- (4.2,2.9) -- (6.4,3.7); \fill[acc] (0.6,0.9) circle (1.8pt); \fill[acc] (2.4,2.1) circle (1.8pt); \fill[acc] (4.2,2.9) circle (1.8pt); \fill[acc] (6.4,3.7) circle (1.8pt); % SU(5) prediction band \draw[black, dashed] (0.4,2.0) -- (7.0,2.0); \draw[black, dashed] (0.4,2.6) -- (7.0,2.6); \node[black, right, font=\scriptsize, align=left] at (4.6,1.6) {minimal $SU(5)$\\excluded}; \node[acc, above left, font=\scriptsize] at (6.4,3.7) {Super-K}; \end{tikzpicture} $$ ## 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](/particle-physics/beyond-standard-model/matter-antimatter-open-questions). 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 $M_X \approx 2\times10^{16}$ GeV. Embedding a fermion generation in the $\bar{\mathbf 5} \oplus \mathbf{10}$ forces the down-quark charge to $-\tfrac13$ 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 $\tau_p > 2.4\times10^{34}$ years excludes minimal $SU(5)$ and pushes the field toward $SO(10)$ and supersymmetric unification.[^gr-gut][^pk-gut] [^tong-anom]: The requirement that gauge anomalies cancel generation by generation, correlating quark and lepton charges, is developed in Tong, _The Standard Model_ (Cambridge Part III), §5.1.1, [damtp.cam.ac.uk/user/tong/standardmodel.html](http://www.damtp.cam.ac.uk/user/tong/standardmodel.html); the same cancellation is automatic once quarks and leptons share an $SU(5)$ multiplet. [^gut-run]: The one-loop coefficients $(b_1,b_2,b_3)=(41/10,-19/6,-7)$ and the near-unification of the couplings are standard; the exact meeting under a TeV-scale supersymmetric spectrum is discussed in Perkins, Ch. 12, and Thomson, Ch. 18. [^gr-gut]: Griffiths, Ch. 12 (Afterword), presents the $SU(5)$ embedding of a generation, the tracelessness argument for charge quantization, and the proton-decay estimate $\tau_p \sim M_X^4/\alpha_G^2 m_p^5$. [^pdg-proton]: Proton-decay partial-lifetime limits, including $\tau/B(p\to e^+\pi^0) > 2.4\times10^{34}$ yr from Super-Kamiokande, are compiled in the Particle Data Group review, [pdg.lbl.gov](https://pdg.lbl.gov). [^pk-gut]: Perkins, Ch. 12, treats coupling unification, the $SU(5)$ and $SO(10)$ groups, and the water-Cherenkov proton-decay searches; Thomson, Ch. 18, covers the unification extrapolation and its supersymmetric completion.