Proton decay
Every atom you are made of rests on a quiet assumption: that the proton at the heart of each nucleus lasts forever. Nothing you can do to a hydrogen atom — heat it, ionise it, slam it into another — has ever turned its proton into anything lighter. The textbook reason is baryon-number conservation. But look closely at the Standard Model and a surprise appears: baryon number was never imposed. It is an accidental symmetry, a coincidence of the fact that no gauge-invariant, renormalisable term you can write from the known fields happens to change the net number of quarks. The proton is stable the way a marble is stable at the bottom of a bowl — held by the shape of the theory, not by any deep law. Grand unification takes that coincidence apart. If the three Standard-Model forces are low-energy shards of one simple gauge group, then quarks and leptons share multiplets, and new superheavy gauge bosons must exist that carry a quark directly into a lepton. The accident is broken. The proton becomes unstable, with `p → e⁺π⁰` the flagship channel — but because the mediating boson weighs some 10¹⁵–10¹⁶ GeV, the decay is suppressed by four powers of that colossal mass, stretching the lifetime past 10³⁴ years, a billion trillion times the age of the Universe. That very slowness is the gift: a tank of water watched for a decade probes energies no collider will ever reach. Proton decay is grand unification's sharpest, most direct experimental prediction, and the reason vast underground detectors sit waiting for a single quark to betray its origin.
- In the Standard Model baryon number B is an accidental symmetry — no renormalisable gauge-invariant operator violates it — so the proton is exactly stable.
- Grand Unified Theories place quarks and leptons in shared multiplets; superheavy `X, Y` gauge bosons then mediate baryon-number violation and the proton decays.
- The effect is a dimension-six four-fermion operator suppressed by `1/M_X²`, giving a rate `Γ ≈ α_GUT² m_p⁵ / M_X⁴` and a lifetime `τ ≈ M_X⁴ / (α_GUT² m_p⁵)`.
- Minimal SU(5) predicts `τ(p → e⁺π⁰) ≈ 10³¹` yr and is excluded; Super-Kamiokande now requires `τ/B > 2.4×10³⁴` yr.
- These operators conserve B − L, fixing the antilepton final state `e⁺` (not `e⁻`) and the neutron partner channel `n → e⁺π⁻`.
- Key idea: the proton's stability is not a fundamental law but an accident of the Standard Model's symmetry structure; any theory that unifies quarks with leptons generically breaks that accident and makes the proton decay — enormously slowly, because the mediator is superheavy.
The proton's stability is an accident of the Standard Model's symmetry structure, not a fundamental law; unifying quarks with leptons breaks that accident through a dimension-six operator whose `1/M_X⁴` suppression stretches the lifetime past 10³⁴ years — turning a tank of water into a probe of the grand-unification scale.
Where this lives today
The search is alive and industrial. Super-Kamiokande — 50 kton of ultrapure water viewed by some 11,000 photomultipliers a kilometre under Mount Ikeno — has watched its protons for over two decades and set the standard-bearing limits: `τ/B > 2.4×10³⁴` yr for `p → e⁺π⁰` and `> 5.9×10³³` yr for `p → ν̄K⁺`. The `e⁺π⁰` channel is almost background-free: a proton at rest decays to a positron and a `π⁰ → γγ`, giving two back-to-back electromagnetic showers whose invariant mass reconstructs to `m_p` with near-zero net momentum, a signature atmospheric neutrinos rarely fake. The next generation is under construction: Hyper-Kamiokande (~260 kton, data expected late this decade) will push `e⁺π⁰` sensitivity toward `10³⁵` yr, and DUNE's 40 kton of liquid argon excels at the `p → ν̄K⁺` channel favoured by supersymmetric GUTs, cleanly imaging the kaon. A detection would be epochal — a direct glimpse of `10¹⁶` GeV physics and the first sighting of baryon-number violation, a Sakharov ingredient for why the Universe contains matter at all. Continued silence keeps raising `M_X`, strangling the simplest unified models.
The physical picture
Inside the proton three valence quarks — two up, one down — are held together by gluons. In the Standard Model that is the end of the story: there is no vertex that lets a quark become a lepton, so the trio is trapped forever. Grand unification adds exactly such a vertex. A superheavy gauge boson, the `X` or `Y` leptoquark, couples a quark line to a lepton line. When two of the proton's quarks approach within a distance of order `1/M_X`, they can exchange this boson: one quark is converted into a positron, another into an antiquark, and the lone spectator quark dresses that antiquark into a pion. The proton is gone; in its place fly a positron and a `π⁰`. The first figure freezes that instant.
The key features of that picture drive everything quantitative. First, the vertex lives at the mass `M_X`: the boson is virtual, and its huge mass makes the effective four-fermion coupling minuscule, `~ 1/M_X²` in the amplitude. Second, the process changes baryon number by one unit and lepton number by one unit in the same direction, so `B − L` — the specific combination the unified gauge symmetry protects — is untouched. That selection rule is why the final lepton is a positron, not an electron. Third, the only mass scale left in the decay kinematics is the proton mass, `m_p ≈ 0.938` GeV, which by dimensional analysis supplies the five powers of energy the rate needs. The second figure is the plot an experimentalist reads: how the lifetime climbs with `M_X`, and where the Super-Kamiokande limit slices the plane.
The core equations
Because the mediating boson is far too heavy to produce, its only trace at low energy is a contact interaction among four fermions — a dimension-six operator obtained by "integrating out" the boson. This single operator encodes the whole phenomenon: its coefficient fixes the rate, and its field content fixes the selection rules.
where: `i, j, k` — colour indices contracted by the antisymmetric `ε` (so the operator makes a colour singlet); `u, d, e` — quark and lepton fields, the superscript `c` denoting charge conjugation; `M_X` — superheavy mediator mass; `c ∼ g_X²` — dimensionless coupling. Reading it: four spin-½ fields give mass-dimension six, so the coefficient must carry dimension `mass⁻²`, the origin of every suppression that follows. The field content carries `ΔB = ΔL = −1`, hence conserves `B − L`. The mass `M_X` is not free — grand unification predicts it. Run the three Standard-Model gauge couplings up in energy with their renormalisation-group equations and demand that they meet at a single point; that meeting scale is `M_X`.
where: `α_i = g_i²/4π` — the `U(1)`, `SU(2)`, `SU(3)` couplings; `b_i` — one-loop beta-function coefficients; `M_Z` — the `Z` mass where the couplings are measured; `α_GUT` — the common value at unification. Reading it: the logarithmic running is slow, so the couplings only converge at an enormous scale — `M_X ≈ 2×10¹⁶` GeV with low-energy supersymmetry, or a rougher `~10¹⁴–10¹⁵` GeV in the minimal non-supersymmetric case, with `α_GUT ≈ 1/40` to `1/25`. With the operator's scale set, the decay rate follows from dimensional analysis. The proton is the only mass in the final-state kinematics, and a four-fermion coupling squared multiplies phase space that, on dimensional grounds, must supply five powers of energy.
where: `Γ_p` — the partial decay rate; `α_GUT² = (g_X²/4π)²` — from two powers of the vertex coupling; `m_p⁵` — the five powers of energy demanded by dimensions (as in muon and neutron beta decay); `M_X⁴` — from the operator coefficient squared. Reading it: the rate scales as the inverse fourth power of the mediator mass, so proton decay is exquisitely sensitive to physics at the unification scale. Inverting the rate gives the lifetime; the experiment converts a null result — no decays seen — into a lower bound on that lifetime.
where: `N_p` — number of protons watched; `T` — running time; `ϵ` — detection efficiency for the channel; `B` — its branching ratio; `N_90 ≈ 2.3` — the Poisson 90 %-confidence upper limit on signal counts when zero (background-free) events are seen. Reading it: sensitivity grows linearly with the exposure `N_p T`, which is why the search is a contest to build ever-larger detectors and watch them for decades.
The aim is to show that proton stability in the Standard Model is not assumed but derived — an accident of what terms the gauge symmetry and renormalisability allow — and to locate exactly where that accident can first break. We enumerate the allowed operators by mass dimension, find that nothing up to the renormalisable ceiling can change baryon number, and then identify the lowest-dimension operator that can, reading off its suppression and its selection rules. The payoff is a precise statement of what would have to be added to the theory to let a proton die.
Step 1. Write the most general Lagrangian consistent with the gauge symmetry `SU(3)×SU(2)×U(1)` and the field content of one generation, keeping only renormalisable terms (mass dimension `≤ 4`): gauge kinetic terms, covariant-derivative couplings, and Yukawa couplings to the Higgs. Assign each quark baryon number `B = 1/3` and each lepton `B = 0`. Why: renormalisability caps the operator dimension at four, and gauge invariance dictates exactly which field combinations may appear — this is the whole menu the Standard Model offers.
Step 2. Check every term on that menu against baryon number. Gauge and Yukawa vertices always pair a quark with its own conjugate, so each conserves `B` term by term; there is simply no dimension-`≤4`, gauge-invariant, Lorentz-invariant operator built from Standard-Model fields with `ΔB ≠ 0`. Baryon number is therefore conserved — but only accidentally, as a by-product of the menu, never having been imposed. Why: an accidental symmetry is fragile precisely because nothing protects it; extend the field content or the dimension and it can vanish.
Step 3. Relax the renormalisability ceiling and organise the corrections as an effective theory, `ℒ = ℒ_SM + (1/M) O₅ + (1/M²) O₆ + ⋯`. The unique dimension-five operator (Weinberg) violates lepton number and gives neutrino masses, not proton decay. Baryon-number violation first appears at dimension six, in operators of the schematic form `qqql` — three quarks and a lepton, as in equation (1). Why: the effective-field-theory expansion sorts new physics by how strongly a heavy scale `M` suppresses it, and `qqql` is the leading window onto `ΔB ≠ 0`.
Step 4. Inspect those dimension-six operators: every gauge-invariant one carries `ΔB = ΔL = −1`, so all conserve `B − L`. This fixes the phenomenology — the proton must decay to an antilepton plus mesons (`p → e⁺π⁰`, `p → ν̄K⁺`), never to `e⁻`, and the bound neutron follows suit with `n → e⁺π⁻`. Their `1/M²` coefficient means the amplitude, and hence the whole enterprise, is controlled by one huge scale `M ≈ M_X`. Why: the selection rule and the suppression are read directly off the operator, before any model of what lives at `M_X` is specified.
Go deeper · Derivation B · 2 minFrom a superheavy boson to a number of years
The goal is to turn the dimension-six operator into an actual lifetime, using only dimensional analysis plus the measured proton mass and a guess for the unification scale, and to expose how violently the answer depends on `M_X`. We build the effective four-fermion coupling, use it and the single available mass scale to fix the rate by dimensions, then substitute numbers and convert to years — the same logic Fermi used for beta decay, transplanted to the unification scale. The result is the classic minimal-SU(5) prediction and the reason it was testable at all.
Step 1. Integrate out the heavy boson: exchanging a particle of mass `M_X` between two currents produces, at energies far below `M_X`, a contact interaction with coupling `G_X ∼ g_X²/M_X²`, exactly analogous to the Fermi constant `G_F ∼ g²/M_W²` of weak decay. The amplitude for `p → e⁺π⁰` is proportional to `G_X`. Why: this is the low-energy shadow of a heavy mediator — a four-fermion coupling of dimension `mass⁻²`, the numerical heart of equation (1).
Step 2. The decay rate goes as `|amplitude|²` times phase space. With `Γ ∝ G_X²` and `G_X²` carrying dimension `mass⁻⁴`, dimensional consistency forces five compensating powers of the only mass in the problem, the proton mass, so `Γ_p ∼ G_X² m_p⁵ = α_GUT² m_p⁵ / M_X⁴` (dropping an `O(1)` hadronic matrix element). Why: the `m⁵` law is universal to four-fermion decays — muon decay gives `Γ ∼ G_F² m_μ⁵` — and here it makes the rate scale as `M_X⁻⁴`.
Step 3. Substitute the minimal-SU(5) values `M_X = 10¹⁵` GeV, `α_GUT = 1/40`, `m_p = 0.938` GeV. Then `α_GUT² m_p⁵ = (6.25×10⁻⁴)(0.727) = 4.5×10⁻⁴` GeV⁵ and `M_X⁴ = 10⁶⁰` GeV⁴, giving `Γ_p ≈ 4.5×10⁻⁶⁴` GeV. Convert with `ħ = 6.58×10⁻²⁵` GeV·s: `τ = ħ/Γ ≈ 1.5×10³⁹` s `≈ 4.6×10³¹` yr. Why: this is a genuine prediction — order 10³¹ years — sitting only a few orders of magnitude above ordinary radioactivity's reach, which is what made a search conceivable.
Step 4. Note the leverage: because `τ ∝ M_X⁴`, raising `M_X` by a single decade lengthens the lifetime by four decades. The `O(1)` hadronic matrix element (now computed on the lattice) and the precise `M_X` from coupling running shift the answer by a couple of orders of magnitude, but the conclusion is robust — minimal SU(5) lands near `10³¹` yr, far below today's bound of `2.4×10³⁴` yr, and is dead. Why: the fourth-power sensitivity is a double-edged sword — it makes the proton a superb probe of `M_X`, but it also means the prediction must be pinned with real matrix elements, not dimensional analysis alone.
Go deeper · Derivation C · 2 minTurning zero events into a lifetime bound
The aim is to show precisely how watching a large mass of matter and seeing nothing becomes a quantitative lower limit on the proton lifetime — the inference at the core of every proton-decay experiment. We start from the radioactive-decay law applied to an enormous, fixed population of protons, fold in detector efficiency, then use Poisson statistics for a null (background-free) count to set a confidence limit, and finally insert the Super-Kamiokande numbers. The output is a bound in years from an experiment that ran for years, exposing why progress is slow and detectors must be huge.
Step 1. For a population `N_p` of protons each with lifetime `τ`, the expected number of decays in time `T ≪ τ` is `N_p(1 − e^{−T/τ}) ≈ N_p T/τ`. A detector catches a fraction `ϵ` of the target channel (efficiency times branching ratio), so the expected observed count is `μ = ϵ N_p T / τ`. Why: over a human timescale the exponential is utterly linear — you are sampling the flat early slope of a decay curve whose half-life dwarfs the run.
Step 2. Observe zero candidate events with negligible background. Poisson statistics then bound the true mean: `P(0 | μ) = e^{−μ} ≥ 0.10` requires `μ ≤ 2.3` at 90 % confidence, so `N_90 = 2.3`. Setting `μ < N_90` and solving for the lifetime gives `τ/B > N_p ϵ T / N_90` — equation (4). Why: a null result cannot prove `τ = ∞`, but it excludes any lifetime short enough to have produced events you would have seen — that exclusion is the measurement.
Step 3. Insert Super-Kamiokande: its 22.5-kton fiducial volume of pure water holds `N_p ≈ 7.5×10³³` protons (free hydrogen plus bound protons in oxygen, ten per H₂O molecule). Take `ϵ ≈ 0.4` for `p → e⁺π⁰` and an exposure `T ≈ 10` yr. Then `τ/B > (7.5×10³³)(0.4)(10)/2.3 ≈ 1.3×10³⁴` yr — squarely in the range of the published limit `2.4×10³⁴` yr. Why: two entirely independent numbers — a molecular proton count and a Poisson factor — combine to probe `10¹⁶` GeV physics.
Step 4. To gain one order of magnitude in `τ` you must raise the exposure `N_p T` tenfold. A single detector cannot simply run ten times longer, so the frontier moves to larger masses: Hyper-Kamiokande's ~260 kton and DUNE's 40 kton of liquid argon. Why: the linear exposure scaling is unforgiving, which is exactly why each generation of proton-decay search is a leap in detector volume, not merely patience.
Limiting cases
| Case | Result | Meaning |
|---|---|---|
| Standard Model alone | `τ = ∞` | baryon number accidental; no `ΔB ≠ 0` operator, proton exactly stable |
| Minimal SU(5), `M_X ∼ 10¹⁵` GeV | `τ(e⁺π⁰) ∼ 10³¹` yr | below Super-K bound — excluded |
| SUSY-GUT dim-6, `M_X ∼ 2×10¹⁶` GeV | `τ(e⁺π⁰) ∼ 10³⁶–10³⁷` yr | `e⁺π⁰` beyond even Hyper-K reach |
| SUSY dim-5 (colour-triplet Higgsino) | `τ(ν̄K⁺) ∼ 10³⁴` yr | `p → ν̄K⁺` dominant, testable now |
| `B − L`-violating `(ΔB = 2)` | neutron–antineutron oscillation | different operator class (dim-9), no lepton in final state |
| `M_X → ∞` (decoupling) | `Γ → 0` as `M_X⁻⁴` | unified physics hidden; proton effectively stable again |
Worked examples
1 · Bookkeeping the decay: charge, B − L, and the neutron partner (confidence builder)
Verify that `p → e⁺π⁰` conserves electric charge and `B − L`, and use the `B − L` rule to predict the analogous decay of a bound neutron.
Step 1. Electric charge. The proton has `Q = +1`; the final state is `e⁺` (`Q = +1`) plus `π⁰` (`Q = 0`), total `+1`. Charge is conserved, as it must be for a gauged symmetry.
Step 2. Baryon and lepton number. Initial: `B = +1`, `L = 0`, so `B − L = +1`. Final: the positron is an antilepton, `L = −1`; the pion carries `B = L = 0`. So `B = 0`, `L = −1`, and `B − L = 0 − (−1) = +1`. Thus `ΔB = −1`, `ΔL = −1`, and `B − L` is conserved — the signature of the dimension-six operators.
Step 3. The neutron partner. A neutron is `udd`; running the same quark-level vertex converts two quarks to `e⁺` plus an antiquark, and charge conservation (`Q = 0` initially) forces a `π⁻` (`Q = −1`) to balance the `e⁺`. The prediction is `n → e⁺π⁻`, with the same `B − L`-conserving structure. Both nucleons decay; oxygen nuclei in a water detector supply the bound neutrons.
2 · Estimating a SUSY-GUT lifetime and asking who can see it (fully worked)
A supersymmetric grand unified theory has `M_X = 2×10¹⁶` GeV and `α_GUT = 1/25`. Estimate `τ(p → e⁺π⁰)` from the dimension-six operator and compare it with the Super-Kamiokande bound and the Hyper-Kamiokande reach.
Step 1. Assemble the pieces of equation (3). `α_GUT² = (1/25)² = 1.6×10⁻³`. `m_p⁵ = (0.938)⁵ = 0.727` GeV⁵. `M_X⁴ = (2×10¹⁶)⁴ = 1.6×10⁶⁵` GeV⁴.
Step 2. Form the rate: `Γ_p = α_GUT² m_p⁵ / M_X⁴ = (1.6×10⁻³)(0.727)/(1.6×10⁶⁵) = 7.3×10⁻⁶⁹` GeV.
Step 3. Convert to a lifetime with `ħ = 6.58×10⁻²⁵` GeV·s: `τ = ħ/Γ = 6.58×10⁻²⁵ / 7.3×10⁻⁶⁹ = 9.0×10⁴³` s. Divide by `3.16×10⁷` s/yr: `τ ≈ 2.9×10³⁶` yr.
Step 4. Compare. The Super-K bound is `2.4×10³⁴` yr and Hyper-K aims for `~10³⁵` yr, so this `~10³⁶`-yr prediction sits beyond the reach of even Hyper-Kamiokande for the `e⁺π⁰` channel. The lesson: because `τ ∝ M_X⁴`, the higher SUSY scale pushes `e⁺π⁰` out of sight — which is why SUSY GUTs are hunted instead through their dimension-five operators, favouring `p → ν̄K⁺` near `10³⁴` yr, right at today's frontier. `τ(e⁺π⁰) ≈ 3×10³⁶` yr — safe from current searches; watch the kaon channel.
Physics of the portrait
Proton decay is the signature prediction of grand unification, and grand unification in its sharp, minimal, falsifiable form is the work of Howard Georgi — who, with Sheldon Glashow, wrote down the theory that first said the proton must die, and then, with Helen Quinn and Steven Weinberg, showed how to compute where.
Origins · 1 minHoward Georgi and how this was discovered
Howard Georgi is the architect of grand unification's minimal realisation. In 1974, with Sheldon Glashow, he proposed that the Standard-Model gauge group `SU(3)×SU(2)×U(1)` is embedded in a single simple group `SU(5)`, placing quarks and leptons of one generation into just two representations. The construction was breathtakingly economical — but it came with a price the authors stated plainly: the extra gauge bosons of `SU(5)` connect quarks to leptons, so the proton cannot be stable. That same year, with Helen Quinn and Steven Weinberg, Georgi turned the idea quantitative, showing that the three Standard-Model couplings, run to high energy, converge near `10¹⁵` GeV, and predicting the weak mixing angle `sin²θ_W` in rough agreement with experiment — the first real evidence that unification might be more than numerology. The `SU(5)` prediction of `τ(p → e⁺π⁰) ∼ 10³¹` years launched the great underground experiments of the 1980s, IMB and Kamiokande. Those detectors saw no decays, and in doing so falsified minimal `SU(5)` — a textbook case of a beautiful theory killed by data. Georgi went on to help build the wider unification landscape (`SO(10)`, the survival hypothesis, effective field theory itself), but his enduring mark is having made "does the proton decay?" one of the most consequential experimental questions in physics.
Going further
Texts. Peskin & Schroeder, An Introduction to Quantum Field Theory, §20.2–20.3 (unification and B-violation) · Peskin, Concepts of Elementary Particle Physics, Ch. 18 · Langacker, The Standard Model and Beyond, 2nd ed., Ch. 10 (grand unification and proton decay) · Georgi, Lie Algebras in Particle Physics, 2nd ed. (SU(5) group theory) · Ross, Grand Unified Theories · Mohapatra, Unification and Supersymmetry. Key results to keep in mind: baryon number is accidental at dimension `≤ 4` and first violated at dimension six, conserving `B − L`; the rate formula is `Γ_p ≈ α_GUT² m_p⁵/M_X⁴`, giving `τ ≈ 4.6×10³¹` yr for `M_X = 10¹⁵` GeV and `α_GUT = 1/40`; the SUSY-GUT estimate yields `τ ≈ 3×10³⁶` yr for `M_X = 2×10¹⁶` GeV and `α_GUT = 1/25`; the Super-K sensitivity is `τ/B > N_p ϵ T/N_90 ≈ 1.3×10³⁴` yr for `N_p = 7.5×10³³`, `ϵ = 0.4`, `T = 10` yr; and the published Super-Kamiokande bound is `2.4×10³⁴` yr for `e⁺π⁰`.
Prerequisites. Standard Model Overview · Grand Unification · The Local Gauge Principle. Leads to. Matter–Antimatter Asymmetry · Open Questions in Particle Physics.
Ask the tutor: Ask me why `B − L` (not `B` alone) is the combination grand unification protects, or to redo the lifetime estimate for any `M_X` and coupling you choose.
If the proton decays, does that mean all matter is doomed?
In principle yes, but on a timescale that makes the word "doom" almost meaningless. A lifetime beyond `10³⁴` years is more than `10²⁴` times the age of the Universe. In a body's worth of atoms — roughly `10²⁸` nucleons — you would expect only a handful of decays over a human lifespan, negligible beside natural radioactivity. Proton decay, if real, would only matter for the far future of the cosmos, after stars burn out and black holes evaporate. It is a statement about the ultimate fate of the Universe, not about anything you will ever touch.
Has proton decay ever actually been observed?
No. Despite four decades of dedicated searches in ever-larger detectors, not a single confirmed proton-decay event has been seen. Every result is a lower limit on the lifetime, currently `τ/B > 2.4×10³⁴` yr for the flagship `p → e⁺π⁰` channel from Super-Kamiokande. These null results are far from worthless: each one raises the minimum allowed unification scale `M_X` and rules out models — most famously the original minimal `SU(5)`, whose predicted `~10³¹`-yr lifetime was excluded in the 1980s. The field lives in the productive tension of a firmly predicted effect that stubbornly refuses to appear, steadily narrowing the space of viable grand unified theories.
Why the specific channel p → e⁺π⁰, and are there others?
`p → e⁺π⁰` is the "golden channel" for gauge-mediated (dimension-six) decay: it is what the `X, Y` boson exchange most naturally produces, and it is experimentally clean — a positron and two photons reconstructing to the proton mass with near-zero net momentum, hard for backgrounds to mimic. But it is not the only mode. `B − L` conservation also allows `p → μ⁺π⁰`, `p → e⁺η`, and, importantly, `p → ν̄K⁺`. That last kaon channel dominates in supersymmetric GUTs, where dimension-five operators (mediated by colour-triplet Higgsinos) often outrun the dimension-six ones. Different theories predict different branching ratios, so the pattern of channels — not just whether the proton decays — is a fingerprint of the underlying unification.
The neutron is already unstable — how is this different?
A free neutron undergoes ordinary beta decay, `n → p e⁻ ν̄`, in about 15 minutes, but that process conserves baryon number: a neutron (`B = 1`) becomes a proton (`B = 1`). It is a Standard-Model weak decay, no new physics required. Proton decay, and the analogous bound-neutron decay `n → e⁺π⁻`, are entirely different: they violate baryon number, turning a baryon into mesons and an antilepton with `ΔB = −1`. That is forbidden in the Standard Model and requires grand-unification-scale operators. So "the neutron decays" and "the proton might decay" describe processes separated by 33 orders of magnitude in rate and by the presence or absence of baryon-number violation.
How can a tank of water possibly probe 10¹⁶ GeV physics?
By trading energy for exposure. A collider must produce a heavy particle, needing centre-of-mass energy near its mass — hopelessly far from `10¹⁶` GeV. Proton decay instead probes the `X, Y` boson virtually: it never appears on-shell, but leaves a tiny permanent contact interaction among light quarks and leptons, suppressed by `1/M_X⁴`. You cannot make the effect bigger, so you compensate with sheer numbers — watch `10³⁴` protons for a decade and even a `10⁻³⁴`-per-year probability yields a countable handful. The `1/M_X⁴` suppression that makes the proton nearly immortal is exactly what lets a patient, enormous experiment reach an energy scale forever beyond any accelerator.
What is the difference between dimension-six and dimension-five proton decay?
Dimension-six operators (`qqql/M_X²`) are the generic, gauge-mediated route: they arise in any GUT from `X, Y` boson exchange, are suppressed by two powers of the unification mass, and favour `p → e⁺π⁰`. Dimension-five operators (`qqql/M_X`, schematically) appear specifically in supersymmetric theories, where colour-triplet Higgsino exchange generates them; being suppressed by only one power of `M_X` (dressed by a loop and squark/gaugino masses), they can dominate and typically drive `p → ν̄K⁺`. So the operator dimension tells you both the expected lifetime and the favoured channel: seeing `e⁺π⁰` would point to gauge mediation, while seeing `ν̄K⁺` would strongly hint at low-energy supersymmetry.
Does proton decay violate any other conservation laws we trust?
Electric charge, colour, and angular momentum are all exactly conserved in `p → e⁺π⁰` — as they must be, since charge and colour are gauged. What is violated is baryon number `B` and lepton number `L`, each by one unit, while the combination `B − L` is preserved by the leading operators. This is not arbitrary: `B − L` is the unique anomaly-free global combination that can be gauged, and groups like `SO(10)` do gauge it, protecting it while breaking `B` and `L` separately. Processes that violate `B − L` itself — such as neutron–antineutron oscillation with `ΔB = 2` — need different, higher-dimension operators and are searched for separately.