Baryon Number Violation: Instantons, Sphalerons, and Anomalies

1. Classical Baryon Number Conservation

In the SM Lagrangian, one can assign baryon number \(+1/3\) to each quark field and \(0\) to all others. Under the global U(1)_B symmetry \(q \to e^{i\alpha/3}\,q\), the kinetic and Yukawa terms of the quark sector are invariant. Noether's theorem provides the classical conserved current

Similarly for lepton number L. At the classical, perturbative level, both B and L are individually conserved. This is why the proton is absolutely stable in the classical Standard Model, and why simple scattering or decay processes cannot create net baryon number.

But quantum mechanics changes the story dramatically via the chiral anomaly.

2. The Chiral Anomaly & Non-Conservation of B+L

The triangle diagram with one axial current and two vector currents yields the Adler-Bell-Jackiw anomaly. For the SU(2)_L gauge fields coupling only to left-handed fermions, the baryon and lepton currents satisfy

where \(N_f = 3\) is the number of fermion generations, \(g\) is the SU(2)_L coupling, and \(\tilde W^{a\,\mu\nu} = \frac{1}{2}\epsilon^{\mu\nu\rho\sigma}W^a_{\rho\sigma}\)is the dual field strength.

Crucially:

• B and L are each anomalously non-conserved.
• B−L is conserved: the anomaly coefficients are equal and opposite for B and L−inverted currents? No, actually they are equal, so B−L has zero anomaly coefficient: \(\partial_\mu (J^\mu_B - J^\mu_L) = 0\).
• B+L is violated: \(\partial_\mu (J^\mu_B + J^\mu_L) \propto W\tilde W\).

Integrating the anomaly equation over spacetime:

where \(\Delta n_{CS}\) is the change in the Chern-Simons number between vacuum states. Each unit of CS number change produces \(\Delta B = \Delta L = 3\), i.e. three quarks and three leptons appear or disappear simultaneously.

3. Instantons: Tunneling Between Vacua

The electroweak vacuum is not unique: the gauge group SU(2) has a periodic structure of topologically distinct pure-gauge vacua labeled by integer winding number \(n_{CS}\). The ‘t Hooft instanton is the semiclassical tunneling amplitude between adjacent vacua, with action

The tunneling rate, by Euclidean saddle-point, is suppressed by \(e^{-S_{inst}} = e^{-2\pi/\alpha_W} \sim 10^{-165}\) — absolutely negligible at zero temperature. Proton decay via instantons is therefore stable over any observable timescale.

4. Sphalerons: Thermal B+L Violation

At finite temperature, the exponential instanton suppression is replaced by thermal activation over the potential barrier separating vacua. The top of this barrier is a static, unstable classical field configuration called the sphaleron (from Greek “ready to fall”, Klinkhamer & Manton 1984).

where \(v(T)\) is the temperature-dependent Higgs vev and \(B \sim 1.9\) is a dimensionless function of the Higgs self-coupling. At \(T = 0\), \(E_{sph} \approx 9\) TeV.

Sphaleron rate regimes

Sphalerons remain in equilibrium (rate per particle \(\Gamma/T^3 > H\)) roughly from the electroweak scale up to \(T \sim 10^{12}\) GeV, then drop below Hubble. They therefore convert any primordial B+L asymmetry to zero, while preserving B−L.

5. Diagram: Vacuum Landscape & Sphaleron Energy

The electroweak vacuum is periodic: successive vacua differ by one unit of Chern-Simons number. Tunneling (instanton) or thermal activation (sphaleron) connects them, each event producing \(\Delta B = \Delta L = 3\).

6. Proton Decay: GUT Predictions and Bounds

Beyond the SM, grand unified theories like SU(5) or SO(10) contain gauge bosons that mediate genuine tree-level proton decay. The dominant channel in minimal SU(5) is \(p \to e^+ \pi^0\) through X-boson exchange. The rate scales as

For \(M_X \sim 10^{16}\) GeV, this gives \(\tau_p \sim 10^{35}\) yr. Super-Kamiokande current bound: \(\tau(p\to e^+\pi^0) > 1.6\times 10^{34}\) yr (Super-K 2020). Future Hyper-K will extend to \(\sim 10^{35}\) yr.

Minimal SU(5) is already ruled out by this bound combined with gauge coupling unification. SO(10), flipped SU(5), and supersymmetric GUTs remain viable with different preferred channels such as \(p \to \bar\nu K^+\).

7. The Crucial Role of B−L

Because sphalerons conserve B−L while violating B+L, any successful baryogenesis mechanism operating after \(T \sim 10^{12}\) GeV must generate a net B−L asymmetry — not merely a B asymmetry. This distinction is crucial for leptogenesis, which generates a lepton asymmetry (hence B−L) that sphalerons partially convert to baryons.

Consider a purely-B asymmetry \(\Delta B\neq 0,\; \Delta L = 0\) produced at high T. Sphalerons drive the system toward equilibrium by changing B+L. The equilibrium condition is \(\Delta(B+L)_{final} = 0\), so \(\Delta B_{final} + \Delta L_{final} = 0\). Since B−L is exactly conserved, \(\Delta B_{final} - \Delta L_{final} = \Delta B_{init}\). Solving:

So pure B at high T is halved but not erased. Pure L is similarly half-erased. But pure B+L is completely washed out. Generating only B−L is the efficient path.

8b. Effective Action and Tunneling Rate

The semiclassical tunneling rate between adjacent vacua is set by the instanton Euclidean action:

With \(\alpha_W \approx 1/30\), the exponent is \(-2\pi\cdot 30 \approx -188\), giving a suppression \(e^{-188}\sim 10^{-82}\) per volume 1/M_W \(^4\) per time 1/M_W. The rate of B+L violation at T=0 is utterly unobservable on any physical timescale. Only at finite temperature, where thermal fluctuations activate the sphaleron barrier, does this become relevant.

9. Neutron-Antineutron Oscillations

Beyond proton decay, some BSM scenarios predict \(\Delta B = 2\) processes such as neutron-antineutron oscillations \(n \leftrightarrow \bar n\). This is a dimension-9 operator of the form \((udd)(udd)/M^5\) and probes B−L-violating new physics orthogonal to proton decay.

Current bound from nuclear stability (e.g. Super-K neutron disappearance): \(\tau_{n\bar n} > 4.7\times 10^8\) s (Super-K 2021), corresponding to an effective scale \(M \gtrsim 300\) TeV. The ESS-nnbar experiment targets \(\tau > 10^{10}\) s.

A positive detection would favor post-sphaleron baryogenesis scenarios where \(\Delta B = 2\) operators are active at low energies.

Appendix B: Chern-Simons Number and Winding

For an SU(2) gauge field \(A_\mu\) on spatial slice \(\Sigma\), the Chern-Simons number is

On pure-gauge configurations \(A_\mu = i g^{-1}U^{-1}\partial_\mu U\), n_CS equals the integer winding number of the map \(U: S^3 \to SU(2) \cong S^3\). Vacuum-to-vacuum transitions are classified by \(\pi_3(SU(2)) = \mathbb{Z}\). Integrating the anomaly identity \(\partial_\mu J^\mu_B = (g^2 N_f/32\pi^2) W\tilde W\) over a region bounded by vacua at \(t=\pm\infty\):

Each instanton contributes one unit of \(\Delta n_{CS}\), hence \(\Delta B = 3\). This is the topological origin of SM baryon-number violation.

Appendix: The ‘t Hooft 12-fermion Vertex

In the SM with three generations, each generation contributes one quark doublet and one lepton doublet to the anomalous current. An instanton event creates nine quark fields and three lepton fields, giving the famous 12-fermion ‘t Hooft vertex:

Each \(Q_i\) creates one left-handed quark doublet of generation i ( \(u_L, d_L\)), and each \(L_i\) creates a left-handed lepton doublet ( \(\nu_L, e_L\)). This simultaneous 9-quark + 3-lepton creation changes the baryon number by \(\Delta B = 9\times\tfrac{1}{3} = 3\) and lepton number by \(\Delta L = 3\), giving \(\Delta(B-L) = 0\).

At high T, thermal sphaleron transitions populate this operator at a rate \(\Gamma \sim \alpha_W^5\, T^4\), efficient enough to erase any B+L asymmetry in the primordial plasma.

8. Proton Decay Channels: A Closer Look

GUT proton decay proceeds through dimension-6 operators with Wilson coefficients determined by the broken gauge bosons. The 9 possible operators of the form \((qq)(q\ell)/M_X^2\) reduce, after accounting for color and flavor, to a small number of independent structures. Dominant decay channels:

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