GUT Baryogenesis: X-Boson Decays at the Unification Scale

1. Grand Unified Theories

The SM gauge couplings \((g_3, g_2, g_1)\) run with energy and approach each other at \(M_{GUT} \sim 10^{16}\) GeV. This suggests unification into a larger simple group that breaks spontaneously to the SM at \(M_{GUT}\).

Both groups contain baryon-number-violating processes at tree level through X/Y exchange, so Condition 1 (B violation) is automatic.

2. X-Boson Decay Channels

Consider a heavy leptoquark X with mass \(M_X \sim M_{GUT}\). It has two distinct decay channels with different baryon numbers:

And for the CP conjugate:

CPT forces equal total widths: \(r + (1-r) = \bar r + (1-\bar r) = 1\). But \(r\neq\bar r\) is allowed and produces a net baryon number per decaying X+Xbar pair:

3. Origin of \(\varepsilon\): Tree-Loop Interference

At tree level the decay rates of X and Xbar are equal (CPT). An asymmetry requires interference between tree-level and one-loop amplitudes whose absorptive parts generate phases (Cutkosky rules).

Schematically, writing the tree amplitude as \(\mathcal M_{tree} = A\) and the loop amplitude as \(\mathcal M_{loop} = B\, e^{i\delta}\) with \(\delta\) a CP-violating phase and B complex,

The CP asymmetry

Typical magnitude \(\varepsilon \sim 10^{-3}\)– \(10^{-6}\). To reach the observed \(Y_B \sim 10^{-10}\) one needs \(\varepsilon \cdot Y_X^{eq}/g_* \sim 10^{-10}\) with \(Y_X^{eq}\sim 10^{-3}\), so \(\varepsilon \sim 10^{-7}\)– \(10^{-8}\) is viable for unsuppressed GUT scenarios.

4. Diagram: X Decay Channels and SU(5) Representations

5. Yield Calculation

For a single species X decaying out of equilibrium, the Boltzmann equation gives (for \(K \ll 1\))

With \(g_* \approx 100\), matching \(Y_B^{obs} \approx 9\times 10^{-11}\) requires \(\varepsilon \sim 10^{-8}\). For \(K\gg 1\) the efficiency drops exponentially due to washout, making GUT baryogenesis most efficient near the boundary \(K\sim 1\).

6. The Reheating Problem

Inflation solves the horizon and flatness problems, but also dilutes any preexisting relics — including GUT-produced baryons — to zero. For GUT baryogenesis to leave any asymmetry, one needs reheating to re-populate X bosons:

Standard cosmology bounds \(T_{reh} \lesssim 10^{15}\) GeV from gravitino overproduction (in SUGRA) and from the scale of inflation ( \(H_{inf} < 6\times10^{13}\) GeV implied by tensor-to-scalar ratio \(r<0.06\)). This disfavors standard thermal GUT baryogenesis.

7. SO(10) and the Seesaw Connection

The orthogonal group SO(10) is the most economical GUT that unifies all matter of one generation plus a right-handed neutrino into the spinor representation 16:

Breaking SO(10) \(\to\) SU(5) \(\times\) U(1) or via the Pati-Salam route produces Majorana right-handed neutrino masses naturally. This means GUT baryogenesis and leptogenesis become two facets of one framework: heavy SO(10) X-bosons decay, but so do the right-handed neutrinos. Depending on the mass hierarchy, one or the other dominates the final asymmetry.

5b. Flipped SU(5) and Alternatives

Beyond minimal SU(5), several GUT variants remain phenomenologically viable:

• Flipped SU(5) \u00D7 U(1): Different embedding of the SM hypercharge, different proton-decay selection rules. Dominant mode \(p\to \bar\nu\pi^+\).
• Pati-Salam SU(4)_C \u00D7 SU(2)_L \u00D7 SU(2)_R: Unifies quarks and leptons via an enlarged color group, giving leptons “fourth color”.
• Trinification SU(3)\u00B3: Proposes three independent SU(3) factors with cyclic symmetry.
• E_6 and E_8: Exceptional Lie groups considered in heterotic string compactifications.

Each provides a different set of heavy gauge bosons and Yukawa structures, and thus different GUT baryogenesis predictions. The general framework of tree-loop interference asymmetry applies to all of them.

6a. Phase-Space Estimate of the Asymmetry

In the limit of heavy X bosons decaying while marginally out of equilibrium, the yield can be estimated directly:

For \(Y_X^{eq} = 45\zeta(3)/(2\pi^4 g_*) \approx 4\times 10^{-3}/g_*\), and \(g_* = 106.75\), \(Y_B \sim 4\times 10^{-5}\,\varepsilon\). Demanding \(Y_B = 9\times 10^{-11}\) requires \(\varepsilon \sim 2\times 10^{-6}\). This is readily achieved with one-loop interference diagrams in any GUT with complex Yukawa couplings.

6b. Preheating and Non-Thermal Production

If the inflaton \(\phi\) couples to a massive scalar X via \(\tfrac{1}{2}g^2\phi^2 X^2\), the X effective mass oscillates as \(m_X^2(t) = g^2\phi^2(t)\). When \(m_X^2\) crosses zero adiabaticity fails and X quanta are produced non-perturbatively via parametric resonance. The resulting abundance can exceed the naive thermal value by many orders of magnitude.

The Mathieu-like equation for X modes has exponentially growing instability bands; occupation numbers grow as \(n_k \sim e^{2\mu_k t}\). For couplings of order unity, X with mass \(10^{14}\) GeV can be copiously produced even when the reheating temperature is only \(10^{10}\) GeV.

This revitalized GUT baryogenesis in the 2000s. The subsequent X decay then proceeds in the standard way, yielding baryon number via CP-violating branching ratios.

7. Gauge Coupling Unification

The Standard Model gauge couplings \(g_1, g_2, g_3\) (for U(1)_Y, SU(2)_L, SU(3)_C) run with energy according to their renormalization-group equations. At one loop,

with SM beta coefficients \((b_1, b_2, b_3) = (41/10, -19/6, -7)\) in GUT normalization ( \(g_1^{GUT} = \sqrt{5/3}\,g'\)). Starting from LEP values at \(M_Z\) and running upward:

• SM alone: couplings don't meet at a single scale (miss by ~5%).
• MSSM: \((b_1, b_2, b_3) = (33/5, 1, -3)\); couplings meet at \(\sim 2\times 10^{16}\) GeV.
• This is a strong motivation for low-energy SUSY and for SUSY GUTs as the framework for GUT baryogenesis and proton decay predictions.

The unification scale M_GUT is therefore a prediction of the framework, not a free parameter. Proton decay experiments directly probe this scale.

8. Representation Theory of SU(5)

SU(5) contains the SM gauge group as the subgroup \(SU(3)\times SU(2)\times U(1)\) with the identification

One SM generation fits into the anti-fundamental plus antisymmetric representations:

The 24 adjoint (gauge) representation decomposes under SM as \(8+3+1+(3,2)+(\bar 3,2) = \mathrm{gluons} + W + B + X + Y\). The 12 new X,Y bosons mediate the baryon-number violation crucial for GUT baryogenesis.

Yukawa structure: the SU(5)-symmetric Yukawas \(10\cdot\bar5\cdot\bar 5_H\) and \(10\cdot 10\cdot 5_H\) generate quark and lepton masses. Generation of lepton asymmetries in SU(5) is unfortunately constrained: the symmetry forbids it at dim-6.

9. Dimension-6 Operators and the GUT Scale

Integrating out the heavy X, Y bosons produces effective four-fermion operators of the form

These operators are \(\Delta B = \Delta L = 1\), preserving B−L. They mediate proton decay with rate \(\Gamma \sim m_p^5/M_X^4\), but at \(T \gtrsim M_X\) they drive X production/decay in equilibrium. The baryogenesis phase is the narrow window around \(T\sim M_X\) where the X population is dropping out of equilibrium.

References

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