Beyond the Standard Model

The Standard Model is the most precisely tested theory in the history of science. The magnetic moment of the electron agrees with theoretical prediction to 12 decimal places. The Higgs boson mass was predicted from precision electroweak data before its discovery. And yet, we know with certainty that the Standard Model is incomplete. It does not account for gravity, dark matter, dark energy, neutrino masses, or the matter-antimatter asymmetry of the universe. It contains unexplained patterns, arbitrary parameters, and fine-tuning problems that strongly suggest a deeper theory exists. This lesson surveys what we know we do not know, and the leading ideas for what might lie beyond.

The Hierarchy Problem

The most acute theoretical crisis in particle physics: why is the Higgs boson so light? The measured Higgs mass is $m_H \approx 125$ GeV. But in quantum field theory, the Higgs mass receives radiative corrections from virtual particles at all energy scales:

$$\delta m_H^2 \sim \frac{\Lambda^2}{16\pi^2}$$

where $\Lambda$ is the highest energy scale where the Standard Model is valid. If the SM is valid up to the Planck scale ($M_{Pl} \sim 10^{19}$ GeV), then quantum corrections to $m_H^2$ are of order $10^{38}$ GeV$^2$: thirty-four orders of magnitude larger than the observed value of $(125 \text{ GeV})^2$.

To get the right answer, the bare Higgs mass and the quantum corrections must cancel to one part in $10^{34}$. This extraordinary fine-tuning seems deeply unnatural. Either new physics enters below the Planck scale to stabilize the Higgs mass, or there is a mechanism we do not yet understand.

The Strong CP Problem

The QCD Lagrangian allows a CP-violating term:

$$\mathcal{L}_\theta = \bar{\theta}\frac{g_s^2}{32\pi^2}G^a_{\mu\nu}\tilde{G}^{a\mu\nu}$$

where $\bar{\theta}$ combines the bare theta parameter with a phase from the quark mass matrix. This term would generate an electric dipole moment for the neutron. Experiments constrain $|\bar{\theta}| < 10^{-10}$. But $\bar{\theta}$ is a free parameter with no reason to be small, why is it essentially zero?

The most popular solution is the Peccei-Quinn mechanism, which promotes $\bar{\theta}$ to a dynamical field, the axion. The axion relaxes to $\bar{\theta} = 0$ dynamically. If the axion exists, it would be extremely light and weakly coupled, making it an excellent dark matter candidate. Experiments like ADMX and ABRACADABRA are actively searching for axions.

Matter-Antimatter Asymmetry

The universe is overwhelmingly made of matter rather than antimatter. The observed baryon-to-photon ratio is approximately $\eta \sim 6 \times 10^{-10}$. Sakharov identified three conditions needed to generate this asymmetry from initially symmetric conditions: baryon number violation, C and CP violation, and departure from thermal equilibrium.

The Standard Model satisfies all three conditions in principle, baryon number is violated by electroweak sphalerons, CP is violated in the CKM matrix, and the electroweak phase transition provides departure from equilibrium. But quantitatively, the SM CP violation is far too small and the electroweak phase transition (with $m_H = 125$ GeV) is not strongly first-order. New sources of CP violation and/or new mechanisms are needed.

Dark Matter and Dark Energy

Astrophysical and cosmological observations provide overwhelming evidence that 27% of the universe's energy density is dark matter and 68% is dark energy. The Standard Model accounts for only about 5% of the total.

Dark matter interacts gravitationally and possibly through weak-scale interactions, but not electromagnetically. Leading particle candidates include:

• WIMPs (Weakly Interacting Massive Particles): particles with masses ~10 GeV to ~10 TeV that naturally produce the observed relic density
• Axions: ultralight particles from the Peccei-Quinn solution to the strong CP problem
• Sterile neutrinos: right-handed neutrinos with keV-scale masses

Dark energy is even more mysterious. Its energy density $\rho_\Lambda \sim (10^{-3} \text{ eV})^4$ is 120 orders of magnitude smaller than the naive quantum field theory estimate of $(M_{Pl})^4$. This is the cosmological constant problem: arguably the worst fine-tuning problem in all of physics.

Gravity

The Standard Model does not include gravity. General relativity is a classical theory that breaks down when quantum effects become important, at the Planck scale $E \sim M_{Pl} \sim 10^{19}$ GeV. At these energies, quantum gravity effects are essential but we do not have a consistent quantum theory of gravity (though string theory and loop quantum gravity are leading candidates).

The Planck scale is so far above accessible energies that this might seem like a purely academic problem. But the hierarchy problem, why is the weak scale $10^{17}$ times smaller than the Planck scale, connects gravity directly to particle physics.

Supersymmetry (SUSY)

Supersymmetry postulates a symmetry between fermions and bosons: every known particle has a "superpartner" with spin differing by 1/2. Electrons get selectrons, quarks get squarks, photons get photinos, and so on.

If SUSY is exact, every superpartner has the same mass as its Standard Model counterpart. Since no selectrons at 0.511 MeV have been observed, SUSY must be broken, pushing superpartner masses higher. The LHC has searched for superpartners up to ~2 TeV without finding any. This puts significant pressure on natural SUSY, though models with heavier superpartners remain viable.

Grand Unification

The Standard Model gauge group $SU(3) \times SU(2) \times U(1)$ has three independent coupling constants. At high energies, thanks to the running of couplings, they approach each other. In the minimal SM, they come close but do not quite meet. With SUSY, they converge beautifully at $M_{GUT} \sim 2 \times 10^{16}$ GeV.

Grand Unified Theories (GUTs) embed the SM gauge group in a single larger group. The simplest is $SU(5)$ (Georgi-Glashow), which unifies quarks and leptons into common multiplets. The representation $\bar{\mathbf{5}} + \mathbf{10}$ of $SU(5)$ contains exactly one generation of SM fermions. Other candidates include $SO(10)$, which additionally accommodates right-handed neutrinos naturally, and $E_6$.

A key prediction of GUTs is proton decay. In $SU(5)$, quarks can transform into leptons via heavy $X$ and $Y$ bosons with mass $\sim M_{GUT}$. The dominant decay channel is $p \to e^+\pi^0$ with a predicted lifetime:

$$\tau_p \sim \frac{M_X^4}{m_p^5} \sim 10^{34\text{-}36} \text{ years}$$

Super-Kamiokande has set a lower limit of $\tau_p > 2.4 \times 10^{34}$ years for this channel, ruling out minimal $SU(5)$ but still consistent with SUSY GUTs (where the dominant mode becomes $p \to K^+\bar{\nu}$). Next-generation experiments like Hyper-Kamiokande and DUNE will push these limits further.

Extra Dimensions

Perhaps the hierarchy problem is not about the Higgs mass being too light, but about gravity being too weak. If there are extra spatial dimensions beyond the three we observe, gravity could dilute over a larger volume, appearing weak in our 3+1 dimensional world.

In the Randall-Sundrum model, a single extra dimension with a warped (anti-de Sitter) geometry can explain the hierarchy. The fundamental Planck scale could be ~TeV, with the apparent weakness of gravity arising from an exponential warping factor. In ADD (Arkani-Hamed, Dimopoulos, Dvali) models, large flat extra dimensions achieve a similar effect.

These models predict deviations from Newtonian gravity at short distances and the production of Kaluza-Klein excitations at the LHC. Searches have found no evidence, constraining the extra dimension scale.

Composite Higgs and Technicolor

An alternative approach to the hierarchy problem: the Higgs is not a fundamental scalar but a composite state, analogous to how pions in QCD are composites of quarks. If the Higgs is a pseudo-Goldstone boson of a new strongly-coupled sector at the ~10 TeV scale, its mass is naturally light, protected by the approximate Goldstone symmetry.

Technicolor, the original version of this idea, postulated a new QCD-like force at the TeV scale. It was largely ruled out by precision electroweak data. Modern composite Higgs models are more sophisticated, using partial compositeness to generate fermion masses and satisfy precision constraints. They predict new resonances in the multi-TeV range that could be discovered at future colliders.

Experimental Searches at the LHC

The Large Hadron Collider at CERN, operating at center-of-mass energies up to 13.6 TeV, is the primary tool for searching for BSM physics. Despite extensive searches, no new particles beyond the Standard Model have been found. Key results include:

• No superpartners (squarks, gluinos) below ~2 TeV
• No $Z'$ or $W'$ bosons below ~5 TeV
• No extra dimension signals
• Higgs properties consistent with SM to ~10% precision
• No significant anomalies in flavor physics (after the B-anomalies resolved)

The absence of new physics at the LHC is itself a profound result. It means that if the hierarchy problem is solved by new physics, it may lie at scales beyond direct LHC reach, requiring future colliders (FCC, CEPC, muon colliders) or indirect probes.

What We Know We Don't Know

Beyond the specific problems above, the Standard Model contains 19 free parameters (or 26 including neutrino masses and mixing) that must be put in by hand: 6 quark masses, 3 lepton masses, 3 neutrino masses, 4 CKM parameters, 4 PMNS parameters, 3 gauge couplings, the Higgs mass, and $\theta_{QCD}$. A deeper theory should explain these numbers.

Some of the deepest open questions include:

• Why three generations? Is there a pattern in the masses and mixing angles?
• Why is the gauge group $SU(3) \times SU(2) \times U(1)$ and not something simpler?
• Is there a unified description of all forces, including gravity?
• What determines the cosmological constant?
• Is the vacuum stable? (The measured Higgs and top masses place us near the edge of metastability)

The Standard Model may be the low-energy effective theory of a much richer structure. Finding that structure is the central challenge of twenty-first century physics.

Key Insights

• The Standard Model is incomplete: neutrino masses, dark matter, dark energy, baryogenesis, and gravity all require new physics
• The hierarchy problem asks why the Higgs mass is $10^{17}$ times smaller than the Planck scale
• The strong CP problem asks why $\bar{\theta} < 10^{-10}$; the axion is the leading solution
• Supersymmetry solves the hierarchy problem but has not been found at the LHC
• Grand unification embeds SM forces in a single group, predicting proton decay
• Extra dimensions and composite Higgs are alternative approaches to the hierarchy problem
• The LHC has found no BSM physics, pushing new particles above ~2-5 TeV
• The SM has ~26 free parameters; a deeper theory should explain them

Looking Ahead

With this lesson, we complete our study of the Standard Model. We have seen its extraordinary successes and its clear limitations. The search for what lies beyond is far from over, it motivates next-generation colliders, underground experiments, gravitational wave detectors, and cosmological surveys. In Phase 9, we turn to cosmology: the physics of the universe as a whole, from the Big Bang to the large-scale structure we observe today.