Beyond the Standard Model

9.1 Supersymmetry (SUSY)

Supersymmetry relates fermions and bosons: every SM particle has a superpartner differing by Half a unit of spin.

• Squarks ($\tilde{q}$), sleptons ($\tilde{l}$): scalar partners of quarks and leptons.
• Gluinos ($\tilde{g}$): fermionic partners of gluons.
• Neutralinos, charginos: partners of the Higgs and gauge bosons.

Motivations:

1. Solves the hierarchy problem by cancelling quadratic divergences in the Higgs mass.
2. Provides a dark matter candidate (lightest supersymmetric particle, LSP).
3. Enables gauge coupling unification.
4. Arises from string theory.

Status: No SUSY particles have been observed at the LHC (as of 2026). If SUSY exists, the Superpartners must be heavier than $\sim 2$ TeV, requiring either fine-tuning or an extended model (e.g., split SUSY).

9.1.1 The Hierarchy Problem

The hierarchy problem is the puzzle of why the Higgs mass ($m_H \sim 125$ GeV) is so much Smaller than the Planck scale ($M_{\mathrm{Pl} \sim 10^{19}}$ GeV).

The problem. In quantum field theory, the Higgs mass receives quadratically divergent Radiative corrections from virtual particles. For a fermion loop (e.g., top quark):

$\delta m_H^2 = -\frac{\lvert y_t\rvert^2}{8\pi^2}\Lambda^2 + \mathrm{finite}$

Where $\Lambda$ is the ultraviolet cutoff. If $\Lambda \sim M_{\mathrm{Pl}}$Then $\delta m_H^2 \sim 10^{38}$ GeV$^2$Requiring an incredible fine-tuning of the bare mass To cancel this and yield $m_H^2 \sim 10^4$ GeV$^2$.

SUSY solution. Each fermion loop contribution $-\lvert y_t\rvert^2\Lambda^2/(8\pi^2)$ is Cancelled by the corresponding scalar (stop) loop contribution $+\lvert y_t\rvert^2\Lambda^2/(8\pi^2)$ Because bosonic and fermionic loops contribute with opposite signs. This cancellation is exact When SUSY is unbroken. With SUSY broken at the TeV scale, the residual correction is only:

$\delta m_H^2 \sim \frac{\lvert y_t\rvert^2}{8\pi^2}m_{\mathrm{SUSY}^2 \sim (100\;\mathrm{GeV})^2}$

Which is of the same order as $m_H^2$Eliminating the fine-tuning.

9.1.2 R-Parity

SUSY models impose R-parity, a discrete symmetry defined as:

$R = (-1)^{3(B-L)+2s}$

All SM particles have $R = +1$And all superpartners have $R = -1$. R-parity conservation has Two important consequences:

1. The lightest supersymmetric particle (LSP) is stable, since there is no lighter $R = -1$ particle for it to decay into. If the LSP is electrically neutral and weakly interacting, it is an excellent dark matter candidate.
2. Superpartners are produced in pairs, making their detection at colliders more challenging (missing energy signatures from the escaping LSPs).

9.2 Grand Unified Theories (GUTs)

GUTs unify the SM gauge groups into a single simple group. The simplest is SU(5):

$\mathrm{SU}(3)_C \times \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y \subset \mathrm{SU}(5)$

Predictions of minimal SU(5):

• $\sin^2\theta_W = 3/8$ at the GUT scale (reasonable).
• Proton decay: $p \to e^+ + \pi^0$ with lifetime $\tau_p \sim 10^{30\pm 1}$ years. Current limit: $\tau_p \gt 1.6 \times 10^{34}$ years (Super-Kamiokande), ruling out minimal SU(5).
• Charge quantisation: $Q(\mathrm{proton}) + Q(\mathrm{electron}) = 0$.

Charge quantisation in SU(5). One of the elegant features of SU(5) is that it explains why Electric charge is quantised. In the SM, the values of the electric charges are inputs. In SU(5), Each generation of fermions fits into a $\bar{\mathbf{5}} \oplus \mathbf{10}$ representation:

$\bar{\mathbf{5}}: \quad \begin{pmatrix} \bar{d}_r \\ \bar{d}_g \\ \bar{d}_b \\ e^- \\ \nu_e \end{pmatrix}, \qquad \mathbf{10}: \quad \begin{pmatrix} 0 & u_r & u_g & u_b & \bar{e}^+ \\ -u_r & 0 & d_r & d_g & \bar{d}_b \\ -u_g & -d_r & 0 & d_b & \bar{d}_g \\ -u_b & -d_g & -d_b & 0 & \bar{d}_r \\ -e^+ & d_b & d_g & d_r & 0 \end{pmatrix}$

The fact that quarks and leptons sit in the same multiplet of a simple gauge group Explains why the proton charge exactly cancels the electron charge.

9.3 String Theory

String theory replaces point particles with one-dimensional objects (strings). The different Vibrational modes of the string correspond to different particles.

Key features:

• includes gravity (the graviton arises as a closed string excitation).
• Requires extra spatial dimensions (10 for superstring theory, 11 for M-theory).
• The extra dimensions are compactified, leading to a vast “landscape” of possible vacua.

Status: No experimental evidence. The Planck length ($\sim 10^{-35}$ m) is far below current Experimental reach.

Key concepts.

1. Five consistent superstring theories: Type I, Type IIA, Type IIB, Heterotic-O(32), and Heterotic-E$_8 \times$ E$_8$. These are related by dualities (S-duality, T-duality) and are believed to be different limits of a single underlying theory, M-theory, in 11 dimensions.

2. Compactification. The six extra spatial dimensions must be compactified on a small manifold (Calabi—Yau manifold) to recover four-dimensional physics. The choice of compactification determines the particle content and coupling constants of the effective four-dimensional theory.

3. Brane world scenarios. In some string theory constructions, our four-dimensional universe is a “brane” embedded in a higher-dimensional “bulk.” Standard Model particles are confined to the brane, while gravity can propagate into the bulk. This can lead to observable signatures such as deviations from Newton”s law at short distances or the production of Kaluza—Klein gravitons at colliders.

4. AdS/CFT correspondence. String theory in anti-de Sitter space is equivalent to a conformal field theory on its boundary. This holographic duality has found applications in QCD (quark-gluon plasma), condensed matter physics, and quantum information theory.

9.4 Quantum Gravity

The reconciliation of general relativity with quantum mechanics remains the central unsolved problem In theoretical physics.

Approaches:

• String theory: As above.
• Loop quantum gravity (LQG): Quantises spacetime geometry directly. Predicts discrete area and volume spectra.
• Causal set theory: Spacetime is fundamentally discrete.
• Asymptotic safety: The renormalisation group flow of gravity has a non-trivial fixed point.

No approach has been experimentally verified or gained universal acceptance.

9.5 Experimental Frontiers

Several experiments are probing physics beyond the Standard Model: