String Theory & Beyond the Standard Model

String theory posits that fundamental particles are one-dimensional vibrating strings. It naturally includes gravity, gauge theories, and produces a finite theory — but requires extra dimensions and supersymmetry. Here we survey its key ideas, the landscape problem, and leading alternative approaches to quantum gravity.

ST.1 Why We Need Something Beyond the Standard Model

The Standard Model (SM) is extraordinarily successful — but incomplete:

1. Gravity: SM does not include quantum gravity. Perturbative quantum gravity (adding gravitons to QFT) is non-renormalizable. The Planck scale E_Pl = √(ℏc⁵/G) ≈ 1.22×10¹⁹ GeV is where quantum gravity effects become O(1).

2. Dark matter and dark energy: SM has no dark matter candidate. Dark energy (Λ) is 123 orders of magnitude smaller than the Planck-scale vacuum energy — the cosmological constant problem.

3. Hierarchy problem: why is M_Higgs = 125 GeV ≪ M_Pl = 10¹⁹ GeV? Quantum corrections to the Higgs mass are quadratically UV-divergent: δm_H² ∝ Λ²_UV. Fine-tuning of 10³⁴ is required unless new physics appears near the TeV scale.

4. Gauge coupling unification: the three gauge couplings (α₁, α₂, α₃) nearly meet at ~10¹⁵ GeV in the MSSM (with supersymmetry) — hinting at grand unification.

ST.2 Supersymmetry

Supersymmetry (SUSY) introduces a symmetry between bosons and fermions. Every SM particle has a superpartner:

Quarks → squarks, Leptons → sleptons, Gluons → gluinos, W/Z → winos/zinos, Higgs → higgsinos, Photon → photino.

SUSY algebra: {Q_α, Q̄_α̇} = 2σ^μ_(αα̇) P_μ — the supercharges Q relate bosons to fermions. SUSY cancels the quadratic divergences in δm_H² because boson loops (+) and fermion loops (−) cancel exactly if SUSY is exact. If SUSY is broken at scale M_SUSY ~ TeV: δm_H² ~ M_SUSY² — acceptable.

The lightest SUSY particle (LSP) — typically the neutralino — is stable (R-parity conservation) and is a natural dark matter candidate (WIMP). LHC searches (2012–2023): no SUSY particles found at energies accessible so far. Naturalness arguments push against M_SUSY > few TeV.

ST.3 String Theory Basics

Replace point particles with 1D strings (length ℓ_s = √(α'), with α' ≈ (10⁻³³ cm)² for strings near the Planck scale). A string can vibrate in many modes — each vibrational mode is a different particle:

Open strings (endpoints free or on D-branes): lowest excitations are gauge bosons (vector representation). Closed strings (loops): lowest excitations include the graviton (spin 2). This is why string theory automatically contains gravity!

For bosonic string: a = 1. Massless states (N=1): gauge bosons, graviton. Tachyon (N=0): M² < 0 — unstable vacuum. Superstring theory eliminates the tachyon via worldsheet supersymmetry.

Critical dimension: conformal anomaly vanishes only in specific spacetime dimensions. Bosonic string: D = 26. Superstring: D = 10. We observe D = 4, so 6 dimensions must be compactified (curled up) at the Planck scale.

ST.4 D-Branes and Dualities

D-branes (Dirichlet branes): hypersurfaces on which open string endpoints are confined. A D_p-brane is p-dimensional. D0-branes = particles, D1-branes = strings, D3-branes = 4D membranes.

N coincident D3-branes have a worldvolume theory: N=4 super-Yang-Mills (SYM) with gauge group U(N). This led to the discovery of:

AdS/CFT correspondence (Maldacena 1997): string theory on AdS₅ × S⁵ is dual to N=4 SYM on its 4D boundary. A theory of quantum gravity in 5D Anti-de-Sitter space is equivalent to a conformal field theory in 4D. The holographic principle: information in a volume is encoded on its boundary.

Applications: quark-gluon plasma at the LHC (strong coupling = weakly coupled gravity), condensed matter (strange metals, high-T_c via holography), entanglement entropy (Ryu-Takayanagi formula: S_EE = Area/(4G_N)).

ST.5 Alternatives and the Landscape

String landscape: compactifying 6D to 4D with different fluxes gives ~10^500 distinct vacua with different low-energy physics. This is the "landscape" — each vacuum is a possible universe. Anthropic argument (Weinberg 1987): we live in a vacuum where Λ allows structure formation. Correct order of magnitude for Λ! But this is controversial — is it science or metaphysics?

Loop quantum gravity (LQG): quantize the gravitational field directly. Space is made of discrete units — spin networks (Penrose 1971). Area and volume operators have discrete spectra: A = 8πγ ℓ_Pl² √(j(j+1)) (Barbero-Immirzi parameter γ). No extra dimensions, no supersymmetry. Problem: low-energy limit not yet fully derived.

Causal dynamical triangulations (CDT): quantum gravity via path integral over piecewise-linear spacetimes. Shows 4D spacetime emerges dynamically.Asymptotic safety: gravity is UV-complete at a non-Gaussian fixed point.