Chapter 14: Supersymmetry

Supersymmetry (SUSY) is a symmetry relating fermions and bosons. It is the only known extension of the Poincare spacetime symmetry consistent with an interacting quantum field theory (the Coleman-Mandula/Haag-Lopuszanski-Sohnius theorems). SUSY elegantly solves the hierarchy problem by canceling quadratic divergences, enables gauge coupling unification, and provides a natural dark matter candidate.

Derivation 1: The Supersymmetry Algebra

The $\mathcal{N}=1$ SUSY algebra extends the Poincare algebra with fermionic generators$Q_\alpha$ and $\bar{Q}_{\dot{\alpha}}$ (two-component Weyl spinors). The defining anticommutation relation is:

$\{Q_\alpha, \bar{Q}_{\dot{\beta}}\} = 2\sigma^\mu_{\alpha\dot{\beta}} P_\mu$

with all other anticommutators vanishing:

$\{Q_\alpha, Q_\beta\} = 0, \quad \{\bar{Q}_{\dot{\alpha}}, \bar{Q}_{\dot{\beta}}\} = 0$

The SUSY generators commute with translations and transform as spinors under Lorentz:

$[Q_\alpha, P_\mu] = 0, \quad [Q_\alpha, M_{\mu\nu}] = (\sigma_{\mu\nu})_\alpha^{\ \beta} Q_\beta$

Key Consequences

Mass degeneracy: Since $[Q, P^2] = 0$, bosonic and fermionic partners have equal mass in unbroken SUSY

Equal number of states: Each supermultiplet has equal numbers of bosonic and fermionic degrees of freedom

Positive energy: From $\{Q, \bar{Q}\} = 2\sigma^\mu P_\mu$, the vacuum energy $\langle 0|H|0\rangle \geq 0$

The anticommutator $\{Q, \bar{Q}\} = 2\sigma^\mu P_\mu$ shows that SUSY is intimately connected to spacetime translations — it is a "square root" of translations. This means that any SUSY-invariant vacuum has zero energy: if $Q|0\rangle = 0$ and$\bar{Q}|0\rangle = 0$, then $\langle 0|H|0\rangle = \langle 0|\{Q, \bar{Q}\}|0\rangle = 0$. Conversely, if $\langle 0|H|0\rangle > 0$, SUSY is spontaneously broken.

Coleman-Mandula theorem (1967): The only Lie algebra extending Poincare is the direct product with internal symmetries. Haag, Lopuszanski, and Sohnius (1975) showed this can be evaded by graded Lie algebras — supersymmetry is the unique such extension. This uniqueness is a powerful theoretical motivation: SUSY is not just one possibility among many, but the only way to extend spacetime symmetry in an interacting QFT.

Derivation 2: Superfields and Superspace

Superspace extends spacetime with anticommuting coordinates $\theta_\alpha, \bar{\theta}_{\dot{\alpha}}$. A superfield $\Phi(x, \theta, \bar{\theta})$ contains an entire supermultiplet. The chiral superfield (satisfying $\bar{D}_{\dot{\alpha}}\Phi = 0$) expands as:

$\Phi = \phi(y) + \sqrt{2}\theta\psi(y) + \theta\theta F(y)$

where $y^\mu = x^\mu + i\theta\sigma^\mu\bar{\theta}$, $\phi$ is a complex scalar,$\psi$ is a Weyl fermion, and $F$ is a complex auxiliary field. The SUSY transformation acts as:

$\delta_\xi \phi = \sqrt{2}\xi\psi, \quad \delta_\xi \psi = \sqrt{2}\xi F + i\sqrt{2}\sigma^\mu\bar{\xi}\partial_\mu\phi, \quad \delta_\xi F = i\sqrt{2}\bar{\xi}\bar{\sigma}^\mu\partial_\mu\psi$

Vector Superfield

Gauge fields live in a real vector superfield $V = V^\dagger$, containing in Wess-Zumino gauge:

$V_\text{WZ} = \theta\sigma^\mu\bar{\theta}A_\mu + \theta\theta\bar{\theta}\bar{\lambda} + \bar{\theta}\bar{\theta}\theta\lambda + \frac{1}{2}\theta\theta\bar{\theta}\bar{\theta}D$

This contains a gauge boson $A_\mu$, a gaugino $\lambda$ (Majorana fermion), and an auxiliary field $D$. The field strength superfield is:

$W_\alpha = -\frac{1}{4}\bar{D}\bar{D}D_\alpha V$

The SUSY Lagrangian for a chiral superfield is constructed from two functions: the Kahler potential $K(\Phi^\dagger, \Phi)$ (which determines the kinetic terms) and the superpotential $W(\Phi)$ (which determines the interactions). The F-term auxiliary field is $F_i = -\partial W/\partial\phi_i$, and the scalar potential is:

$V = \sum_i |F_i|^2 + \frac{1}{2}\sum_a D_a^2 = \sum_i \left|\frac{\partial W}{\partial\phi_i}\right|^2 + \frac{g^2}{2}\sum_a(\phi^\dagger T^a \phi)^2$

The scalar potential is completely determined by the superpotential and gauge interactions — there are no free parameters beyond those already present in $W$. This is a remarkable consequence of supersymmetry.

Non-renormalization theorem: The superpotential$W(\Phi)$ receives no perturbative corrections beyond those present at tree level. This powerful result, proven by Seiberg using holomorphy arguments, ensures that SUSY protects the Higgs mass from quadratic divergences to all orders in perturbation theory.

Derivation 3: The Minimal Supersymmetric Standard Model

The MSSM is the minimal supersymmetric extension of the Standard Model. Each SM particle gets a superpartner differing by half a unit of spin. The superpotential is:

$W_\text{MSSM} = y_u \hat{Q}\hat{H}_u\hat{u}^c + y_d \hat{Q}\hat{H}_d\hat{d}^c + y_e \hat{L}\hat{H}_d\hat{e}^c + \mu\hat{H}_u\hat{H}_d$

The MSSM requires two Higgs doublets $H_u, H_d$ (to give mass to both up-type and down-type quarks while maintaining holomorphy and canceling anomalies). The ratio of their VEVs is $\tan\beta = v_u/v_d$.

Particle Content

Squarks: $\tilde{q}_L, \tilde{u}_R, \tilde{d}_R$ — scalar partners of quarks (spin 0)

Sleptons: $\tilde{\ell}_L, \tilde{e}_R, \tilde{\nu}$ — scalar partners of leptons

Gauginos: gluino $\tilde{g}$, winos $\tilde{W}$, bino $\tilde{B}$ — fermion partners of gauge bosons

Higgsinos: $\tilde{H}_u, \tilde{H}_d$ mix with gauginos to form neutralinos $\tilde{\chi}^0_{1-4}$ and charginos $\tilde{\chi}^\pm_{1,2}$

R-parity: The MSSM conserves R-parity,$R = (-1)^{3(B-L)+2s}$, under which SM particles have $R = +1$ and superpartners have $R = -1$. This forbids rapid proton decay and makes the lightest superpartner (LSP) stable — a dark matter candidate.

Derivation 4: Soft SUSY Breaking

Since no superpartners have been observed at the electroweak scale, SUSY must be broken. The breaking must be "soft" — it lifts superpartner masses without reintroducing quadratic divergences. The allowed soft-breaking terms are:

$\mathcal{L}_\text{soft} = -m_{\tilde{f}}^2 |\tilde{f}|^2 - \frac{1}{2}M_a \tilde{\lambda}^a\tilde{\lambda}^a - (A_f y_f \tilde{f}\tilde{f}\tilde{f} + B\mu H_u H_d + \text{h.c.})$

These include scalar masses $m_{\tilde{f}}$, gaugino masses $M_a$ ($a = 1,2,3$), trilinear A-terms, and the B-term. In the most general case, the MSSM has 105 new parameters beyond the Standard Model.

Naturalness Argument

With soft breaking, the residual Higgs mass correction is:

$\delta m_H^2 \approx \frac{3y_t^2}{8\pi^2}m_{\tilde{t}}^2\ln\frac{\Lambda}{m_{\tilde{t}}}$

For $m_H = 125$ GeV to be natural ($\delta m_H^2 \lesssim m_H^2$), the stop mass should satisfy $m_{\tilde{t}} \lesssim 500$ GeV. Current LHC bounds require$m_{\tilde{t}} \gtrsim 1$ TeV, implying at least ~10% fine-tuning.

SUSY breaking mediation: Several mechanisms transmit SUSY breaking from a hidden sector to the MSSM: gravity mediation (soft masses $\sim F/M_\text{Pl}$), gauge mediation (via messenger fields), and anomaly mediation (via the conformal anomaly). Each predicts different patterns of superpartner masses.

Derivation 5: The MSSM Higgs Mass Prediction

In the MSSM, the lightest CP-even Higgs boson mass has a tree-level upper bound:

$m_h^2 \leq m_Z^2 \cos^2 2\beta \leq m_Z^2 \approx (91 \text{ GeV})^2$

This is too light! Radiative corrections from top/stop loops are essential:

$m_h^2 \approx m_Z^2\cos^2 2\beta + \frac{3m_t^4}{4\pi^2 v^2}\left[\ln\frac{m_{\tilde{t}}^2}{m_t^2} + \frac{X_t^2}{m_{\tilde{t}}^2}\left(1 - \frac{X_t^2}{12m_{\tilde{t}}^2}\right)\right]$

where $X_t = A_t - \mu/\tan\beta$ is the stop mixing parameter. The maximal mixing scenario ($X_t \approx \sqrt{6}\,m_{\tilde{t}}$) maximizes $m_h$. Achieving$m_h = 125$ GeV requires either heavy stops ($m_{\tilde{t}} \gtrsim 1$ TeV with moderate mixing) or maximal mixing with lighter stops.

The MSSM Higgs sector has five physical states: $h, H$ (CP-even), $A$ (CP-odd), and $H^\pm$ (charged). In the decoupling limit ($m_A \gg m_Z$), $h$ behaves exactly like the SM Higgs — consistent with LHC measurements. The heavy Higgs states have masses $m_H \approx m_A \approx m_{H^\pm}$ in this limit, and their couplings to down-type fermions are enhanced by $\tan\beta$. LHC searches for heavy Higgs bosons in the $H/A \to \tau\tau$ channel exclude $m_A \lesssim 1$ TeV for large$\tan\beta$.

The MSSM Higgs mass prediction $m_h \leq 135$ GeV (including radiative corrections) was made in the early 1990s, two decades before the LHC discovery. The fact that the observed $m_h = 125$ GeV falls within this window is a nontrivial consistency check. However, achieving 125 GeV typically requires heavy stops ($m_{\tilde{t}} \gtrsim 1$ TeV with maximal mixing, or $m_{\tilde{t}} \gtrsim 5$ TeV without mixing), creating tension with the original naturalness motivation for SUSY.

The 125 GeV coincidence: The observed Higgs mass sits precisely in the narrow window $90\text{-}130$ GeV predicted by the MSSM. This is a nontrivial success — the MSSM predicted $m_h \lesssim 135$ GeV decades before the LHC discovery. However, achieving 125 GeV requires heavy stops, creating tension with naturalness.

Computational Analysis

This simulation demonstrates the cancellation of quadratic divergences in SUSY, displays a benchmark MSSM superpartner spectrum, maps the fine-tuning as a function of stop mass and UV cutoff, and shows the MSSM Higgs mass prediction for different values of $\tan\beta$ and stop mass.

Supersymmetry: MSSM Spectrum, Naturalness & Higgs Mass

Python

script.py220 lines

#!/usr/bin/env python3
"""
supersymmetry.py - SUSY Algebra, Spectrum & Naturalness
Computes MSSM spectrum, mass corrections, and superpartner phenomenology
"""
import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import base64
from io import BytesIO

print("=" * 72)
print("  SUPERSYMMETRY - NUMERICAL ANALYSIS")
print("=" * 72)
print()

# ============================================================
# Part 1: SUSY Cancellation of Quadratic Divergences
# ============================================================

print("Cancellation of Quadratic Divergences:")
print("-" * 50)

# Top quark contribution to delta m_H^2
y_t = 1.0
Lambda = 10000  # cutoff in GeV (10 TeV)
N_c = 3

delta_top = -N_c * y_t**2 / (8 * np.pi**2) * Lambda**2
print(f"  Top quark loop:  delta m_H^2 = {delta_top:.0f} GeV^2")
print(f"  (= -{abs(delta_top)**0.5:.0f} GeV)^2")

# Stop (scalar top partner) contribution
# In SUSY: delta_stop = +N_c * y_t^2 / (8*pi^2) * Lambda^2 (opposite sign!)
delta_stop = +N_c * y_t**2 / (8 * np.pi**2) * Lambda**2
print(f"  Stop squark loop: delta m_H^2 = +{delta_stop:.0f} GeV^2")
print(f"  Sum (exact SUSY): {delta_top + delta_stop:.0f} GeV^2 (EXACT cancellation!)")
print()

# With soft breaking (stop mass m_tilde)
print("With soft SUSY breaking (stop mass m_tilde):")
print("-" * 50)
for m_stop in [500, 1000, 2000, 5000]:
    # Residual correction: delta m_H^2 ~ N_c y_t^2/(8 pi^2) m_stop^2 ln(Lambda/m_stop)
    delta_soft = N_c * y_t**2 / (8*np.pi**2) * m_stop**2 * np.log(Lambda/m_stop)
    fine_tuning = (125**2) / (2*delta_soft) * 100
    print(f"  m_stop = {m_stop:5d} GeV: delta m_H^2 = ({delta_soft**0.5:.0f} GeV)^2, fine-tuning ~ {fine_tuning:.1f}%")

print()

# ============================================================
# Part 2: MSSM Spectrum
# ============================================================

print("MSSM Superpartner Spectrum (benchmark):")
print("-" * 50)

particles = {
    'gluino': 2500,
    'squarks (1st gen)': 2800,
    'stop_1': 1200,
    'stop_2': 2000,
    'sbottom_1': 1800,
    'neutralino_1 (LSP)': 300,
    'neutralino_2': 600,
    'chargino_1': 580,
    'slepton_L': 800,
    'slepton_R': 400,
    'sneutrino': 780,
    'heavy Higgs H': 1500,
    'A (pseudoscalar)': 1490,
    'H+- (charged Higgs)': 1500,
}

for name, mass in particles.items():
    print(f"  {name:25s}: {mass:6d} GeV")

print()

fig, axes = plt.subplots(2, 2, figsize=(14, 11))
fig.patch.set_facecolor('#0f0a1a')

# --- Plot 1: Cancellation of quadratic divergences ---
ax1 = axes[0, 0]
ax1.set_facecolor('#1a1025')
for spine in ax1.spines.values():
    spine.set_color('#4c1d95')
ax1.tick_params(colors='#c4b5fd')
ax1.grid(True, color='#4c1d95', alpha=0.3)

Lambda_range = np.logspace(2, 6, 200)

delta_top_range = N_c * y_t**2 / (8*np.pi**2) * Lambda_range**2
m_stop_vals = [500, 1000, 2000]
colors_s = ['#a78bfa', '#f97316', '#22d3ee']

ax1.plot(np.log10(Lambda_range), np.log10(np.sqrt(delta_top_range)), linewidth=2.5,
         color='#ef4444', label='SM (no SUSY): ~ Lambda')

for ms, col in zip(m_stop_vals, colors_s):
    delta_susy = N_c * y_t**2 / (8*np.pi**2) * ms**2 * np.log(Lambda_range/ms)
    delta_susy = np.maximum(delta_susy, 1)  # avoid log of negative
    ax1.plot(np.log10(Lambda_range), np.log10(np.sqrt(delta_susy)), linewidth=2,
             color=col, label=f'SUSY m_stop = {ms} GeV')

ax1.axhline(y=np.log10(125), color='white', linewidth=1.5, linestyle='--', alpha=0.5, label='m_H = 125 GeV')

ax1.set_xlabel('log10(Lambda / GeV)', color='#a78bfa', fontsize=11)
ax1.set_ylabel('log10(delta m_H / GeV)', color='#a78bfa', fontsize=11)
ax1.set_title('SUSY: Quadratic -> Logarithmic Sensitivity', color='#e9d5ff', fontsize=13, fontweight='bold')
ax1.legend(facecolor='#1a1025', edgecolor='#4c1d95', labelcolor='#c4b5fd', fontsize=9)

# --- Plot 2: MSSM mass spectrum ---
ax2 = axes[0, 1]
ax2.set_facecolor('#1a1025')
for spine in ax2.spines.values():
    spine.set_color('#4c1d95')
ax2.tick_params(colors='#c4b5fd')

categories = ['Gauginos', 'Squarks', 'Sleptons', 'Higgs']
cat_particles = [
    [('chi_1^0', 300), ('chi_2^0', 600), ('chi_1^+', 580), ('gluino', 2500)],
    [('stop_1', 1200), ('stop_2', 2000), ('sbottom', 1800), ('squark', 2800)],
    [('sel_R', 400), ('sel_L', 800), ('snu', 780)],
    [('h', 125), ('H', 1500), ('A', 1490), ('H+-', 1500)],
]
cat_colors = ['#a78bfa', '#f97316', '#22d3ee', '#ef4444']

x_offset = 0
for cat_idx, (cat, parts, col) in enumerate(zip(categories, cat_particles, cat_colors)):
    for i, (name, mass) in enumerate(parts):
        x = x_offset + i * 0.7
        ax2.bar(x, mass, 0.5, color=col, alpha=0.8)
        ax2.text(x, mass + 50, name, ha='center', va='bottom', color='#c4b5fd', fontsize=7, rotation=45)
    x_offset += len(parts) * 0.7 + 0.5

ax2.set_ylabel('Mass (GeV)', color='#a78bfa', fontsize=11)
ax2.set_title('MSSM Benchmark Spectrum', color='#e9d5ff', fontsize=13, fontweight='bold')
ax2.set_xticks([])

# Add category labels
ax2.text(1.0, -300, 'Gauginos', color='#a78bfa', fontsize=10, ha='center')
ax2.text(4.5, -300, 'Squarks', color='#f97316', fontsize=10, ha='center')
ax2.text(7.5, -300, 'Sleptons', color='#22d3ee', fontsize=10, ha='center')
ax2.text(10.0, -300, 'Higgs', color='#ef4444', fontsize=10, ha='center')

# --- Plot 3: Fine-tuning as function of stop mass ---
ax3 = axes[1, 0]
ax3.set_facecolor('#1a1025')
for spine in ax3.spines.values():
    spine.set_color('#4c1d95')
ax3.tick_params(colors='#c4b5fd')
ax3.grid(True, color='#4c1d95', alpha=0.3)

m_stop_range = np.linspace(200, 5000, 500)
Lambda_vals_ft = [1e4, 1e6, 1e16]
colors_ft = ['#a78bfa', '#f97316', '#22d3ee']

for Lam, col in zip(Lambda_vals_ft, colors_ft):
    delta = N_c * y_t**2 / (8*np.pi**2) * m_stop_range**2 * np.log(Lam/m_stop_range)
    ft = delta / (125**2 / 2)  # Delta = 1/fine-tuning
    ax3.plot(m_stop_range, ft, linewidth=2.5, color=col, label=f'Lambda = {Lam:.0e} GeV')

ax3.axhline(y=10, color='#ef4444', linewidth=1.5, linestyle='--', alpha=0.7, label='10% tuning')
ax3.axhline(y=100, color='#ef4444', linewidth=1, linestyle=':', alpha=0.5, label='1% tuning')

ax3.set_xlabel('Stop mass m_stop (GeV)', color='#a78bfa', fontsize=11)
ax3.set_ylabel('Fine-tuning Delta', color='#a78bfa', fontsize=11)
ax3.set_title('Naturalness: Fine-Tuning vs Stop Mass', color='#e9d5ff', fontsize=13, fontweight='bold')
ax3.legend(facecolor='#1a1025', edgecolor='#4c1d95', labelcolor='#c4b5fd', fontsize=9)
ax3.set_yscale('log')
ax3.set_ylim(0.1, 1e4)

# --- Plot 4: MSSM Higgs mass prediction ---
ax4 = axes[1, 1]
ax4.set_facecolor('#1a1025')
for spine in ax4.spines.values():
    spine.set_color('#4c1d95')
ax4.tick_params(colors='#c4b5fd')
ax4.grid(True, color='#4c1d95', alpha=0.3)

m_Z = 91.188
# Tree-level MSSM Higgs mass: m_h <= m_Z |cos(2 beta)|
# Radiative correction: delta m_h^2 ~ (3 m_t^4)/(4 pi^2 v^2) ln(m_stop^2/m_t^2)
m_t = 172.76
v_higgs = 246.22
m_stop_h = np.linspace(200, 10000, 500)

tan_beta_vals = [2, 5, 10, 50]
for tb, col in zip(tan_beta_vals, ['#a78bfa', '#f97316', '#22d3ee', '#ef4444']):
    cos2b = (1 - tb**2) / (1 + tb**2)
    m_h_tree2 = m_Z**2 * cos2b**2
    delta_rad = 3 * m_t**4 / (4 * np.pi**2 * v_higgs**2) * np.log(m_stop_h**2 / m_t**2)
    m_h = np.sqrt(np.maximum(m_h_tree2 + delta_rad, 0))
    ax4.plot(m_stop_h, m_h, linewidth=2, color=col, label=f'tan(beta) = {tb}')

ax4.axhline(y=125.25, color='white', linewidth=2, linestyle='--', alpha=0.7, label='m_h = 125.25 GeV')
ax4.axhspan(124, 127, alpha=0.1, color='white')

ax4.set_xlabel('Stop mass (GeV)', color='#a78bfa', fontsize=11)
ax4.set_ylabel('Lightest Higgs mass m_h (GeV)', color='#a78bfa', fontsize=11)
ax4.set_title('MSSM Higgs Mass vs Stop Mass', color='#e9d5ff', fontsize=13, fontweight='bold')
ax4.legend(facecolor='#1a1025', edgecolor='#4c1d95', labelcolor='#c4b5fd', fontsize=10)
ax4.set_ylim(50, 150)

plt.tight_layout(pad=2.0)

buf = BytesIO()
plt.savefig(buf, format='png', dpi=150, bbox_inches='tight', facecolor='#0f0a1a')
buf.seek(0)
img_b64 = base64.b64encode(buf.read()).decode()
print(f'<img src="data:image/png;base64,{img_b64}" alt="Supersymmetry Calculations" />')
buf.close()

print()
print("Key insight: SUSY converts quadratic divergences to logarithmic ones, solving the")
print("hierarchy problem. The observed m_h = 125 GeV requires multi-TeV stops in the MSSM,")
print("creating tension with naturalness expectations.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

SUSY Dark Matter: The Neutralino

In the MSSM with conserved R-parity, the lightest supersymmetric particle (LSP) is stable. If the LSP is the lightest neutralino $\tilde{\chi}_1^0$ — a mixture of bino, wino, and higgsinos — it is a natural WIMP dark matter candidate.

Neutralino Mass Matrix

The neutralino mass matrix in the $(\tilde{B}, \tilde{W}^0, \tilde{H}_d^0, \tilde{H}_u^0)$basis is:

$M_{\tilde{\chi}^0} = \begin{pmatrix} M_1 & 0 & -m_Z s_W c_\beta & m_Z s_W s_\beta \\ 0 & M_2 & m_Z c_W c_\beta & -m_Z c_W s_\beta \\ -m_Z s_W c_\beta & m_Z c_W c_\beta & 0 & -\mu \\ m_Z s_W s_\beta & -m_Z c_W s_\beta & -\mu & 0 \end{pmatrix}$

The eigenvalues and mixings determine the LSP composition and thus its annihilation cross section, which controls the relic density:

Bino-like: Couples weakly, tends to overproduce dark matter unless co-annihilation or resonance effects are present

Wino-like: $m_{\tilde{\chi}} \approx 2.7$ TeV gives correct relic density; strong annihilation into $W^+W^-$

Higgsino-like: $m_{\tilde{\chi}} \approx 1.1$ TeV gives correct abundance; nearly degenerate with chargino

Well-tempered: Specific bino-higgsino or bino-wino mixtures can achieve $\Omega h^2 = 0.12$ over a wide mass range

Direct detection prospects: Spin-independent neutralino-nucleon scattering proceeds through Higgs exchange and squark exchange. Current experiments (LZ, XENONnT, PandaX) probe cross sections down to$\sim 10^{-47}$ cm$^2$, excluding large portions of the MSSM parameter space. The neutrino fog ($\sim 10^{-49}$ cm$^2$) sets the ultimate background limit.

Extended Supersymmetry and Exact Results

Theories with $\mathcal{N} \geq 2$ supersymmetry are more constrained and admit powerful exact results:

$\mathcal{N}=2$: Seiberg-Witten theory (1994) gives the exact low-energy effective action for $SU(2)$ gauge theory, including the exact prepotential, the spectrum of BPS states, and a proof of confinement via monopole condensation.

$\mathcal{N}=4$: The maximally supersymmetric Yang-Mills theory is exactly conformal (the beta function vanishes to all orders). It is the field theory side of the AdS/CFT correspondence, relating it to type IIB string theory on $AdS_5 \times S^5$.

Seiberg duality: $\mathcal{N}=1$ SQCD with$N_f$ flavors and $SU(N_c)$ gauge group is dual to SQCD with$SU(N_f - N_c)$ and $N_f$ flavors in the infrared. This electric-magnetic duality exchanges strong and weak coupling.

LHC status: As of 2025, no superpartners have been observed at the LHC. Gluinos are excluded below ~2.3 TeV, first/second generation squarks below ~1.8 TeV, and stops below ~1.2 TeV (depending on the decay mode and mass spectrum). The absence of low-energy SUSY creates tension with naturalness, but does not rule out SUSY — it may simply be at higher scales than originally hoped.

SUSY Breaking Mediation Mechanisms

SUSY is broken in a hidden sector and communicated to the visible (MSSM) sector via a mediation mechanism. Different mechanisms predict characteristic patterns of superpartner masses:

Gravity Mediation (mSUGRA/CMSSM)

SUSY breaking is transmitted by gravitational interactions, giving soft masses$m_\text{soft} \sim F/M_\text{Pl}$ where $F$ is the SUSY-breaking F-term. In the constrained MSSM, all scalars share a common mass $m_0$, all gauginos share$m_{1/2}$, and all trilinear couplings share $A_0$ at the GUT scale. RG evolution to the weak scale generates the physical spectrum.

Gauge Mediation (GMSB)

Messenger fields charged under the SM gauge group transmit SUSY breaking. Soft masses are generated at one and two loops:

$M_a \sim \frac{\alpha_a}{4\pi}\frac{F}{M_\text{mess}}, \quad m_{\tilde{f}}^2 \sim \sum_a C_a\left(\frac{\alpha_a}{4\pi}\right)^2\left(\frac{F}{M_\text{mess}}\right)^2$

A key prediction: the gravitino is the LSP with mass $m_{3/2} = F/(\sqrt{3}M_\text{Pl})$, and the NLSP (often a stau or neutralino) decays to it. GMSB automatically suppresses flavor-changing neutral currents because the soft masses are determined by gauge quantum numbers alone.

Anomaly Mediation (AMSB)

SUSY breaking is transmitted through the superconformal anomaly. Gaugino masses are proportional to the beta function:

$M_a = \frac{b_a g_a^2}{16\pi^2}m_{3/2}$

This predicts the wino to be the lightest gaugino (since $b_2 = 1$ is the smallest beta coefficient in the MSSM). However, pure AMSB gives tachyonic sleptons, requiring additional contributions to the scalar masses.

Experimental discrimination: Different mediation mechanisms predict different mass orderings, decay chains, and collider signatures. For example, GMSB predicts photon-rich final states (from $\tilde{\chi}^0_1 \to \gamma\tilde{G}$), while gravity mediation typically predicts missing energy from a stable neutralino LSP. If superpartners are ever discovered, measuring the mass spectrum would identify the SUSY-breaking mechanism.

Summary: Supersymmetry Essentials

SUSY Algebra

$\{Q_\alpha, \bar{Q}_{\dot{\beta}}\} = 2\sigma^\mu_{\alpha\dot{\beta}}P_\mu$. Unique extension of Poincare symmetry. Equal boson-fermion degeneracy in each multiplet.

Naturalness

Boson and fermion loops cancel: $\Lambda^2 \to m_\text{soft}^2\ln(\Lambda/m_\text{soft})$. Quadratic sensitivity eliminated, logarithmic sensitivity remains.

MSSM

Two Higgs doublets, superpartners for all SM particles, R-parity conservation. Lightest neutralino as dark matter candidate. 105 new soft-breaking parameters.

Higgs Mass

Tree level: $m_h \leq m_Z$. Radiative corrections from stops raise it to 125 GeV for $m_{\tilde{t}} \sim 1\text{-}10$ TeV, depending on mixing.

Dark Matter Candidate

With R-parity conservation, the lightest neutralino $\tilde{\chi}_1^0$ is a stable WIMP. Bino, wino, and higgsino admixtures give different relic densities: pure wino at $\sim 2.7$ TeV, pure higgsino at $\sim 1.1$ TeV give correct $\Omega h^2$.

Gauge Coupling Unification

The three SM gauge couplings unify precisely at $M_\text{GUT} \approx 2 \times 10^{16}$ GeV in the MSSM with superpartners at the TeV scale — the most compelling quantitative success of low-energy supersymmetry.

Mediation Mechanisms

Gravity, gauge, and anomaly mediation predict distinct mass spectra and collider signatures. The pattern of superpartner masses, if discovered, would reveal the mechanism of SUSY breaking and its connection to the hidden sector.

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