Table of Contents

1. Introduction
2. The Motivation for Supersymmetry
3. Fermions and Bosons in Quantum Field Theory
4. The Hierarchy Problem
5. Basics of Supersymmetry (SUSY)
6. Supersymmetry Algebra
7. Superpartners and Supermultiplets
8. Chiral and Vector Supermultiplets
9. Superspace and Superfields
10. SUSY Transformations
11. SUSY Lagrangians
12. Wess–Zumino Model
13. Gauge Supersymmetry
14. SUSY Invariant Actions
15. Extended Supersymmetry
16. Supersymmetry Breaking
17. Soft SUSY Breaking
18. Minimal Supersymmetric Standard Model (MSSM)
19. R-Parity and Dark Matter Candidates
20. Phenomenological Implications
21. SUSY and Unification of Forces
22. SUSY and String Theory
23. Experimental Searches for SUSY
24. Challenges and Open Questions
25. Conclusion

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1. Introduction

Supersymmetry (SUSY) is a theoretical framework that proposes a symmetry between fermions (matter particles) and bosons (force carriers). Each particle has a corresponding “superpartner” with different spin statistics. SUSY offers solutions to several major issues in particle physics and is a cornerstone in many extensions of the Standard Model and string theory.

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2. The Motivation for Supersymmetry

Supersymmetry addresses key theoretical issues:

• The hierarchy problem
• Grand unification of forces
• Inclusion in string theory
• Dark matter candidates
• Better UV behavior of quantum field theories

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3. Fermions and Bosons in Quantum Field Theory

Particles are classified as:

• Fermions: half-integer spin, follow Pauli exclusion principle
• Bosons: integer spin, can occupy the same state

Standard Model includes both:

• Fermions: quarks, leptons
• Bosons: photons, W/Z bosons, gluons, Higgs

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4. The Hierarchy Problem

The mass of the Higgs boson receives large quantum corrections, leading to fine-tuning. Supersymmetry cancels divergent contributions from boson and fermion loops, stabilizing the Higgs mass.

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5. Basics of Supersymmetry (SUSY)

SUSY postulates:

• Each boson has a fermionic superpartner
• Each fermion has a bosonic superpartner
• These partners differ by half a unit of spin

Example:

• Electron ↔ Selectron
• Photon ↔ Photino
• Quark ↔ Squark

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6. Supersymmetry Algebra

The SUSY algebra extends the Poincaré algebra with generators \( Q_\alpha \) and \( \bar{Q}_{\dot{\alpha}} \) satisfying:

\[
\{ Q_\alpha, \bar{Q}{\dot{\beta}} \} = 2 \sigma^\mu{\alpha \dot{\beta}} P_\mu
\]

This relates internal spin and spacetime translations.

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7. Superpartners and Supermultiplets

SUSY groups particles into supermultiplets:

• Particles in a supermultiplet differ by spin
• Equal number of bosonic and fermionic degrees of freedom

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8. Chiral and Vector Supermultiplets

• Chiral supermultiplet: scalar and Weyl fermion
• Vector supermultiplet: gauge boson and gaugino

These are building blocks of SUSY field theories.

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9. Superspace and Superfields

SUSY is conveniently formulated in superspace:

• Superspace extends spacetime by adding Grassmann coordinates \( \theta \)
• Fields are combined into superfields with components of different spins

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10. SUSY Transformations

A SUSY transformation shifts bosons into fermions and vice versa:

\[
\delta \phi = \bar{\epsilon} \psi, \quad \delta \psi = i \sigma^\mu \bar{\epsilon} \partial_\mu \phi
\]

Here \( \epsilon \) is a spinor parameter.

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11. SUSY Lagrangians

SUSY Lagrangians are constructed to be invariant under SUSY transformations. They include:

• Kinetic terms for bosons and fermions
• Interaction terms derived from superpotentials

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12. Wess–Zumino Model

The simplest interacting SUSY theory:

• Contains one scalar and one Majorana fermion
• Includes interactions preserving SUSY

This model is the prototype for building more complex SUSY theories.

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13. Gauge Supersymmetry

Gauge interactions can be made supersymmetric by introducing vector superfields and ensuring covariant derivatives respect SUSY.

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14. SUSY Invariant Actions

An action \( S \) is SUSY invariant if:

\[
\delta S = 0 \quad \text{under SUSY transformations}
\]

This ensures the consistency of the theory under quantum corrections.

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15. Extended Supersymmetry

N=1 SUSY: minimal in 4D
N=2, N=4 SUSY: extended theories with multiple supercharges

• Higher N leads to more symmetry and more constraints
• N=4 is finite and conformal in 4D

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16. Supersymmetry Breaking

SUSY must be broken, since no superpartners have been observed. Breaking mechanisms:

• Spontaneous breaking
• Soft breaking terms

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17. Soft SUSY Breaking

Soft terms break SUSY without spoiling renormalizability:

• Scalar masses
• Gaugino masses
• Trilinear couplings

They allow viable SUSY phenomenology.

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18. Minimal Supersymmetric Standard Model (MSSM)

The MSSM is the simplest SUSY extension of the SM:

• Doubles particle content
• Introduces R-parity
• Provides a dark matter candidate (neutralino)

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19. R-Parity and Dark Matter Candidates

R-parity is a discrete symmetry:

\[
R = (-1)^{3(B-L) + 2s}
\]

SM particles have \( R = +1 \), superpartners have \( R = -1 \). The lightest SUSY particle (LSP) is stable and a dark matter candidate.

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20. Phenomenological Implications

• Gauge coupling unification
• Solutions to hierarchy problem
• Radiative electroweak symmetry breaking
• Dark matter predictions

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21. SUSY and Unification of Forces

SUSY improves gauge coupling unification at high energies, supporting Grand Unified Theories (GUTs).

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22. SUSY and String Theory

SUSY is integral to string theory:

• Superstrings require SUSY
• Helps cancel anomalies
• Enables consistent theories of quantum gravity

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23. Experimental Searches for SUSY

SUSY has not been observed at the LHC:

• Limits on squark and gluino masses > TeV
• Searches continue in higher energy colliders and dark matter experiments

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24. Challenges and Open Questions

• Why has SUSY not been observed?
• What is the SUSY breaking scale?
• Is MSSM the correct extension?
• Are there signals in cosmology or neutrino physics?

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25. Conclusion

Supersymmetry is a profound theoretical framework that provides deep insights into the structure of matter and spacetime. Though yet unobserved, it remains a leading candidate for physics beyond the Standard Model. Its implications for unification, quantum gravity, and dark matter continue to drive experimental and theoretical exploration.

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Table of Contents

1. Introduction
2. Classical Symmetries in Field Theories
3. What Are Anomalies?
4. Noether’s Theorem and Conservation Laws
5. Path Integral and Anomalous Jacobians
6. Chiral Anomalies
7. Example: Axial Anomaly in QED
8. Triangle Diagrams and Anomaly Calculations
9. Anomaly Cancellation Conditions
10. Gauge Anomalies vs Global Anomalies
11. Gravitational Anomalies
12. The Adler–Bell–Jackiw (ABJ) Anomaly
13. The Role of γ₅ and Dimensional Regularization
14. Anomalies in the Standard Model
15. Witten’s Global SU(2) Anomaly
16. Index Theorem and Topological Aspects
17. The Atiyah–Singer Index Theorem
18. Anomalies and Instantons
19. Chern–Simons Terms and Anomalous Currents
20. Gauge Invariance and Renormalizability
21. The Green–Schwarz Mechanism
22. Anomalies in String Theory
23. Anomaly Inflow and Brane Dynamics
24. Physical Consequences of Anomalies
25. Conclusion

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1. Introduction

Quantum anomalies occur when a symmetry of the classical action is broken upon quantization. Although the classical theory may have conserved currents corresponding to continuous symmetries, the quantum theory may fail to preserve them due to the regularization or renormalization process.

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2. Classical Symmetries in Field Theories

In classical field theory, symmetries correspond to conserved currents via Noether’s theorem. These conservation laws typically survive quantization — except when an anomaly is present.

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3. What Are Anomalies?

An anomaly is the breaking of a classical symmetry due to quantum effects. Anomalies usually emerge during the regularization of divergent integrals, especially in loop diagrams in perturbation theory.

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4. Noether’s Theorem and Conservation Laws

Noether’s theorem relates continuous symmetries to conserved currents:

\[
\partial_\mu j^\mu = 0
\]

An anomaly manifests as:

\[
\partial_\mu j^\mu = \frac{e^2}{16\pi^2} F_{\mu\nu} \tilde{F}^{\mu\nu}
\]

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5. Path Integral and Anomalous Jacobians

In the path integral formulation, anomalies arise from non-invariant measures. If the functional determinant changes under a symmetry transformation, it indicates an anomaly.

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6. Chiral Anomalies

Chiral anomalies occur when left- and right-handed fermions transform differently under a gauge group. They appear in theories with massless fermions and chiral symmetry.

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7. Example: Axial Anomaly in QED

The axial current \( j_5^\mu = \bar{\psi} \gamma^\mu \gamma^5 \psi \) is classically conserved in massless QED. However, loop calculations reveal:

\[
\partial_\mu j_5^\mu = \frac{e^2}{16\pi^2} F_{\mu\nu} \tilde{F}^{\mu\nu}
\]

This is the famous Adler–Bell–Jackiw (ABJ) anomaly.

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8. Triangle Diagrams and Anomaly Calculations

Anomalies are calculated using triangle Feynman diagrams with one axial and two vector vertices. Regularization ambiguities cause current non-conservation.

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9. Anomaly Cancellation Conditions

Gauge anomalies must cancel for consistency. In the Standard Model, anomalies cancel between different fermion generations due to specific charge assignments.

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10. Gauge Anomalies vs Global Anomalies

• Gauge anomalies: break gauge invariance, render theory inconsistent.
• Global anomalies: affect global symmetries but don’t spoil consistency.

Both are important in building quantum field theories.

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11. Gravitational Anomalies

Occur when fermionic currents couple to gravity. The non-conservation of energy-momentum tensor or gravitational currents can spoil diffeomorphism invariance.

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12. The Adler–Bell–Jackiw (ABJ) Anomaly

One of the first anomalies discovered. It explains why the neutral pion \( \pi^0 \rightarrow \gamma\gamma \) decay occurs much faster than expected from symmetry arguments.

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13. The Role of γ₅ and Dimensional Regularization

Dimensional regularization struggles with the treatment of \( \gamma^5 \) in \( d \neq 4 \). This technical issue makes calculating anomalies subtle and regularization-dependent.

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14. Anomalies in the Standard Model

The SM is anomaly-free due to delicate cancellations:

• \( SU(2)^2 \times U(1) \)
• \( U(1)^3 \)
• \( \text{Gravitational} \times U(1) \)

These cancellations constrain possible fermion representations.

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15. Witten’s Global SU(2) Anomaly

This is a global anomaly in SU(2) involving topological considerations. It occurs if a theory has an odd number of SU(2) doublets, which would spoil gauge invariance.

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16. Index Theorem and Topological Aspects

The Atiyah–Singer Index Theorem relates the difference between zero modes of the Dirac operator and the topological charge of gauge fields, connecting anomalies to topology.

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17. The Atiyah–Singer Index Theorem

\[
n_+ – n_- = \frac{1}{32\pi^2} \int d^4x\, F_{\mu\nu} \tilde{F}^{\mu\nu}
\]

This connects anomalies to instantons and topologically nontrivial configurations.

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18. Anomalies and Instantons

Instantons are non-perturbative solutions that mediate tunneling between vacua. They contribute to anomalous processes that violate otherwise conserved symmetries (e.g., baryon/lepton number).

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19. Chern–Simons Terms and Anomalous Currents

Chern–Simons terms can be added to cancel anomalies via boundary inflow. They are crucial in anomaly inflow mechanisms in string theory and condensed matter systems.

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20. Gauge Invariance and Renormalizability

Uncancelled gauge anomalies destroy the renormalizability and consistency of a quantum field theory. Thus, all gauge anomalies must cancel for the theory to be viable.

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21. The Green–Schwarz Mechanism

In string theory, anomalies are canceled via the Green–Schwarz mechanism, where a two-form field cancels anomalies through higher-dimensional interactions.

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22. Anomalies in String Theory

Consistent string theories (like heterotic strings) must be anomaly-free. This imposes constraints on the allowed gauge groups and dimensions of spacetime.

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23. Anomaly Inflow and Brane Dynamics

Anomalies on D-branes are canceled by bulk contributions flowing into the brane. This anomaly inflow ensures consistency of the combined bulk-brane system.

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24. Physical Consequences of Anomalies

• Proton decay suppression
• \( \pi^0 \rightarrow \gamma\gamma \) decay
• Strong CP problem and axions
• Confinement and chiral symmetry breaking

Anomalies impact observable processes and constrain model building.

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25. Conclusion

Quantum anomalies highlight the delicate interplay between symmetry and quantization. While often subtle, they carry profound implications — from the structure of the Standard Model to string theory and beyond. Understanding and managing anomalies is essential for constructing consistent quantum theories and probing the fundamental structure of nature.

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Table of Contents

1. Introduction
2. The Need for Mass in Gauge Theories
3. Spontaneous Symmetry Breaking
4. The Goldstone Theorem
5. Higgs Field and Its Potential
6. Mexican Hat Potential
7. Classical Field Solution
8. Gauge Symmetry and the Higgs Field
9. Abelian Higgs Mechanism (U(1))
10. Gauge Boson Mass Generation
11. Degrees of Freedom Before and After
12. Non-Abelian Higgs Mechanism (SU(2))
13. Electroweak Symmetry Breaking
14. Higgs Doublet and Vacuum Expectation Value (VEV)
15. Masses of W and Z Bosons
16. Massless Photon and Gauge Invariance
17. Fermion Masses via Yukawa Couplings
18. Prediction and Discovery of the Higgs Boson
19. Properties of the Higgs Boson
20. Experimental Confirmation at the LHC
21. Role in the Standard Model
22. Extensions Beyond the Standard Model
23. Higgs Self-Coupling and Stability
24. Open Questions in Higgs Physics
25. Conclusion

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1. Introduction

The Higgs mechanism is a process by which gauge bosons in the Standard Model acquire mass through interactions with a scalar field, while preserving gauge invariance. It solves the problem of how particles like the W and Z bosons can be massive without explicitly breaking the symmetries of the electroweak theory.

2. The Need for Mass in Gauge Theories

Mass terms like

\[
\frac{1}{2} m^2 A_\mu A^\mu
\]

break gauge invariance. However, observations confirm that weak interaction mediators (W and Z) are massive. Thus, an alternative mechanism is required—one that introduces mass dynamically without breaking local gauge symmetry.

3. Spontaneous Symmetry Breaking

Spontaneous symmetry breaking (SSB) occurs when the Lagrangian of a system is symmetric, but the ground state (vacuum) is not. This results in physical phenomena not manifest in the equations themselves but in the solutions.

4. The Goldstone Theorem

In systems with continuous global symmetry that undergo SSB, massless scalar particles—Goldstone bosons—appear. However, in gauge theories, these massless scalars are “eaten” by gauge bosons to give them mass, leading to the Higgs mechanism.

5. Higgs Field and Its Potential

A complex scalar field \( \phi \) with a self-interaction potential:

\[
V(\phi) = \mu^2 \phi^\dagger \phi + \lambda (\phi^\dagger \phi)^2
\]

When \( \mu^2 < 0 \), the potential takes a “Mexican hat” shape and the field acquires a nonzero vacuum expectation value (VEV).

6. Mexican Hat Potential

The shape of the potential implies a ring of degenerate minima. Any point on the circle of minima can be chosen as the vacuum, breaking the original symmetry:

\[
\langle \phi \rangle = \frac{v}{\sqrt{2}} e^{i\theta}
\]

Choosing a specific \( \theta \) breaks the symmetry spontaneously.

7. Classical Field Solution

By selecting a vacuum (say, \( \theta = 0 \)):

\[
\phi(x) = \frac{1}{\sqrt{2}}(v + h(x))
\]

The field is expanded around the vacuum, introducing a real scalar field \( h(x) \)—the Higgs boson.

8. Gauge Symmetry and the Higgs Field

In a gauge theory (local symmetry), introducing a scalar doublet field and coupling it to gauge fields modifies the dynamics. The would-be Goldstone bosons are absorbed by the gauge bosons, making them massive.

9. Abelian Higgs Mechanism (U(1))

Consider a U(1) gauge theory with a complex scalar field. After symmetry breaking:

• The scalar acquires a VEV.
• The gauge field becomes massive.
• A real scalar particle (the Higgs boson) remains.

This toy model demonstrates the core features of the Higgs mechanism.

10. Gauge Boson Mass Generation

The kinetic term:

\[
|D_\mu \phi|^2 = \left| \left( \partial_\mu + igA_\mu \right) \phi \right|^2
\]

Generates a mass term for \( A_\mu \) after \( \phi \) acquires a VEV:

\[
m_A = gv
\]

The gauge field \( A_\mu \) now has three degrees of freedom: two transverse, one longitudinal (formerly the Goldstone boson).

11. Degrees of Freedom Before and After

Particle Type	Degrees of Freedom Before	Degrees of Freedom After
Gauge boson \( A_\mu \)	2 (massless)	3 (massive)
Complex scalar \( \phi \)	2	1 (Higgs)
Total	4	4

The degrees of freedom are conserved. The longitudinal polarization of the gauge boson comes from the broken scalar degree of freedom.

12. Non-Abelian Higgs Mechanism (SU(2))

In the Standard Model, the Higgs is an SU(2) doublet:

\[
\phi = \begin{pmatrix} \phi^+ \ \phi^0 \end{pmatrix}
\]

This setup breaks the electroweak symmetry:

\[
SU(2)L \times U(1)_Y \rightarrow U(1){\text{EM}}
\]

Three Goldstone bosons are absorbed to give mass to \( W^\pm \) and \( Z^0 \), while one scalar (Higgs) remains.

13. Electroweak Symmetry Breaking

The vacuum expectation value:

\[
\langle \phi \rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 \ v \end{pmatrix}
\]

Breaks the symmetry while preserving gauge invariance. This explains the existence of a massless photon and massive W and Z bosons.

14. Higgs Doublet and VEV

From the kinetic term:

\[
|D_\mu \phi|^2 \rightarrow m_W = \frac{1}{2} gv, \quad m_Z = \frac{1}{2} \sqrt{g^2 + g’^2} v
\]

Where \( v \approx 246\, \text{GeV} \) is measured from the Fermi constant.

15. Masses of W and Z Bosons

\[
m_W \approx 80.4\, \text{GeV}, \quad m_Z \approx 91.2\, \text{GeV}
\]

These match experimental results and were among the major triumphs of the Standard Model before the discovery of the Higgs.

16. Massless Photon and Gauge Invariance

The unbroken U(1) symmetry corresponds to electromagnetism. The photon, being the gauge boson of this subgroup, remains massless.

17. Fermion Masses via Yukawa Couplings

Fermions acquire mass through their interaction with the Higgs:

\[
\mathcal{L}_Y = – y_f \bar{\psi}_L \phi \psi_R + \text{h.c.}
\]

After SSB:

\[
m_f = \frac{y_f v}{\sqrt{2}}
\]

The mass of each fermion is proportional to its Yukawa coupling.

18. Prediction and Discovery of the Higgs Boson

The existence of the Higgs boson was predicted in the 1960s. Its discovery at the Large Hadron Collider (LHC) in 2012 by the ATLAS and CMS collaborations confirmed the Higgs mechanism.

19. Properties of the Higgs Boson

• Spin-0, CP-even scalar
• Mass ~ 125 GeV
• Couples to particles proportionally to their mass
• Decays: \( h \rightarrow \gamma\gamma, ZZ, WW, b\bar{b}, \tau\tau \)

20. Experimental Confirmation at the LHC

Data showed excess events consistent with Higgs decays. The discovery completed the Standard Model particle spectrum and led to the 2013 Nobel Prize for François Englert and Peter Higgs.

21. Role in the Standard Model

The Higgs mechanism:

• Provides mass to gauge bosons and fermions
• Preserves gauge invariance
• Ensures unitarity of electroweak theory
• Allows perturbative consistency

22. Extensions Beyond the Standard Model

Possible extensions include:

• Two-Higgs-Doublet Models (2HDM)
• Supersymmetry (MSSM: 5 Higgs particles)
• Higgs as a composite state
• Higgs portal to dark matter

23. Higgs Self-Coupling and Stability

The potential:

\[
V(\phi) = \lambda (\phi^\dagger \phi – v^2/2)^2
\]

Determines triple and quartic Higgs couplings. Measuring these helps probe the shape of the Higgs potential and test vacuum stability.

24. Open Questions in Higgs Physics

• Is the Higgs elementary or composite?
• Why is the Higgs mass stable (hierarchy problem)?
• Does the Higgs connect to dark matter?
• Are there more scalar fields?

25. Conclusion

The Higgs mechanism elegantly solves the problem of mass generation in gauge theories. By introducing a scalar field whose vacuum breaks electroweak symmetry, the theory explains the masses of W, Z, and fermions while preserving gauge invariance. The experimental discovery of the Higgs boson has confirmed this picture, but deeper questions remain—driving theoretical exploration into new physics beyond the Standard Model.

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