4.5 Scattering amplitudes and cross-sections
5 min read•august 14, 2024
Scattering amplitudes are the backbone of particle physics, describing how particles interact. They connect theory to experiment by relating the probability of particle interactions to measurable quantities like cross-sections. Understanding these amplitudes is crucial for testing theories and predicting outcomes in particle colliders.
The bridges the gap between abstract field theory and observable scattering processes. It allows us to calculate scattering amplitudes from correlation functions, linking the mathematical framework of quantum fields to real-world particle interactions we can measure in experiments.
Scattering amplitudes and observables
Definition and properties of scattering amplitudes
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- Scattering amplitudes are complex-valued functions describing the probability of a particular scattering process occurring given the initial and final states of the particles involved
- Contain information about the underlying interactions and symmetries of the particles involved in the scattering process
- The modulus squared of the is proportional to the , a measurable quantity describing the probability of detecting scattered particles in a particular direction
- The phase of the scattering amplitude is not directly observable but can provide insights into the nature of the interaction
- Scattering amplitudes are crucial for understanding the behavior of particles in high-energy physics experiments (particle colliders)
Relationship between scattering amplitudes and cross-sections
- The , another observable quantity, is obtained by integrating the differential cross-section over all possible scattering angles
- Differential cross-section provides detailed information about the angular distribution of scattered particles
- Total cross-section represents the overall probability of a scattering event occurring, regardless of the scattering angle
- Scattering amplitudes serve as a bridge between the theoretical description of particle interactions and the measurable quantities in experiments
- By comparing the predicted cross-sections derived from scattering amplitudes with experimental data, researchers can test and refine their theoretical models (Standard Model of particle physics)
Cross-sections from amplitudes
Calculating differential cross-sections
- The differential cross-section is given by the modulus squared of the scattering amplitude multiplied by a factor accounting for the density of final states
- In the center-of-mass frame, the differential cross-section is typically expressed as a function of the scattering angle, the angle between the initial and final momenta of the scattered particles
- Differential cross-sections are often presented in units of area per solid angle (barns per steradian)
- The angular dependence of the differential cross-section provides information about the nature of the interaction (isotropic scattering, forward peaking, backward peaking)
- Differential cross-sections can be measured experimentally using detectors that cover a range of scattering angles (multi-purpose detectors at particle colliders)
Obtaining total cross-sections
- The total cross-section is obtained by integrating the differential cross-section over all possible scattering angles, taking into account the appropriate Jacobian factors for the chosen coordinate system
- Dimensional analysis can determine the units of the total cross-section, typically expressed in terms of area (barns or square meters)
- Total cross-sections are essential for determining the overall probability of a scattering process and the rate of events in a particle physics experiment
- The energy dependence of the total cross-section can reveal important features of the interaction, such as resonances or thresholds (e.g., the Z boson resonance in electron-positron collisions)
- Measuring total cross-sections requires detectors with a large angular acceptance to capture scattered particles over a wide range of angles (e.g., forward detectors at particle colliders)
Optical theorem and unitarity
Statement and consequences of the optical theorem
- The relates the imaginary part of the forward scattering amplitude (scattering amplitude at zero scattering angle) to the total cross-section of the scattering process
- Provides a useful consistency check for scattering calculations and can constrain the possible forms of scattering amplitudes
- The optical theorem is a powerful tool for relating the real and imaginary parts of the scattering amplitude
- Allows for the determination of the total cross-section from the forward scattering amplitude, which is often easier to calculate theoretically
- The optical theorem has been used to make precise predictions for total cross-sections in various scattering processes (proton-proton, electron-positron)
Connection to unitarity and probability conservation
- The optical theorem is a consequence of the unitarity of the S-matrix, which ensures that probability is conserved in the scattering process
- Unitarity is a fundamental principle in quantum mechanics, stating that the sum of probabilities for all possible outcomes must equal one
- The optical theorem can be derived by considering the conservation of probability current in the scattering process and applying the unitarity condition to the S-matrix
- Unitarity imposes constraints on the possible forms of scattering amplitudes, ensuring that they behave consistently with probability conservation
- Violations of unitarity can indicate the presence of new physics or the breakdown of the theoretical framework at high energies (e.g., unitarity violations in the Standard Model without the Higgs boson)
LSZ reduction formula
Relating correlation functions to scattering amplitudes
- The LSZ (Lehmann-Symanzik-Zimmermann) reduction formula relates the time-ordered correlation functions of quantum fields to the scattering amplitudes of the corresponding particles
- Correlation functions are vacuum expectation values of time-ordered products of field operators, containing information about the propagation and interaction of particles
- The LSZ formula involves taking the Fourier transform of the correlation function and multiplying it by appropriate wave function renormalization factors and pole factors for each external particle
- Wave function renormalization factors account for the normalization of the particle states, ensuring that the scattering amplitude is properly normalized
- Pole factors ensure that the scattering amplitude is evaluated on the mass shell of the external particles, i.e., when their energies and momenta satisfy the relativistic dispersion relation
Application in perturbative quantum field theory
- The LSZ formula allows one to calculate scattering amplitudes from the correlation functions obtained using perturbative quantum field theory techniques, such as
- In perturbative calculations, the correlation functions are expanded in powers of the coupling constant, representing the strength of the interaction
- Feynman diagrams provide a systematic way to calculate the contributions to the correlation functions at each order in
- By applying the LSZ formula to the perturbatively calculated correlation functions, one obtains the corresponding scattering amplitudes order by order in the coupling constant
- The LSZ formula is a key tool in connecting the abstract formalism of quantum field theory to the observable quantities measured in particle physics experiments (cross-sections, decay rates)
Key Terms to Review (20)
Bosons: Bosons are a category of particles that obey Bose-Einstein statistics and have integer spin values (0, 1, 2, etc.). Unlike fermions, which adhere to the Pauli exclusion principle, bosons can occupy the same quantum state, allowing them to mediate forces and enable phenomena such as superfluidity and Bose-Einstein condensation.
Conservation Laws: Conservation laws are fundamental principles in physics stating that certain quantities remain constant within isolated systems over time. These laws provide critical insights into physical processes, enabling predictions about outcomes in various scenarios, such as particle interactions and field dynamics.
Differential Cross-Section: The differential cross-section is a measure of the likelihood of a specific scattering process occurring, represented as the probability per unit solid angle. It provides important information about how particles scatter off each other and is typically expressed in units of area, such as barns. By detailing how scattering events depend on angles and energies, the differential cross-section connects scattering amplitudes to observable outcomes in particle physics experiments.
Elastic Scattering: Elastic scattering is a process where particles collide and scatter without any change in their internal energy states, meaning the total kinetic energy before and after the interaction remains constant. This type of scattering is important in understanding interactions in particle physics, as it helps in calculating scattering amplitudes and cross-sections, which describe the probability of such interactions occurring during collisions.
Experimental cross-section: The experimental cross-section is a measure of the probability of a scattering event occurring between particles, often represented in units of area. It is a critical concept that connects the theoretical predictions of scattering amplitudes with actual measurements obtained in particle physics experiments. By quantifying how likely particles are to interact, the experimental cross-section helps physicists understand fundamental interactions and assess the validity of quantum field theories.
Fermions: Fermions are a class of particles that follow Fermi-Dirac statistics and are characterized by having half-integer spin, such as 1/2, 3/2, etc. These particles include electrons, protons, and neutrons, and they obey the Pauli exclusion principle, which states that no two identical fermions can occupy the same quantum state simultaneously. This unique property is crucial for understanding the structure of matter and the behavior of systems at low temperatures.
Feynman diagrams: Feynman diagrams are pictorial representations of the interactions between particles in quantum field theory. They simplify complex calculations in particle physics by visually depicting the paths and interactions of particles, facilitating the understanding of processes like scattering and decay.
High-energy limit: The high-energy limit refers to the regime in scattering processes where the energy of the incoming particles is significantly larger than the mass scales involved in the interaction. In this limit, certain simplifications occur, allowing for clearer insights into scattering amplitudes and cross-sections, leading to predictions that can be more easily analyzed within quantum field theory frameworks.
Inelastic scattering: Inelastic scattering refers to a type of collision between particles where the kinetic energy is not conserved, leading to a change in the internal energy states of the particles involved. This process can result in the excitation of particles or the production of new particles, reflecting how energy is transformed during interactions. Understanding inelastic scattering is crucial for calculating scattering amplitudes and cross-sections, as it contributes to the overall probability and characteristics of particle interactions.
Low-energy theorem: The low-energy theorem is a principle in quantum field theory that relates scattering amplitudes to the properties of particles involved in the interaction, particularly when the energies are much lower than the masses of the particles. This theorem allows for simplified calculations and predictions of physical observables, as it connects cross-sections and amplitudes directly to underlying symmetries and conservation laws at low energies.
LSZ Reduction Formula: The LSZ reduction formula is a crucial tool in quantum field theory that relates the S-matrix elements to the time-ordered correlation functions of quantum fields. It serves to connect the scattering amplitudes calculated from interaction terms with the actual observable scattering processes, allowing for the extraction of physical quantities such as cross-sections from theoretical predictions. This formula highlights the importance of defining initial and final states in terms of asymptotic fields, bridging the gap between abstract field theory and measurable phenomena.
Matrix Element: A matrix element is a component of a mathematical object that describes the transition amplitude between different quantum states in a quantum field theory. This concept is crucial when calculating scattering amplitudes, as it connects the initial and final states of particles involved in interactions, enabling the computation of observable quantities like cross-sections.
Optical Theorem: The optical theorem is a principle in quantum mechanics that relates the total scattering cross-section to the forward scattering amplitude. It states that the imaginary part of the forward scattering amplitude is directly proportional to the total cross-section, providing a powerful tool for analyzing scattering processes in particle physics. This theorem connects the observable quantity of cross-section with the theoretical scattering amplitudes, making it essential for understanding interactions between particles.
Perturbation theory: Perturbation theory is a mathematical technique used in quantum mechanics and quantum field theory to approximate the behavior of a system that is subject to small disturbances or interactions. It allows for the calculation of physical quantities by treating the interaction as a small perturbation of a solvable system, providing a powerful method to understand complex systems and their dynamics.
Phase Space: Phase space is a mathematical concept used to describe all possible states of a physical system, where each state is represented by a unique point in a multidimensional space. It encompasses both the positions and momenta of all particles in the system, allowing physicists to analyze dynamic systems and their evolution over time. This framework is crucial for understanding scattering processes, where particles interact and exchange energy and momentum, influencing observable quantities such as scattering amplitudes and cross-sections.
Reaction Rate: The reaction rate refers to the speed at which a specific reaction occurs, typically measured by the change in concentration of reactants or products over time. In particle physics, particularly in scattering processes, the reaction rate is directly linked to the likelihood of interactions occurring and can be expressed through cross-sections and scattering amplitudes, which quantify how likely two particles are to scatter off each other during a collision.
Scattering Amplitude: Scattering amplitude is a complex number that quantifies the probability amplitude for a specific scattering process to occur between particles. It serves as a fundamental element in calculating observable quantities like cross-sections, and plays a critical role in connecting theoretical predictions with experimental results through techniques like Feynman diagrams. Understanding scattering amplitudes is essential for studying interactions in quantum field theories, especially in determining how particles scatter in various forces such as electromagnetism and the strong force.
Symmetry Principles: Symmetry principles refer to fundamental features of physical systems that remain unchanged under certain transformations, such as rotations, translations, or reflections. These principles play a crucial role in understanding the conservation laws and interactions in quantum field theory, especially in analyzing scattering processes and calculating scattering amplitudes and cross-sections.
Total Cross-Section: The total cross-section is a measure of the probability of a scattering event occurring between particles, representing the effective area that quantifies the likelihood of interaction. This concept connects the scattering amplitude, which describes the quantum mechanical probability of a particular outcome, to observable quantities in scattering experiments, helping to bridge theoretical predictions and experimental results.
Unitarity Theorem: The unitarity theorem is a principle in quantum field theory that states that the total probability of all possible outcomes of a quantum process must equal one, ensuring the conservation of probability. This theorem is crucial in scattering processes as it relates the scattering amplitudes to observable quantities like cross-sections, providing a fundamental consistency check for quantum theories and their predictions.