FR05
Scattering amplitudes and the S-matrix bootstrap
A scattering amplitude can be far simpler than the diagrams used to calculate it. Start with a detector count, learn how spin and singularities organize an amplitude, then use probability and analytic structure to test which scattering functions can survive.
Choose a starting point
Chapters 1–2 begin with probability and angular patterns. Chapter 3 introduces the spinor variables needed for the four-dimensional massless theory; Chapters 4–6 compare calculations, take singular limits and reconstruct a tree amplitude. Chapters 7–8 start a separate massive theory in one spatial dimension. Their prerequisite is Chapter 2, so the bootstrap branch can be read independently.
Each chapter states its prerequisites and includes a worked example, a prediction task and explained exercises. For wider background, use the common textbook plan and the field-theory and mathematical-method routes as needed. Textbook plan · FR01 · FR41
L1 · 1
How a scattering amplitude becomes a detector count
Derive differential cross sections from probability current and use interference to see why amplitudes are added before probabilities.
L1 · 2
How a circle limits the strength of scattering
Decompose an angular pattern into partial waves, derive the unitarity circle and calculate a resonance cross section.
L2 · 3
Why split a momentum into two spinors?
Factor a massless momentum matrix into spinors, check helicity weights and use color ordering to identify allowed poles.
L3 · 4
Why four-gluon scattering reduces to one line
Compare a four-gluon diagram sum with Parke–Taylor, then test polarization-reference changes, Ward identities and cancellation errors.
L3 · 5
Why a soft gluon forgets most of the collision
Derive soft and collinear factors from five-point amplitudes, build conserving scan paths and separate limiting errors from numerical errors.
L3 · 6
Reconstruct a tree amplitude from its poles
Build an on-shell conserving BCFW shift, calculate a complete four-point residue and identify the boundary contribution at infinity.
L4 · 7
Put two-dimensional scattering in a complex strip
Use rapidity to open the massive physical sheet, derive crossing and calculate the mass and residue of a fixed CDD pole.
L4 · 8
Shrink a family of S matrices and reproduce a published curve
Solve a finite rational bootstrap analytically, compare crossing and pole constraints, and reproduce a specified Paulos et al. reference curve.
Three experiments along the route
- Change a potential and follow its angular pattern and unitarity circle
- Drag a gluon direction, compare diagrams and approach soft or collinear limits
- Propose an S matrix and watch consistency conditions remove coefficients
The first experiment uses a nonrelativistic central potential. The second uses four-dimensional massless Yang–Mills trees. The third tests necessary conditions in a specified massive two-dimensional elastic-element family. Each has its own normalization and model limits.
Sources and reproduced scope
Worked examples and schematic figures are authored for this course. Numerical values are calculated model references. The published comparison recalculates a specified analytic CDD curve; it does not copy experimental data, source artwork or the paper’s numerical optimization points. Source locations and reproduced scope are listed below.
- David Tong, Topics in Quantum Mechanics, Chapter 10, Scattering Theory
Sections 10.2.1–10.2.3: three-dimensional asymptotic waves, cross sections and partial waves. We derive the current-to-area relation and unitarity-circle examples for the course; the linked square-well experiment retains its own model and units.
- Lance J. Dixon, A brief introduction to modern amplitude methods, arXiv:1310.5353v1 (2013)
Section 2, equation (5), color decomposition; section 3.1, equations (14), (19), (21)–(27), spinors and all-outgoing conventions; section 3.3, equations (41)–(46), polarizations; section 4.1, equations (55)–(57), soft factors; section 4.3, equations (68)–(70), MHV and collinear factors; sections 4.5 and 5.1, equations (74)–(84), complex three-point seeds and recursion. Our conserving scan and four-point residue example are derived extensions.
- R. Britto, F. Cachazo, B. Feng and E. Witten, Direct Proof Of Tree-Level Recursion Relation In Yang-Mills Theory, hep-th/0501052v2; Physical Review Letters 94, 181602 (2005)
Section 2 introduces null-momentum spinors, constructs the shift in equation (2.1), and derives recursion in equation (2.7); section 3 controls large-parameter behavior; section 4 proves the MHV construction. The course keeps Dixon’s overall amplitude phase and specializes the original shift to legs one and four. Its soft scan uses the MHV function as input and derives the conserving real-momentum path separately.
- M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees and P. Vieira, The S-matrix Bootstrap II: Two Dimensional Amplitudes, arXiv:1607.06110v2; JHEP 11 (2017) 143
Section 2 equations (1)–(3) fix crossing, elastic unitarity and coupling normalization. Equations (9), (13)–(14), Figures 5a and 6 give the sine-Gordon reference, CDD factor and physical strip. We recalculate the analytic Figure 5a reference at external mass one and exchanged mass square root of three, its pole residue and the complex value at energy squared six. We do not reproduce the dispersive optimization of section 2.1/Appendix A. The rational three-parameter family is our extension.
- Patrick Dorey, Exact S-matrices, arXiv:hep-th/9810026v1 (1998)
Section 3, printed pages 12–16, derives the rapidity map, physical strip, real analyticity, matrix unitarity and crossing. Section 4, pages 16–21, relates imaginary rapidity to bound-state masses and discusses sine-Gordon poles, including the limits of identifying every pole with a new particle. The course uses the scalar identical-particle specialization; its finite rational ansatz is separately defined.