S-Matrix Unitarity, Analyticity, and Resonances

Probability conservation relates an amplitude’s absorptive part to physical intermediate states. Analytic continuation then connects channels and exposes threshold cuts and resonance poles. This chapter explains those connections without assuming that every formal continuation or high-energy limit is physically justified.

Enter this chapter

Begin with S-Matrix Unitarity, then derive the optical theorem before interpreting a cut or partial-wave bound.

Topics

1. S-Matrix Unitarity — Derives the transition-operator identity on the physical state space.
2. The Optical Theorem and Cut Interpretation — Relates forward absorption to an inclusive total rate.
3. Partial-Wave Unitarity — Turns angular-momentum decomposition into channel-by-channel constraints.
4. Analyticity and Crossing of Amplitudes — Treats physical channels as boundary values of analytic amplitudes.
5. Resonance Poles, Riemann Sheets, and Unstable States — Distinguishes continued poles from peaks and approximate line shapes.
6. Causality, Growth, and Analytic Domains — States the causal and growth assumptions behind analytic claims.
7. High-Energy and Regge Limits — Separates fixed-angle and fixed-transfer high-energy behavior.

How the ideas fit together

With $S=1+iT$, unitarity gives $-i(T-T^\dagger)=T^\dagger T$. Its forward matrix element inserts a complete sum of physical intermediate states, producing the optical theorem and its nonnegative total probability. The same states create thresholds and discontinuities in analytically continued amplitudes. See Schwartz 2014, Chapter 24.

Analyticity adds information beyond the real-axis identity, including sheet choices and growth assumptions. An unstable resonance is diagnosed by a suitable continued pole, while a threshold cusp or interference can also create a visible peak. A simple Breit–Wigner fit need not reproduce invariant pole data.

Review the chapter

Why does a nonzero real tree amplitude not contradict the optical theorem’s positive right-hand side?

Solution

Unitarity is an order-by-order relation. The product of leading amplitudes contributes at a higher coupling order than the tree amplitude itself. Its required absorptive part appears in loop corrections at that order; comparing unequal perturbative orders creates the apparent contradiction.

Where to go next

Continue to Singularities, Cuts, and Integrand Reconstruction to apply on-shell intermediate states to loop calculations.

References

• Matthew D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014. DOI.