January 11, 2001 Physics 841 1

Cross-Sections and Decay Rates

• There are three types of measurements one can make– static properties of particles (mass, charge, etc.)– decay rates (lifetimes/line-widths, branching

ratios)– cross-sections (inclusive, semi-inclusive, exclusive)

• We will focus our attention on the dynamical properties of particles (decay rates and cross-sections), not because they are intrinsically more interesting than the static properties (mass, charge, etc.) but, rather, because they are more directly related to understanding CP violation.

• Understanding the masses of particles, and the patterns of masses, is a key problem in particle physics, and we will address it, in part, next quarter.

January 11, 2001 Physics 841 2

Differential Cross-Sections

• Classically, scattering from a point-like object, or a spherically symmetric potential, can be described in terms of the impact parameter of the incident particle with respect to the target and the (asymptotic) scattering angle. If the impact parameter is known, and the potential is known, the scattering angle is unique. While this is not true in the quantum mechanical world, understanding the classical picture is useful in developing intuition.

• Cross-sections measure the strengths of interactions. We expect that– “larger” objects should have larger cross-

sections– “stronger” potentials (or their equivalent)

should have larger cross-sections

January 11, 2001 Physics 841 3

Classical Scattering

January 11, 2001 Physics 841 4

Hard Sphere Scattering

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b

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R

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α

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α

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α

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ϑ

January 11, 2001 Physics 841 5

Classical Rutherford Scattering

January 11, 2001 Physics 841 6

More Rutherford Scattering

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January 11, 2001 Physics 841 7

Fermi’s Golden Rule for Scattering

nota bene:

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h=1

January 11, 2001 Physics 841 8

Lorentz Invariant Phase Space

January 11, 2001 Physics 841 9

Lorentz Invariant Phase Space, Continued

January 11, 2001 Physics 841 10

Transition Matrix Elements in Perturbation Theory - I

January 11, 2001 Physics 841 11

Transition Matrix Elements in Perturbation Theory - II

January 11, 2001 Physics 841 12

Transition Matrix Elements in Perturbation Theory - III

January 11, 2001 Physics 841 13

nota bene:

Transition Matrix Elements in Perturbation Theory - IV

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E⇔ ω