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Understanding Lifetimes, Decay Rates, Cross-Sections, and Matrix Elements in Particle Physics

This document explores key concepts in particle physics, particularly focusing on lifetimes, decay rates, cross sections, and matrix elements. It explains the relationships among these parameters, including natural widths, branching ratios, and the application of the Golden Rule in various decay processes. Through examples, including two-body scattering and Feynman diagrams, the complexities of particle interactions and the calculation of scattering cross sections are examined. This material serves as a foundational guide for students in physics, particularly those interested in particle physics theory.

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Understanding Lifetimes, Decay Rates, Cross-Sections, and Matrix Elements in Particle Physics

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  1. Lifetimes, Cross-Sectionsand Matrix Elements • Decay Rates and Lifetimes • Cross Sections • Matrix Elements for a Toy Model Brian Meadows, U. Cincinnati.

  2. Lifetime and Decay Rate and Natural Width • Decay rate is W W = - (dN/dt) / N • N(t) = N(0) e-t/ • Lifetime  (time for population to decrease by factor e)  = 1/W • Natural width  = ~/ = ~ W (uncertainty principal) Brian Meadows, U. Cincinnati

  3. Lifetime and Decay Rate and Natural Width • Particles may decay in several different modes: K+ +0,  +,  ++-, etc. • Partial widths  different for each mode  =  • Branching ratios/ also provide information on interaction between the decay products • AND on the interaction causing the decay • NOTE – the width of the parent particle is G, NOT Ga Brian Meadows, U. Cincinnati

  4. Golden Rule for Lifetimes (Relativistic) • The decay rate (0  1 + 2 + … + n) is given by: • The total width is therefore the integral. • For a two-body decay Usually a function of the pi and their spins Brian Meadows, U. Cincinnati

  5. Example 0   : • We have • Work in CM so that E1 = E2 = E = |p| c (= M0 c2 / 2) • Integrating over d3p2: • Matrix element is scalar (depends only on |p|) so, : Brian Meadows, U. Cincinnati

  6. Example M m1+m2 : • This time, in CM, we have • Integrating over d3p2: • Matrix element is scalar (depends on |p|) so, : Using Where p0 is the CM momentum of m1 or of m2 Brian Meadows, U. Cincinnati

  7. Golden Rule for Scattering (Relativistic) • The cross-section (1 + 2  3 + 4 + … + n) is given by: • In the CM frame, where p1 = -p2 = pin: Brian Meadows, U. Cincinnati

  8. Example - Two-Body Scattering • We have • Integrating over p4 in the CM frame, we simply set: • Therefore Brian Meadows, U. Cincinnati

  9. Example - Two-Body Scattering • The matrix element may depend on out and out • To integrate over pout we compute the differential cross-section(d / d cos out d out = d / dout) • Using property of  function and dEi/dpout = pout/Ei • Since E1+E2 = E3+E4 in the CM: Brian Meadows, U. Cincinnati

  10. Evaluating . • Sometimes possible to do this from a Feynman diagram for the process. For example: • Can compute using the “Feynman Rules” for a given set of spin alignment • Usually want answer independent of spin alignments: • Average over initial spin-states • Sum over final spin-states Pair annihilation e+ + e-   +  Time Brian Meadows, U. Cincinnati

  11. Feynman Rules for Toy Model • Suppose we ignore spins at first. Here are the rules: Label: • Label each external line with 4-momenta pi using arrows to indicate the positive direction. Label internal lines with 4-momenta qk • For each vertex write a factor –ig where g is a coupling constant (/ e for electro-magnetic interaction) • Write a propagator factor for each internal line Brian Meadows, U. Cincinnati

  12. Feynman Rules for Toy Model Now conserve momentum (at each vertex) 4. Include a d function to conserve momentum at each vertex. where the k's are the 4-momenta entering the vertex 5. Integrate over all internal 4-momenta qj. I.e. write a factor For each internal line. • Cancel the d function. Result will include factor • Erase the d function and you are left with Brian Meadows, U. Cincinnati

  13. Example – Decay of 0  0 + 0 • Label diagram (no internal lines) • Rules 2 and 4: = -ig(2)44(p1-p2-p3) • Rule 6: = -ig  = g • So  = g2|p|/(8~ M2c and  = 1/ = 8~ M2c / (g2|p|) p2 p1 Time p3 Brian Meadows, U. Cincinnati

  14. Example: Spin-less Scattering 1 + 2  3 + 4 • There are two diagrams of same order: (labelled) p3 p4 p3 p4 q Time -ig -ig -ig -ig q C C p1 p2 p1 p2 rule 5 Similar, but p3 p4 p4  p3 rule 4 rules 1-3 Cancel - rule 7 Brian Meadows, U. Cincinnati

  15. Cross-Section • So resulting matrix element is • In CMS |p1|=|p2|=pin and |p1|=|p2|=pout • If m1=m2=m3=m3=m and mc=0 then p=pin=pout and In general(p3-p2)2 = m32 c4+ m22 c4 -2 (E2E3 + |pin||pout|c2 cos) and(p4-p2)2 = m42 c4 + m22 c4 -2 (E2E3 - |pin||pout|c2 cos)  Brian Meadows, U. Cincinnati

  16. A Loop Diagram • Consider q2 p4 p3 q1 q4 Time p2 p1 q3 Set q4=p4-p2 Set q1=p1-p3 Set q2=p1-p3-q3 Brian Meadows, U. Cincinnati

  17. Renormalization – Very Sketchy • Introduce factor -L2c2/(q2-L2c2) into the integral and integrate (by parts). As L 1, this factor  1 • In effect, this splits the fundamental parameters of the theory (e.g. QED) into two parts: mphysical = m +  m ; gphysical = g +  g • In QED, e.g., the sum of the lowest order and loop uses • If renormalizable, a theory has one additional “d” per type of divergence encountered • Can think of basing perturbation on equations of the type d’s depend on L1 Still get 1’s, but each type cancels Brian Meadows, U. Cincinnati

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