Lifetimes, Cross-Sections and Matrix Elements

1. Lifetimes, Cross-Sectionsand Matrix Elements • Decay Rates and Lifetimes • Cross Sections • Matrix Elements for a Toy Model Much of the content of these slides is acknowledged to come from “Introduction to Elementary Particles” By David Griffiths, Wiley, 1987

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) Much of the content of these slides is acknowledged to come from “Introduction to Elementary Particles” By David Griffiths, Wiley, 1987

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 Much of the content of these slides is acknowledged to come from “Introduction to Elementary Particles” By David Griffiths, Wiley, 1987

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 Much of the content of these slides is acknowledged to come from “Introduction to Elementary Particles” By David Griffiths, Wiley, 1987

5. Simple Example 0   :(2-body decay to massless particles) • 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, : Much of the content of these slides is acknowledged to come from “Introduction to Elementary Particles” By David Griffiths, Wiley, 1987

6. Example 2: 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 Much of the content of these slides is acknowledged to come from “Introduction to Elementary Particles” By David Griffiths, Wiley, 1987

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: Much of the content of these slides is acknowledged to come from “Introduction to Elementary Particles” By David Griffiths, Wiley, 1987

8. Example - Two-Body Scattering • We have • Integrating over p4 in the CM frame, we simply set: • Therefore Much of the content of these slides is acknowledged to come from “Introduction to Elementary Particles” By David Griffiths, Wiley, 1987

9. Example - Two-Body Scattering • The matrix element may depend on out and out • To integrate over pout we compute the differential cross-section(d2 / 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: Much of the content of these slides is acknowledged to come from “Introduction to Elementary Particles” By David Griffiths, Wiley, 1987

10. Evaluating . • Sometimes, it is possible to do this from a Feynman diagram for the process. For example: • We can compute this using the “Feynman Rules” for a given set of spin alignments for the particles • Usually, we want an answer independent of spin alignments: • Average over initial spin-states • Sum over final spin-states Pair annihilation e+ + e-   +  Time This may not be obvious Much of the content of these slides is acknowledged to come from “Introduction to Elementary Particles” By David Griffiths, Wiley, 1987

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 “Off the mass shell” Much of the content of these slides is acknowledged to come from “Introduction to Elementary Particles” By David Griffiths, Wiley, 1987

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 Much of the content of these slides is acknowledged to come from “Introduction to Elementary Particles” By David Griffiths, Wiley, 1987

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 Much of the content of these slides is acknowledged to come from “Introduction to Elementary Particles” By David Griffiths, Wiley, 1987

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 Much of the content of these slides is acknowledged to come from “Introduction to Elementary Particles” By David Griffiths, Wiley, 1987

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)  Much of the content of these slides is acknowledged to come from “Introduction to Elementary Particles” By David Griffiths, Wiley, 1987

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 Much of the content of these slides is acknowledged to come from “Introduction to Elementary Particles” By David Griffiths, Wiley, 1987

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 Much of the content of these slides is acknowledged to come from “Introduction to Elementary Particles” By David Griffiths, Wiley, 1987