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Lifetimes, Cross-Sections and Matrix Elements

Lifetimes, Cross-Sections and Matrix Elements. Decay Rates and Lifetimes Cross Sections Matrix Elements for a Toy Model. Lifetime and Decay Rate and Natural Width. Decay rate is W W = - ( dN/dt) / N N ( t ) = N (0) e - t/ 

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Lifetimes, Cross-Sections and Matrix Elements

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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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