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Lifetimes, Cross-Sections and LIPS

Lifetimes, Cross-Sections and LIPS. Decay Rates and Lifetimes Cross Sections Lorentz Invariant Phase Space (LIPS) Dalitz Plots. Studying Interactions. Three ways for a pair of particles a and b to interact: Scatter: a + b  c + d Energy large enough, or force repulsive

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Lifetimes, Cross-Sections and LIPS

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  1. Lifetimes, Cross-Sectionsand LIPS • Decay Rates and Lifetimes • Cross Sections • Lorentz Invariant Phase Space (LIPS) • Dalitz Plots Brian Meadows, U. Cincinnati.

  2. Studying Interactions • Three ways for a pair of particles a and b to interact: • Scatter: a + b  c + d Energy large enough, or force repulsive • Form bound state X: a + b  X Energy < binding energy (attractive force) • Decay: Y  a + b Energy (mass of X in CMS) > binding energy • Measurement of ’s, masses and  ’s provides information Rate characterized by “cross-section”  Binding energy  mass spectrum of X’s Decay rate  and lifetime  Brian Meadows, U. Cincinnati

  3. 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

  4. 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 natural width of the parent particle does NOT depend on its decay mode - • Natural width isG, NOTGa Brian Meadows, U. Cincinnati

  5. Cross-Sections • These are / probability for a scatter to occur: • Conceptually, target b has a cross-section  • Particles a and b may change in several different modes: e.g. K- + p K- + p ,  0 + 0,  - + K+ + K0, etc. Partial cross-sections  different for each mode • Total cross-section is tot =  c v a + b c + d a b d Brian Meadows, U. Cincinnati

  6. Cross-Sections • Suppose: • ais density of particlesa •  is the effective cross-section of each particleb • vithe velocity for particlesa(relative to bthat is at rest) • Incident flux through target is  = aviparticles area-1¢time-1 • Number of reactions W =   (area-1¢time-1¢target¢particle-1) • So for one incident particle:  = W / vi (reaction rate ¥ rel. speed) • Unit for  is the barn 1b = 10-24 cm2 (approx. area of nucleus Z=100) Brian Meadows, U. Cincinnati

  7. Fermi’s (NR) Golden Rule • For an interaction where final state f arises from initial state i, the reaction rate is given non-relativistically by • Mi f is matrix element for interaction potential U. • Can depend on spin-states of initial and final particles • ’s so that Mif has units energy per unit volume • dN/dE is density of phase space states (# per unit energy) • It is proportional to V • So units for W are (fractional) transitions per unit time Interaction volume Brian Meadows, U. Cincinnati

  8. Decay Rate • For decays, natural width is given (non-relativistically) by So If you can compute Mif then you can predict lifetimes and widths. Brian Meadows, U. Cincinnati

  9. Non-relativistic Phase Space Factor • Number of final states for n final state particles is • Spin-state factor S: • If final state spins DO NOT matter  S =  (2s+1) • If NEITHER initial NOR final spin states matter average over initial spins AND sum over final spins • Normalize  ’s so that V cancels (can set V = 1) Brian Meadows, U. Cincinnati

  10. Relative speed of c and d An example – 2-body system in CMS • Compute d2N/dEdW for a two-body system in CMS Solution: In CMS (PCMS = 0): Integration over p2 selects p1 = -p2 = p and E1+E2 = E Leads to Brian Meadows, U. Cincinnati

  11. d pf a b CMS pf c An example … • Compute d / d for the reaction a + b  c + d given the interaction matrix Mif Solution: Compute everything in CMS: Recall:  = W / vi Brian Meadows, U. Cincinnati

  12. A Problem … • N (so far) is non-relativistic, so is Mif . BUT Mif is most often computed as a 4-tensor • If s, ds/dW, etc. are not to depend on frame of reference • We need Lorentz-invariant phase space • And Lorentz-invariant matrix elements. Brian Meadows, U. Cincinnati

  13. Lorentz Invariant Phase Space (LIPS) • Relativistic Mif usually involves 4-tensors, so need to replace 3-vectors by 4-vectors: ie by Ensures that particle  remains “on its mass shell”. Brian Meadows, U. Cincinnati

  14. Lorentz Invariant Phase Space (LIPS) • Integrate over the energies using • So we obtain • If total energy is E, then in the CMS (where P=0): Brian Meadows, U. Cincinnati

  15. LIPS for 2-Body System • Compute in CMS • Integrate over p2 (simply defines p2=-p1=p) • Integrate over|p| using (E1+E2 -E) / p = p(1/E1+1/E2) Brian Meadows, U. Cincinnati

  16. n-1 1, n-1 1, n-3 1, n-2 n n-2 Recurrence Relation • LIPS for n particles with total energy E is • This can be factorized: • … and again 1, n , etc. … Brian Meadows, U. Cincinnati

  17. 2 p0 p3 p3 0 p0 3 1 Example: 3-Body System • Let masses be m1, m2 and m3. Invariant masses m12, etc. • In {12} CMS, the momentum of 2 is p0 where we know • Also, using the identity obtain (In overall CMS) Brian Meadows, U. Cincinnati

  18. 2 p0 p3 p3 0 p0 3 1 3-Body System (cont’d) • Lorentz transform 2 into 3-body CM where p3 is defined • Therefore • If overall orientation (ied3) is not important: Brian Meadows, U. Cincinnati

  19. s13 s12 Dalitz Plot • Therefore, if we plot E2 vs. E3 then, if Mif is constant, we expect to see a uniform population of events. • Effects of Mif can be seen directly • NOTE – the limits on E3 (CMS) are: m3 < E3 < (E-m1-m2 ) • The number of phase space states available for this three-body decay is proportional to the area of the Dalitz plot N(E, 3) /ss dE2dE3 Brian Meadows, U. Cincinnati

  20. s13 s12 Dalitz Plot • It is more common to plot squared invariant masses sij These pin-point effects of resonances more clearly • If M is mass of system decaying to 3 bodies then • But • So that • Therefore Brian Meadows, U. Cincinnati

  21. m1 Parent m2 m3 Resonance mass Resonance width Resonances • Two of the final state particles can form a resonance This resonance, the f0 meson, is spin 0. Brian Meadows, U. Cincinnati

  22. m1 Parent m2 m3 Resonance mass Resonance width Resonances • Sometimes the resonance looks different • There is a “spin factor” This resonance, the r0 meson, is spin 1. Brian Meadows, U. Cincinnati

  23. Resonances • There can be several resonances • They usually interfere with each other f= 0 deg. f= 90 deg. f= 180 deg. Resonance A + eif x Resonance B Brian Meadows, U. Cincinnati

  24. A Dalitz Plot • Example of a Dalitz plot • D+ K-++ • If K - and + did not interact, then density of points would be uniform over the plot • Can conclude that Mif depends on position on the plot. Brian Meadows, U. Cincinnati

  25. s12 s13 Kinematics of Dalitz Plots • Number of degrees of freedom N • Three particles (4-vectors) !N=12 variables • All three are on their mass shell !N=9 • Conserve E and p!N=5 • Three Euler angles defining orientation of the three body decay plane: • If decaying system is polarized N=5. • For decay of J=0 parent: are arbitrary N=2 • So for D or B decays N=2 s Brian Meadows, U. Cincinnati

  26. p l q q L p Boundary of Dalitz Plot • Important figure: 2 Momenta in CM Frame of 1 & 2. 3 1 p = l (s12, m12, m22) / \/ s12 q = l(s, s12, m32) / \/s12 NOTE – in 3-body CM: Q = l(s, s12, m32) /\/ s = q s12 / s • (a, b, c) = [(a-b-c)2-4bc]1/2 / 2 Brian Meadows, U. Cincinnati

  27. So at a given value for s12 the value of s13 can be computed s13 = m12+m32 + 2(E1E3-pq cosq) Therefore the limits of s13 are given by cos q= § 1 Notice that ds13 = d(cos q) S12 S13 Boundary of Dalitz Plot cos q = -1 cos q = +1 Brian Meadows, U. Cincinnati

  28. Helicity – Spin Conservation • If decaying particle is spin zero (e.g. D or B) then helicity of resonance must be same as that of the bachelor: p l q q L p Brian Meadows, U. Cincinnati

  29. Spin Factor (spinless daughters) • Consider decay as if it is in two steps: • Decay to resonance and bachelor • followed by decay of resonance. Y1/ QL YLM “solid harmonic” L Q Q M / pl QL Ylm YLM “Spin Factor” p l q q L p No preferred Direction so ! 1.0 Brian Meadows, U. Cincinnati

  30. p l q q L p Helicity and Spin Factor • If decaying products are pseudoscalar then l=L (to conserve spin): • Then M / pl qlYlm(cosq) (m = 0.) ! |M|2/ |Yl0(cosq)|2 Brian Meadows, U. Cincinnati

  31. Another Example • Ds+!fp+ ! K*K+ • f and K* are vectors So l = 1 • Therefore |M|2/ cos2q ! K-K+p+ Brian Meadows, U. Cincinnati

  32. K*(890) ! K0+ 15,753 events 23 fb-1 K*(890) ! K0- ? D0 K0p+p- - Interesting Case • B+ D0 K+ and B+  D0K+ • are related by a phase that can be measured by looking at the Dalitz plot BECAUSE … • D0 and D0 Dalitz plots interfere with each other! Brian Meadows, U. Cincinnati

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