1 / 32

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

lenore-holder
Télécharger la présentation

Lifetimes, Cross-Sections and LIPS

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related