Particle Lifetime. Subatomic Physics: Particle Physics Lecture 3. Measuring Decays, Scatterings and Collisions. N(t) = N 0 exp( t/τ) = N 0 exp( Γt/)

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1 Sutomic Physics: Prticle Physics Lecture 3 Mesuring Decys, Sctterings n Collisions Prticle lifetime n with Prticle ecy moes Prticle ecy kinemtics Scttering cross sections Collision centre of mss energy Lots of exmples use toy. Lern the concepts, not the etils out the prticulr ecys or sctterings! 1 Prticle Lifetime Prticle lifetime,!, is the time tken for the smple to reuce to 1/e of originl smple. Different forces hve ifferent typicl lifetimes. lso efine totl ecy with,! " /!. N t = N τ = Γ N N(t) = N 0 exp( t/τ) = N 0 exp( Γt/) In its own rest frme prticle trvels v! = #c! efore ecying. In the l, time is ilte y $. If! is lrge enough, energetic prticles trvel mesurle istnce L = $#c! in l. exmple of recent mesurement of prticle lifetime Force Strong Typicl Lifetime 10 "0-10 "3 s Electromg 10 "0-10 "16 s Wek 10 " s c!= this experimentl t ws use to etermine!(") = (1.30 ± 0.15) # 10!1 s

2 Decy Moes Prticles cn hve more thn one possile finl stte or ecy moe. e.g. The KS meson ecys 99.9% of the time in one of two wys: K S π + π, K S π 0 π 0 Ech ecy moe hs its own mtrix element, M. Fermi s Golen Rule gives us the prtil ecy with for ech ecy moe: Γ(K S π + π ) M(K S π + π ) Γ(K S π 0 π 0 ) M(K S π 0 π 0 ) The totl ecy with is equl to the sum of the ecy withs for ll the llowe ecys. Γ(K S ) = Γ(K S π 0 π 0 ) + Γ(K S π + π ) The rnching rtio, BR, is the frction of time prticle ecys to prticulr finl stte: BR(K S π + π ) = Γ(K S π + π ) Γ(K S ) This is n exmple. Lern the concepts, not the etils out this prticulr ecy. BR(K S π 0 π 0 ) = Γ(K S π 0 π 0 ) Γ(K S ) 3 Prticle Decy Kinemtics Most prticles ecy. e.g. KS meson cn ecy s: K S π + π Reconstruct the mss of prticle from the moment of the ecy proucts: pinitil = p finl p(k S ) = p(π + ) + p(π ) squring ech sie... (M(K S )) = p(π + ) + p(π ) = p(π + ) + p(π ) + p(π + ) p(π ) Invrint mss from KS!! +! " ecy. Not perfect ue to limite ccurcy of mesurements = m(π + ) + m(π ) + E(π + )E(π ) p (π + ) p (π ) M(KS) is reconstructe invrint mss of KS 4

3 Decy of n unstle prticle t rest:! Decy Kinemtics II Before M p p = (M, 0) p = (E, p ) p = (E, p ) m fter m p Four-momentum conservtion: p = p + p p = p p p = p (p ) = (p ) + (p ) p p = M + m M E = m E = M + m m M For moving prticles, pply pproprite Lorentz oost. Exmple: π + µ + ν µ work in rest frme of pion. m" % 0 E µ = m π + m µ m π = MeV p ν = p µ = E µ m µ = 9.8 MeV/c 5 Cross Section see JH D&R We hve em of prticles incient on trget (or nother em). (E, p ) Flux of incient em, f : numer of prticles per unit re per unit time. Bem illumintes N prticles in trget. We mesure the scttering rte, w/#, numer of prticles scttere in given irection, per unit time per unit soli ngle, #. w Ω = fn σ Ω (E, p ) (E, p ) &/# is ifferentil cross section Integrte over the soli ngle, rte of scttering: Define luminosity, L = f N Scttering rte w = Lσ w = fnσ 6

4 Collision Centre of n Mss Energy, s For collision efine Lorentz-invrint quntity, s: squre of sum of fourmomentum of incient prticles: s = (p + p ) (p + p ) = (p ) + (p ) + p p = m + m + (E E p p cos θ)!s=ecm is the energy in centre of momentum frme, energy ville to crte new prticles! Fixe Trget Collision, is t rest. E >> m, m s = m + m + E m E m E CM = E m (E, p ) Collier Experiment, with E = E = E >> m, m, ' = ( s = 4E E CM = E (E, p ) (E, p ) 7 Scttering With Lifetimes cn e very short, e.g. lifetime of $ ++ ryon (uuu) is 5#10 $4 s. Heisenurg Uncertinty Principle: E t Very short lifetime gives smll *t *E % /*t is significnt! mesurle with!"#$$%$&'()#*%+,-.!!!!!!!!!!!!!!!!!! "mx 10 "mx/e 10 ( + p scttering!! π + p elstic Mss of short live prticles (e.g. $ ++ ) is not fixe "s GeV #p )s, collision energy (GeV) Mss hs most-likely vlue, ut cn tke on rnge of vlues. nother exmple. Lern concepts, not etils! ( + p!$ ++!( + p scttering From nlysing the t in the ove plot it ws foun: m($ ++ ) ~ 13 MeV/c!($ ++ ) ~ 118 MeV 8

5 Collision Exmples (in Nturl Units) The previous LEP collier t CERN collie electrons n positrons he-on with E(e! ) = E(e + ) = 45.1 GeV. s = p(e + ) + p(e ) = m e + (E p e + p e cos θ) 4E (E + p e + p e ) E CM = E = 91. GeV Cross section to crete muons pirs ws $(e + e! %µ + µ! )=1.9 n t ECM = 91. GeV Totl integrte luminosity &L t = 16 p!1 Nevts(e + e #!µ + µ # ) = 16,000 # 1.9 = 307,800 To mke hrons, 45.1 GeV electron em ws fire into Beryllium trget. Electrons collie with protons n neutrons in Beryllium. s = p(e ) + p(p) = m e + m p +(E e E p p e p p cos θ) (E e m p ) E CM = E e m p = = 9.5 GeV In fixe trget electron energy is wste proviing momentum to the CM system rther thn to mke new prticles. 9 Summry of Lecture 3 Fermi s Golen Rule Rtes of ecys n sctterings yiel informtion out the forces n prticles involve through the mtrix element, M. M cn e relte to the quntum escription of the Stnr Moel. T = π M ρ Prticle Decy Lifetime,!, time tken for smple to ecrese y 1/e. Prtil with of ecy moe,!("x) # M(+x) Totl with is sum of ll possile ecy withs,!= /! Brnching rtio, proportion of ecys to given finl stte, BR ("x) =!("x)/! Prticle Scttering Cross section, &, proility for ecy to hppen. Mesure in = 10 "8 m. Luminosity, L is property of em (s) Integrte luminosity,!lt. Numer of events: N = &!Lt Two types of scttering experiment: collier n fixe trget. Nturl Units: set =c=1 Mesure energies, moment n mss in GeV Reltivistic Kinemtics p = (E, p x, p y, p z ) = (E, p ) p = E p = m Centre of Mss energy s = (p + p ) E CM = s 10