Introduction: Measurements in Particle Physics

Transcription

1 Sutomic Physics: Prticle Physics Lecture 2: Prcticl Prticle Physics 3rd Novemer 2009 Wht cn we mesure t the LHC, nd how do we interpret tht in term of fundmentl prticles nd interctions? Prticle Properties nd Quntum Numers Nturl Units Reltivistic kinemtics Decy Properties Scttering Properties 1 Introduction: Mesurements in Prticle Physics We sw lst time tht prticles generlly do two things: Sctter e.g. collisions of protons on protons t LHC collision of cosmic rys in tmosphere Decy e.g. decy of prticles produced t LHC decy of cosmic ry muons! Mesurements of sctterings nd decys re used to infer properties of the prticles nd interctions.! Cn lso mesure directly some properties of the longer-lived prticles. Sttic Prticle Properties Mss, m, Chrge, Q Mgnetic moment Spin nd Prity, J! Prticle Decys Prticle lifetime,!, nd decy width, " Allowed nd foridden decys conservtion lws Prticle Scttering Totl cross section,!. Differentil cross section, d!/d" Collision Luminosity, L Event Rte, N 2

2 Tools of Quntum Mechnics from introductory Sutomic slides Ech prticle cn e descried s quntum stte,!! The electromgnetic, wek nd strong forces cting on these sttes cn e represented y (three different) quntum opertors, Ô Rtes of interctions such s prticle lifetimes nd scttering cross sections re given y Fermi s Golden rule: Trnsition etween n initil stte!i! nd finl stte!f! re relted to the mtrix element M = Vfi = "!f Ô!i!: Trnsition proility,t = 2π M 2 ρ T is relted to the cross section of scttering, ". e.g. "(e + e # $µ + µ # ) # M (e + e # $µ + µ # ) 2. T is relted to the inverse lifetime of decy, %. e.g. %(µ # $e # &'e &µ) # 1/ M (µ # $e # &'e &µ) 2. We will see how to clculte M in future lectures 3 Qurk nd Lepton Flvour Quntum Numers Lepton numer, L: Totl numer of leptons! totl numer of nti-leptons! Electron numer, Le! Muon numer, Lµ! Tu numer, L% L = Le + Lµ + L% L e = N(e ) N(e + ) + N(ν e ) N( ν e ) L µ = N(µ ) N(µ + ) + N(ν µ ) N( ν µ ) L τ = N(τ ) N(τ + ) + N(ν τ ) N( ν τ ) Qurk Numer, Nq: Totl numer of qurks! totl numer of nti-qurks! Up qurk numer, Nu: e.g. N u = N(u)! N(u )! Chrm qurk numer, Nc! Down qurk numer, Nd! Strnge qurk numer, Ns Nq = Nu + Nd + Ns + Nc + N + Nt! Bottom qurk numer, N! Top qurk numer, Nt Lepton numer, electron numer, muon numer nd tu numer (L, Le, Lµ, L%) re conserved in ll interctions: strong, electromgnetic nd wek. Qurk numer (Nq) is lso conserved in ll interctions. Individul qurk numers (Nu, Nd, Ns, Nc, N, Nt) re conserved in strong nd electromgnetic interctions. They re not (necessrily) conserved in wek interctions. 4

3 Conservtion Lws Noether s Theorem: Every symmetry of nture hs conservtion lw ssocited with it, nd vice-vers. Energy, Momentum nd Angulr Momentum! Conserved in ll interctions! Symmetry: trnsltions in time nd spce; rottions in spce Chrge conservtion Q! Conserved in ll interctions! Symmetry: guge trnsformtion - underlying symmetry in QM description of electromgnetism Lepton Numer Le, Lµ, L" nd totl qurk numer Nq! Conserved in ll interctions! Symmetry: mystery! Qurk Flvour Numer, Nu, Nd, Ns, Nc, N, Nt, Prity, #! Conserved in strong nd electromgnetic interctions! Violted in wek interctions! Symmetry: unknown! Emmy Noether kg m s SI units: [M] [L] [T] Nturl Units I GeV c Nturl units: [Energy] [velocity] [ction] For everydy physics SI units re nturl choice: M(SH student)~75kg. Not so good for prticle physics: Mproton~10!27 kg PP chooses different sis - Nturl Units, sed on: " quntum mechnics ( ); " reltivity (c); " pproprite unit of energy 1 GeV = 10 9 ev = 1.60 " 10!10 J Energy GeV Time (GeV/ )!1 Momentum GeV/c Length (GeV/ c)!1 Mss GeV/c 2 Are (GeV/ c)!2 6

4 Nturl Units II Simplify even further y mesuring speeds reltive to c mesuring ction/ngulr momentum/spin reltive to Equivlent to setting c = = 1! All quntities re expressed in powers of GeV Energy GeV Time GeV!1 Momentum GeV Length GeV!1 Mss GeV Are GeV!2 Convert to SI units y reintroducing missing fctors of nd c Exmple: Are = 1 GeV!2 [L] 2 = [E] 2 [] n [c] m = [E] 2 [E] n [T ] n [L] m [T ] m n = 2, m = 2 Are (in SI units) = 1 GeV!2 " 2 c 2 = 3.89 " 10!32 m 2 = m Other common units: Msses nd energies mesured in MeV cross section mesured in rn, # 10!28 m 2 Two useful reltions: c = 197 MeV fm lengths in fm = 10!15 m electric chrge in units of e = MeV s 7 Review: Reltivistic Dynmics Plese review JH D&R 14 Two importnt quntities for Lorentz trnsformtions: β = v/c γ(v) = 1/ 1 β 2 Four-momentum of prticle: p = (E/c, p x, p y, p z ) = (E/c, p ) Energy of prticle E 2 = p 2 c 2 + m 2 c 4 E = γmc 2 Sclr product of 4-momentum: (p) 2 = (E/c) 2 p 2 = m 2 c 2 Prticles with m=0 trvel t the speed of light Nturl Units Lorentz oosts: γ = E/m γβ = p /m β = p /E Four momentum: p = (E, p x, p y, p z ) = (E, p ) Invrint mss (p) 2 = E 2 p 2 = m 2 8

5 Prticle Decy Introduction Most fundmentl prticles nd hdrons decy. We cn mesure: prticle lifetime, ": verge time tken prticle to decy decy width, # $ /", mesured in units of energy decy length, L: verge distnce trvelled efore decying rnching rtio, decy modes: prticles in finl stte, how often given finl stte occurs decy kinemtics pinitil = p finl Look t the following exmples: decy of! + meson into muon: π + µ + ν µ decy of KS mesons into two pions: K S π + π,k S π 0 π 0 Trcks left y chrged prticles Here: one invisile (neutrl) prticle hs decyed into two prticles 9 Prticle Lifetime Prticle lifetime, %, the time tken for the smple to reduce to 1/e of originl smple. Force Different forces hve different typicl lifetimes. Also define totl decy width, # ( /%. Strong dn dt = N τ = Γ N N(t) = N 0 exp( t/τ) = N 0 exp( Γt/) Typicl Lifetime 10!20-10!23 s Electromg 10!20-10!16 s Wek 10! s In its own rest frme prticle trvels v! = )c% efore decying. In the l, time is dilted y *. If % is lrge enough, energetic prticles trvel mesurle distnce L = *)c% in l. exmple of recent mesurement of prticle lifetime c!= $(%) = (1.30 ± 0.15) " 10 "12 s 10

6 Decy Modes Prticles cn hve more thn one decy mode. e.g. The KS meson decys 99.9% of the time in one of two wys: K S π + π, K S π 0 π 0 Ech decy mode hs its own mtrix element, M. Fermi s Golden Rule gives us the prtil decy width for ech decy mode: Γ(K S π + π ) M(K S π + π ) 2 Γ(K S π 0 π 0 ) M(K S π 0 π 0 ) 2 The totl decy width is equl to the sum of the decy widths for ll the llowed decys. 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 decys to prticulr finl stte: BR(K S π + π ) = Γ(K S π + π ) Γ(K S ) BR(K S π 0 π 0 ) = Γ(K S π 0 π 0 ) Γ(K S ) 11 Prticle Decy Kinemtics Most prticles decy. e.g. KS meson cn decy s: K S π + π Reconstruct the mss of prticle from the moment of the decy products: pinitil = p finl p(k S ) = p(π + ) + p(π ) squring ech side... 2 (M(K S )) 2 = p(π + ) + p(π ) 2 2 = p(π + ) + p(π ) + 2p(π + ) p(π ) Invrint mss from KS$! +! " decy. Not perfect due to limited ccurcy of mesurements = m(π + ) 2 + m(π ) E(π + )E(π ) 2 p (π + ) p (π ) M(KS) is reconstructed invrint mss of KS 12

7 Decy Kinemtics II Four-momentum conservtion: p = p + p p = p p A d A d A Decy of n unstle d prticle t rest: m md MA A & d p = (M A, 0) p = (E, p ) p = (E d, p d ) A d Before p After pd p = p d (p ) 2 = (p A ) 2 + (p d ) 2 2p A p d = M 2 A + m 2 d 2M A E d = m 2 E d = M 2 A + m2 d m2 2M A For moving prticles, pply pproprite Lorentz oost. Exmple: π + µ + ν µ work in rest frme of pion. m& + 0 E µ = m2 π + m 2 µ 2m π = MeV p ν = p µ = E 2 µ m 2 µ = 29.8 MeV/c 13 Scttering Consider collision etween two prticles: nd. Elstic collision: nd sctter off ech other ". e.g. e + e # $e + e # Inelstic collision: new prticles re creted " c d... e.g. e + e # $µ + µ # Two min types of prticle physics experiment: Collider experiments ems of nd re rought into collision. Often p = p (E, p ) (E, p ) LHC p-p collider Fixed Trget Experiments: A em of re ccelerted into trget t rest. sctters off in the trget. (E, p ) NA48 Fixed Trget: p+be K 14

8 Mesuring Scttering The cross section, ", mesures the how often scttering process occurs. " is chrcteristic of given process (force) from Fermi s Golden Rule " # M 2 nd energy of the colliding prticles. " mesured in units of re. Normlly use rn, 1 = 10!28 m 2. Luminosity, L, is chrcteristic of the em. Mesured in units of inverse re per unit time. Integrted luminosity,,ldt is luminosity delivered over given period. Mesured in units of inverse re, usully!1. Wht, nd how often, prticles re creted in the finl stte. Force Strong Electromg Wek Event rte: w = Lσ Typicl Cross Sections 10 m 10!2 m 10!13 m Totl numer of events: N = σ Ldt 15 Cross Section see JH D&R 2 We hve em of prticles incident on trget (or nother em). (E, p ) (E, p ) (E, p ) Flux of incident em, f : numer of prticles per unit re per unit time. Bem illumintes N prticles in trget. We mesure the scttering rte, dw/d', numer of prticles scttered in given direction, per unit time per unit solid ngle, d'. dw dω = fn dσ dω d"/d' is differentil cross section Integrte over the solid ngle, rte of scttering: Define luminosity, L = f N Scttering rte w = Lσ w = fnσ 16

9 Collision Centre of n Mss Energy, s For collision define Lorentz-invrint quntity, s: squre of sum of fourmomentum of incident prticles: s = (p + p ) (p + p ) = (p ) 2 + (p ) p p = m 2 + m 2 + 2(E E p p cos θ)!s=ecm is the energy in centre of momentum frme, energy ville to crte new prticles! Fixed Trget Collision, is t rest. E >> m, m s = m 2 + m 2 + 2E m 2E m (E, p ) E CM = 2E m Collider Experiment, with E = E = E >> m, m, - =. s = 4E 2 E CM = 2E (E, p ) (E, p ) 17 Prticle Width Lifetimes cn e very short, e.g. lifetime of ( ++ ryon (uuu) is 5#10!24 s. Heisenurg Uncertinty Principle: E t Very short lifetime gives smll 0t 0E + /0t is significnt $ mesurle width!"#$$%$&'()#*%+,-.!!!!!!!!!!!!!!!!!! "mx 10 2 "mx/e 10 #. + p scttering! Mss of short lived prticles (e.g. ( ++ ) is not fixed. Mss hs most-likely vlue, ut cn tke on rnge of vlues. π + p elstic "s GeV #p /s, collision energy (GeV) 18

10 Collision Exmples The previous collider t CERN collided electrons nd positrons hed-on with E(e % ) = E(e + ) = 45.1 GeV. 2 s = p(e + ) + p(e ) = 2m 2 e + 2(E 2 p e + p e cos θ) 4E 2 2(E 2 + p e + p e ) E CM = 2E = 91.2 GeV #(e + e " $µ + µ " )=1.9 n t ECM = 91.2 GeV Totl integrted luminosity %L dt = 400 p "1 Nevts(e + e % &µ + µ % ) = 400,000 & 1.9 = 760,000 To mke hdrons, 45.1 GeV electron em ws fired into Beryllium trget. Electrons collide with protons s = nd neutrons in Beryllium. 2 p(e ) + p(p) = m 2 e + m 2 p +2(E e E p p e p p cos θ) 2(E e m p ) E CM = 2E e m p = = 9.5 GeV In fixed trget electron energy is wsted providing momentum to the CM system rther thn to mke new prticles. 19 Lecture 2 Summry Nturl Units: set =c=1 Mesure energies in GeV Every quntity is mesured s power of energy Prticle Decy Lifetime, %, time tken for smple to decrese y 1/e. Prtil width of decy mode, #(A%x) # M(A$x) 2 Totl width is sum of ll possile decy widths, #= /% Brnching rtio, proportion decys to given finl stte, BR (A%x) = #(A%x)/# Conserved quntum numers tell us out the underlying symmetries Prticle Scttering Cross section, ", proility for decy to hppen. Mesured in = 10!28 m 2. Luminosity, L is property of em (s) Integrted luminosity,!ldt. Numer of events: N = "!Ldt Two types of scttering experiment: collider nd fixed trget. Reltivistic Kinemtics p = (E, p x, p y, p z ) = (E, p ) 2 p = E 2 p 2 = m 2 Centre of Mss energy s = (p + p ) 2 E CM = s 20