Particle Physics. Dr M.A. Thomson. e + γ. 1 q 2. e - Part II, Lent Term 2004 HANDOUT II. Dr M.A. Thomson Lent 2004

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1 Particl Physics Dr M.A. Thomson µ q 2 µ Part II, Lnt Trm 2004 HANDOUT II Dr M.A. Thomson Lnt 2004

2 Quantum Elctrodynamics 2 QUANTUM ELECTRODYNAMICS: is th quantum thory of th lctromagntic intraction. CLASSICAL PICTURE: Action at a distanc : forcs aris from ~E and ~B filds. Particls act as sourcs of th filds! V (~r). Q.E.D. PICTURE: Forcs aris from th xchang of virtual fild quanta. p q 2 p Although a complt drivation of th thory of Q.E.D. and Fynman diagrams is byond th scop of this cours, th main faturs will b drivd. Dr M.A. Thomson Lnt 2004

3 Intraction via Particl Exchang 3 NONEXAMINABLE FERMI S GOLDEN RULE for Transition rat, fi : fi = 2ß ~ jm fij 2 ρ(e f ) ρ(e f ) = dnsity of final stats. From st ordr prturbation thory, matrix lmnt M fi : M fi = hψ f j ^H 0 jψ i i whr ^H 0 is th oprator corrsponding to th prturbation to th Hamiltonian. This is only th st ordr trm in th prturbation xpansion. In 2 nd ordr X prturbation thory: M fi! M fi jm fj j jm ji j E j6=i i E j whr th sum is ovr all intrmdiat stats j, and E i and E j ar th nrgis of th initial and intrmdiat stat For scattring, th st and 2 nd ordr trms can b viwd as: f f i V fi V fj i j V ji Dr M.A. Thomson Lnt 2004

4 Considr th particl intraction a b! c d which involvs th xchang a particl X. This could b th lastic scattring of lctrons and protons,.g. p! p whr X is an xchangd photon. On possibl spactim pictur for this procss is 4 Spac a ( i ) (p i ) b V ji X V V fj i j f c ( f ) (p f ) d Tim Initial Stat, i: ab Final Stat, f : cd Intrmdiat Stat, j: bcx Th Tim Ordrd intraction consists of a! c X followd by b X! d. For xampl i p i! f p f has th lctron mitting a photon ( i! f fl) followd by th photon bing absorbd by th proton (p i fl! p f ). Th corrsponding trm in 2 nd ordr PT: M ab fi = hψ fj ^H 0 jψ j ihψ j j ^H 0 jψ i i E i E j = hψ dj ^H 0 jψ X ψ b ihψ c ψ X j ^H 0 jψ a i (E a E b ) (E c E X E b ) = hψ dj ^H 0 jψ X ψ b ihψ c ψ X j ^H 0 jψ a i (E a E c E X ) Dr M.A. Thomson Lnt 2004

5 Bfor w go any furthr som commnts: Th suprscript ab on M ab fi whr a intracts with X bfor b indicats th tim ordring consquntly th rsults ar not Lorntz Invariant i.. dpnd on rst fram. 5 Momntum is consrvd in a! c X and b X! d. Th xchangd particl X is ON MASS SHELL: E 2 X p2 X = m2 X Th matrix lmnts hψ d j ^H 0 jψ X ψ b i and hψ c ψ X j ^H 0 jψ a i dpnd on th strngth of th intraction..g. th strngth of th fl and flp intraction which dtrmins th probability that an lctron(proton) will mit(absorb) a photon. For th lctromagntic intraction: hψ k j ^H 0 jψ j i = ffl 0 hψ k jzjψ j i for a photon with polarization in th zdirction. (s Dr Ritchi s QM II lctur 0) Nglcting spin (i.. for assuming all particls ar spin0 i.. scalars) th ME bcoms: hψ d j ^H 0 jψ X ψ b i = Mor gnrally, hψ d j ^H 0 jψ X ψ b i = g, whr g is th intraction strngth. Dr M.A. Thomson Lnt 2004

6 Spac Now considr th othr tim ordring b! d X followd by a X! b a ( i ) (p i ) b V ji X V V fj i j f c ( f ) (p f ) d Tim Spac Th corrsponding trm in 2 nd ordr PT: a b X V V fj V ji i j f c d Tim M ba fi = hψ cj ^H 0 jψ X ψ a ihψ d ψ X j ^H 0 jψ b i (E a E b ) (E d E X E a ) = hψ cj ^H 0 jψ X ψ a ihψ d ψ X j ^H 0 jψ b i (E b E d E X ) = hψ cj ^H 0 jψ X ψ a ihψ d ψ X j ^H 0 jψ b i (E b E d E X ) Assum a common intraction strngth, g, at both vrtics, i.. hψ c j ^H 0 jψ X ψ a i = hψ d ψ X j ^H 0 jψ b i = g ) M ba g 2 fi = (E b E d E X ) 2E X 6 WARNING : I hav introducd an (unjustifid) factor of 2EX. This ariss from th rlativistic normalization of th wavfunction for particl X (s appndix). For initial/final stat particls th normalisation is canclld by corrsponding trms in th flux/phasspac. For th intrmdiat particl X no such canclation occurs. Dr M.A. Thomson Lnt 2004

7 Now sum ovr two tim ordrd transition rats M fi = M ab fi Mba fi = g 2 E a E c E X E b E d E X 2E X 7 giving: M fi = g 2 sinc E a E b = E c E d ) E b E d = E c E a E a E c E X = g 2 E a E c E X = g 2 2E X (E a E c ) 2 E 2 X From th first tim ordring: thrfor M fi = M fi = E 2 X = (~p a ~p c ) 2 m 2 X E c E a E X E a E c E X g 2 2E X 2E X (E a E c ) 2 (~p a ~p c ) 2 m 2 X g 2 q 2 m 2 X with q 2 = q μ q μ = E 2 j~pj 2 whr (E; j~pj) ar nrgy/momntum carrid by th virtual particl. Th SUM of timordrd procsss dpnds on q 2 and is thrfor Lorntz invariant! Th invariant mass of th xchangd particl, X, m 2 inv = E2 j~pj 2, is NOT th REST MASS, m X. 2E X Dr M.A. Thomson Lnt 2004

8 Th trm q 2 m 2 8 is calld th PROPAGATOR It corrsponds to th trm in th matrix lmnt arising from th xchang of a massiv particl which mdiats th forc. For masslss particls.g. photons : q 2 NOTE: q 2 is th 4momntum of th xchangd particl (q 2 = q μ q μ = E 2 j~pj 2 ) Prviously (pag 35 of HANDOUT ) w obtaind th matrix lmnt for lastic scattring in th YUKAWA potntial: M Y UK fi = g2 (m 2 j~pj 2 ) For lastic scattring E X = 0, and q 2 = j~pj 2 M Y UK fi! g2 q 2 m 2 Which is xactly th xprssion obtaind on th prvious pag. Hnc, lastic scattring via particl xchang in 2nd ordr P.T. is quivalnt to scattring in a Yukawa potntial using st ordr P.T. Dr M.A. Thomson Lnt 2004

9 Action at a Distanc 9 NEWTON :...that on body can act upon anothr at a distanc, through a vacuum, without th mdiation of anything ls,..., is to m a grat absurdity In Classical Mchanics and nonrlativistic Quantum Mchanics forcs aris from potntials V (~r) which act instantanously ovr all spac. In Quantum Fild thory, forcs ar mdiatd by th xchang of virtual fild quanta and thr is no mystrious action at a distanc. Mattr and Forc dscribd by particls Dr M.A. Thomson Lnt 2004

10 Fynman Diagrams 0 Th rsults of calculations basd on a singl procss in TimOrdrd Prturbation Thory (somtims calld oldfashiond, OFPT) dpnd on th rfrnc fram. Howvr, th sum of all tim ordrings is not fram dpndnt and provids th basis for our rlativistic thory of Quantum Mchanics. Th sum of tim ordrings ar rprsntd by FEYNMAN DIAGRAMS Spac Spac Tim Tim = Fynman Diagram Enrgy and Momntum ar consrvd at th intraction vrtics But th xchangd particl no longr has m 2 X = E2 X p2 X,itisVIRTUAL Dr M.A. Thomson Lnt 2004

11 Virtual Particls Spac Spac Tim Tim = Fynman Diagram Virtual Particls: Forcs du to xchangd particl X which is trmd VIRTUAL. Th xchangd particl is off massshll, i.. for th unobsrvabl xchangd VIRTUAL particl E 2 6= p 2 m 2 X. i.. m 2 = E 2 X p2 X dos not giv th physical mass, m X. Th mass of th virtual can b v or v. particl m 2 = E 2 X p2 X Qualitativly: th propagator is invrsly proportional to how far th particl is offshll. Th furthr offshll, th smallr th probability of producing such a virtual stat. Dr M.A. Thomson Lnt 2004

12 Undrstanding Fynman Diagrams 2 Fynman diagrams ar th languag of modrn particl physics. Thy will b usd xtnsivly throughout this cours. Th Basic Building Blocks Tim lctron positron photon Not : th positron ( ) lin is drawn as a ngativ nrgy lctron travling backwards in tim Th ± photon intractions Radiation Annihilation Pair Prod. Not: non of ths procsss ar allowd in isolation : Forbiddn by (E; ~p) consrvation. Th strngth of th intraction btwn th virtual photon and frmions is calld th coupling strngth. For th lctromagntic intraction this is proportional to lctric charg. Dr M.A. Thomson Lnt 2004

13 Th Elctromagntic Vrtx 3 Th lctromagntic intraction is dscribd by th photon propagator and th vrtx: Elctromagntic vrtx,µ,τ Q α q α= 2 /4π,µ,τ q COUPLING strngth proportional to th frmion charg. All lctromagntic intractions can b dscribd in trms of th abov diagram Always consrv nrgy and momntum (angular momntum, charg) QED Vrtx NEVER changs flavour i..! fl but not! μ fl QED Vrtx also consrvs PARITY Qualitativly : Q p ff can b thought of th probability of a chargd particl mitting a photon, th probability is proportional to =q 2 of th photon. Dr M.A. Thomson Lnt 2004

14 Physics with Fynman Diagrams 4 Scattring cross sctions calculatd from: Frmion wav functions Vrtx Factors : coupling strngth Propagator Phas Spac Elctron Currnt q 2 Propagator p p Proton Currnt Matrix lmnt M factoriss into 3 trms : im = hu jifl μ ju i Elctron Currnt igμν q 2 Photon Propagator hu p jifl ν ju p i Proton Currnt Th factors fl μ and g μν ar 4 4 matrics which account for th spinstructur of th intraction (dscribd in th lctur on th Dirac Equation). Dr M.A. Thomson Lnt 2004

15 Pur QED Procsss 5 Compton Scattring propagator M ο 2 ff ο jmj 2 ο 4 ff ο (4ß) 2 ff 2 Brmsstrahlung nuclus Pair Production nuclus ß 0 Dcay M ο Z:: jmj 2 ο Z 2 6 ff ο (4ß) 3 Z 2 ff 3 M ο ::Z jmj 2 ο Z 2 6 ff ο (4ß) 3 Z 2 ff 3 u M ο Q u :Q u π 0 jmj u 2 ο Q 4 u 4 ff ο (4ß) 2 Q 4 u ff2 u Dr M.A. Thomson Lnt 2004

16 ElctronProton Scattring p α q 2 α p M ο : jmj 2 ο 4 ff ο (4ß) 2 ff 2 6 Annihilation J=ψ! μ μ q q M ο :Q u jmj 2 ο Q 2 u 4 ff ο (4ß) 2 Q 2 u ff2 J/ψ c c µ µ M ο Q c : jmj 2 ο Q 2 c 4 ff ο (4ß) 2 Q 2 c ff2 Coupling strngth dtrmins ordr of magnitud of matrix lmnt. For particls intracting/dcaying via lctromagntic intraction: typical valus for cross sctions/liftims ff m ο 0 2 mb fi m ο 0 20 s Dr M.A. Thomson Lnt 2004

17 Scattring in QED 7 EXAMPLE Calculat th spinlss cross sctions for th two procsss: lctronproton scattring lctronpositron annihilation µ p q 2 p q 2 Hr w will considr th cas whr all particls ar spin0, (s lctur on Dirac Equation for complt tratmnt) Frmi s Goldn rul and Born Approximation: dff dω = 2ßjMj2 dρ(e f )=dω For both procsss writ th SAME matrix lmnt M = 2 q = 4ßff 2 q 2 Howvr, th fourmomntum transfr (q 2 = E 2 ~q 2 )is vry diffrnt (~q is th 3momntum of th virtual photon) Elastic proton scattring : q = (0; ~q) q 2 = j~qj 2 annihilation : q = (2E; 0) q 2 = 4E 2 µ Dr M.A. Thomson Lnt 2004

18 p q 2 Spinlss p Scattring p From Handout, pags 334: dff dω p i M = 2 = 2ßjMj2 E2 (2ß) 3 p f θ q 2 = 4ßff q 2 = 2ß (4ßff)2 E 2 q 4 (2ß) = 4ff2 E 2 3 q 4 q 2 is th fourmomntum transfr: q 2 = q μ q μ = (E f E i ) 2 (~p f ~p i ) 2 8 = E 2 f E2 i 2E f E i ~p f 2 ~p i 2 2:~p f :~p i = 2m 2 2E f E i 2j~p f jj~p i j cos nglcting lctron mass: i.. m 2 = 0 and j~p fj = E f q 2 = 2E i E f ( cos ) q 2 = 4E i E f sin 2 2 Thrfor for ELASTIC scattring E i = E f dff dω = ff 2 4E 2 sin 4 2 i.. th Ruthrford scattring formula (Handout p.36) Dr M.A. Thomson Lnt 2004

19 9 Spinlss Annihilation µ M = 4ßff q 2 µ q 2 dff dω (4ßff)2 E 2 = 2ß q 4 (2ß) = 4ff2 E 2 3 q 4 sam formula, but diffrnt fourmomntum transfr: q 2 = q μ q μ = (E E ) 2 (~p ~p ) 2 Assuming w ar in th cntrofmass systm E = E = E ~p = ~p! q 2 = (2E) 2 = s dff dω = 4ff2 E 2 = 4ff2 E 2 q 4 6E 4 Intgrating givs total cross sction: = ff2 s ff = 4ß ff2 s Dr M.A. Thomson Lnt 2004

20 This is not quit corrct bcaus w hav nglctd spin. Th actual cross sction (s lctur on Dirac Equation) is dff dω = ff2 4s ( cos2 ) ff(! μ μ ) = 4ßff2 3s Natural Units Exampl cross sction at p s = 22 GV i.. GV lctrons colliding with GV positrons. ff!μ μ = 4ßff2 3s = 4ß =4:6 0 7 GV 2 =4:6 0 7 (~c) 2 =(:6 0 0 ) 2 m 2 =: m 2 = 0:8 nb 20 Dr M.A. Thomson Lnt 2004

21 Th DrllYan Procss 2 Can also annihilat qq as in th DrllYan procss.g. ß p! μ μ hadrons p d u u u µ π u d u µ ff(ß p! μ μ hadrons) / Q 2 u ff2 (s Qustion 3 on th problm sht) Dr M.A. Thomson Lnt 2004

22 Exprimntal Tsts of QED 22 QED is an incrdibly succssful thory Exampl Magntic momnts of ± ; μ ± ~μ = g 2m ~s For a pointlik spin /2 particl : g=2 Howvr highr ordr trms induc an anomalous magntic momnt i.. g not quit 2. (g 2) 2 (g 2) 2 = 59652:869 ± 0: EXPT = 59652:3 ± 0:3 0 0 THEORY Agrmnt at th lvl of in 0 8 Q.E.D. provids a rmarkably prcis dscription of th lctromagntic intraction! Dr M.A. Thomson Lnt 2004

23 Highr Ordrs 23 So far only considrd lowst ordr trm in th prturbation sris. Highr ordr trms also contribut Lowst Ordr: µ µ jmj 2 / ff 2 ο 37 2 Scond Ordr: µ µ... Third Ordr: µ µ jmj 2 / ff 4 ο jmj 2 / ff 6 ο 37 6 Scond ordr supprssd by ff 2 rlativ to first ordr. Providd ff is small, i.. prturbation is small, lowst ordr dominats. Dr M.A. Thomson Lnt 2004

24 Running of ff 24 ff = 2 4ß btwn an lctron and photon. BUT ff isn t a constant spcifis th strngth of th intraction Considr a fr lctron: Quantum fluctuations lad to a cloud of virtual lctron/positron pairs this is just on of many (an infinit st) such diagrams. Th vacuum acts lik a dilctric mdium Th virtual pairs ar polarizd At larg distancs th bar lctron charg is scrnd. At larg R tst charg ss scrnd charg Tst Charg At small R tst charg ss bar charg Tst Charg Dr M.A. Thomson Lnt 2004

25 Running of ff 25 Masur ff(q 2 ) from! μ μ tc. 4πα q 2 4πα µ µ α (Q) TOPAZ µµ/µµ: α (0) qq: Fits to lptonic data from: DORIS, PEP, PETRA, TRISTAN α SM (Q) OPAL Q / GV ff incrass with th incrasing q 2 (i.. closr to th bar charg). At q 2 = 0: ff = =37 At q 2 = (00 GV) 2 : ff = =28 Dr M.A. Thomson Lnt 2004

26 Appndix: Rlativistic Phas Spac 26 NONEXAMINABLE Prviously normalizd wavfunctions to particl in a box of sid L (s Handout, pags 3334 ). Rst Fram particl/v Lab. Fram particl/(v/) In rlativity, box will b Lorntz Contractd by a factor of fl = i:: V 0 = V ( m E ) i.. E=m particls pr volum V p v2 =c = E 2 m NEED to adjust normalization volum with nrgy Convntional choic: N = p 2E In most scattring procss th factors of p 2E in th wavfunction normalization cancl with corrsponding factors in th xprssions for flux and dnsity of stats, just as th factors of L 3 wr cancld prviously (Handout, pags 35) Dr M.A. Thomson Lnt 2004