Particle Physics HANDOUT III. Part II, Lent Term q g. e + e - Dr M.A. Thomson. α S. Q q. 1 q 2. Dr M.A. Thomson Lent 2004

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

2 QCD 2 QUANTUM ELECTRODYNAMICS: is the uantum theory of the electromanetic interaction. mediated by massless photons photon couples to electric chare, e Strenth of interaction : hψ f j ^Hjψ i i / p ff. ff = e2 4ß QUANTUM CHROMO-DYNAMICS: is the uantum theory of the stron interaction. mediated by massless luons, i.e. 1= 2 propaator luon couples to stron chare Only uarks have non-zero stron chare, therefore only uarks feel stron interaction Basic QCD interaction looks like a stroner version of QED, ff S > ff EM QED Q α α = e 2 /4π ~ 1/137 γ QCD α S α S = S 2/4π ~ 1 ( subscript em is sometimes used to distinuish the ff em of electromanetism from ff S ).

3 COLOUR 3 In QED: Chare of QED is electric chare. Electric chare - conserved uantum number. In QCD: Chare of QCD is called COLOUR COLOUR is a conserved uantum number with 3 VALUES labelled red, reen and blue Quarks carry COLOUR Anti-uarks carry ANTI-COLOUR r b r b Leptons, fl, W ±, Z 0 DO NOT carry colour, i.e. have colour chare zero! DO NOT participate in STRONG interaction. Note: Colour is just a label for states in a non-examinable SU(3) representation r = ψ 10 0! = ψ 01 0! b = ψ 00 1!

4 GLUONS 4 In QCD: Gluons are MASSLESS spin-1 bosons Consider a red uark scatterin off a reen uark. Colour is exchaned but always conserved. r luon r r UNLIKE QED: Gluons carry the chare of the interaction. Gluons come in different colours. Expect 9 luons (3 colours 3 anti-colours) rb, r, r, b, b, br rr,, bb However: Real luons are orthoonal linear combinations of the above states. The combination p 1 3 (rr + + bb) is colourless and does not take part in the stron interaction.

5 Colour at Work 5 EXAMPLE: Annihilation r r Normally do not show colour on Feynman diarams - colour is conserved. QED POTENTIAL: V QED = ff r QCD POTENTIAL: At short distances QCD potential looks similar V QCD = 4 3 ff S r apart from 4 3 colour factor. Note: the colour factor (4/3) arises because more than one luon can participate in the process!. Obtain colour factor from averain over initial colour states and summin over final/intermediate colour states

6 6 SELF-INTERACTIONS At this point, QCD looks like a stroner version of QED. This is true up to a point. However, in practice QCD behaves very differently to QED. The similarities arise from the fact that both involve the exchane of MASSLESS spin-1 bosons. The bi difference is that GLUONS carry colour chare. GLUONS CAN INTERACT WITH OTHER GLUONS: 3 GLUON VERTEX 4 GLUON VERTEX EXAMPLE: Gluon-Gluon Scatterin! + + e.. r+ b rr+ rb

7 CONFINEMENT 7 NEVER OBSERVE: sinle FREE uarks/luons uarks are always confined within hadrons This is a conseuence of the stron self-interactions of luons. Qualitatively, picture the colour field between two uarks. The luons mediatin the force act as additional sources of the colour field - they attract each other. The luon-luon interaction pulls the lines of colour force into a narrow tube or STRING. In this model the strin has a tension and as the uarks separate the strin stores potential enery. Enery stored per unit lenth ο constant. V (r) / r Reuires infinite enery to separate two uarks. Quarks always come in combinations with zero net colour chare: CONFINEMENT.

8 How Stron is Stron? 8 QCD Potential between uarks has two components: COULOMB -LIKE TERM : 4 3 ff S r LINEAR TERM : +kr V QCD (GeV) V = -4α s 3r +kr V = -4α s 3r α s =0.2 k=1 GeV/fm r(fm) Force between two uarks at separated by 10 m: V QCD = ff S r + kr with k ß 1 GeV=fm F = dv dr = ff S r 2 at lare r F = k = + k = N 1: Euivalent to the weiht of approximately 65 Widdecombes. N

9 JETS 9 Consider the pair produced in e + e! : e + e - γ As the uarks separate, the enery stored in the colour field ( strin ) starts to increase linearly with separation. When E stored > 2m new pairs can be V QCD (GeV) -1-2 created V = -4α s 3r +kr V = -4α s 3r α s =0.2 k=1 GeV/fm r(fm) as enery decreases... hadrons freeze out

10 As uarks separate, more pairs are produced from the potential enery of the colour field. This process is called HADRONIZATION. Start out with uarks and end up with narrowly collimated JETS of HADRONS 10 SPACE TIME π + (ud) etc... π + π 0 π - K + π - π 0 π 0 π+ p π 0 e + e - γ e - e + Typical e + e! Event The hadrons in a uark(anti-uark) jet follow the direction of the oriinal uark(antiuark). Conseuently e + e! is observed as a pair of back-to-back jets of hadrons

11 aside ALEPH DALI Run=56698 Evt= Gev EC 6 Gev HC Y" RO TPC 1cm 0 1cm 1cm 0 1cm X" Z0<5 D0<2 (φ 175)*SIN(θ) x o x oo x o o o x o x x o o o xo x o x x o x o o x o x x o x o x o x o o o o x o x o x x o x ox x o x Made on 30-Au :24:02 by konstant with DALI_F1. Filename: DC056698_007455_000830_1723.PS 15 GeV θ=180 θ=0 You will now reconize the His event from the cover of Handout I as e + e! somethin!

12 Runnin of ff S 12 ff S specifies the strenth of the stron interaction BUT just as in QED, ff S isn t a constant, it runs In QED the bare electron chare is screened by a cloud of virtual electron-positron pairs. In QCD a similar effect occurs. In QCD uantum fluctuations lead to a cloud of virtual pairs one of many (an infinite set) such diarams analoous to those for QED. In QCD the luon self-interactions ALSO lead to a cloud of virtual luons one of many (an infinite set) such diarams. Here there is no analoy in QED, photons don t have self-interactions since they don t carry the chare of the interaction.

13 Colour Anti-Screenin Due to the luon self-interactions bare colour chare is screened by both virtual uarks and virtual luons The cloud of virtual luons carries colour chare and the effective colour chare INCREASES with distance! At low eneries (lare distances) ff S becomes lare! can t use perturbation theory (not a weak perturbation) 13 α EM QED α S QCD 1/128 1/137 M Z M p Hih Enery M Z Low Enery Hih Enery Low Enery α s 1 α s 1 M Z 0 M p lo 10 ( 2 /GeV 2 ) lo 10 (r/m) At Hih eneries (short distances) ff S is small. In this reime treat uarks as free particles and can use perturbation theory ASYMPTOTIC FREEDOM At p s = 100 GeV, ff S = 0:12.

14 14 Scatterin in QCD EXAMPLE: Hih enery proton-antiproton scatterin Jet alon beam direction p u α S u 1 2 p Visible jet in direction of u α S M ο 1 2 p ffs p ffs ) dff dω ο (ff S) 2 sin 4 =2 u The upper points are the Geier and Marsden data (1911) for the elastic scatterin of ff particles as they traverse thin old and silver foils. The lower points show the anular distribution of the uark jets observed in proton-antiproton scatterin at 2 = 2000GeV 2 Both follow the Rutherford formula for elastic scatterin: sin 4 2.

15 EXAMPLE: pp vs ß + p scatterin 15 p π + p p Calculate ratio of ff(pp) total to ff(ß + p) total QCD does not distinuish between uark flavours, only COLOUR chare of uarks matters. At hih enery (E fl bindin enery of uarks within hadrons) ratio of pp and pß total cross sections depends on number of possible uark-uark combinations. Predict ff(ßp) ff(pp) = = 2 3 Experiment ff(ßp) ff(pp) ß 24 mb 38 mb ß 0:63

16 16 QCD in e + e Annihilation Direct evidence for the existence of colour comes from e + e Annihilation. Compare e + e! μ + μ, e + e! : R μ = ff(e+ e! hadrons) ff(e + e! μ + μ ) e + e - α γ 1 2 α µ + µ - e + e - α γ 1 2 Q α If we nelect the masses of the final state uarks/muons then the ONLY difference is the chare of the final state particles (Q μ = 1, Q = or 1 3 ) Start by calculatin the cross section for the process ff(e + e! ff).(ff represent a fermion-antifermion pair e.. μ + μ or ). see Handout II for the case where ff = μ + μ

17 e + e - α p µ 2 p µ 1 γ 1 2 f Q f α Electron/Positron beams alon z-axis f e - θ e + f f 17 p μ 1 = (E; p x ; p y ; p z ) p μ 1 = (E; 0; 0; E) nelectin m e p μ 2 = (E; 0; 0; E) μ = p μ 1 + pμ 2 = (2E; 0; 0; 0) 2 = 4E 2 = s where s is (centre-of-mass enery) 2. Fermi s Golden rule and Born Approximation dff dω = dρ(e f ) 2ßjMj2 dω Matrix element M : M = hv e +jq e eju e i 1 2 hv f jq f eju f i = 4ßffQ eq f 2 with ff = e2 4ß dff dω = 2ß ( 4ßffQ eq f )2 E (2ß) 2 4 (1 + cos2 ) = ff2 Q 2 f 4s (1 + cos2 )

18 (1 + cos 2 ) comes from spin-1 photon decayin to two spin-half fermions. see lecture on Dirac euation Total cross section for e + e! ff ff = = Z dff dω dω Z 2ß 0 Z ß 0 = ßff2 Q 2 f 2s ff 2 Q 2 f 4s (1 + cos2 )sin d dffi Z +1 1 (1 + y 2 )dy (y = cos ) 18 = 4ßff2 Q 2 f 3s ff(e + e! μ + μ ) = 4ßff2 3s ff(e + e! μ + μ ) for e + e collider data at centre-of-mass eneries 8-36 GeV

19 Back to R μ = ff(e+ e! hadrons) ff(e + e! μ+ μ ) 19 For a sinle uark flavour of a iven colour R = Q 2 However, we measure e + e! jets not e + e! uu. A jet from a u-uark looks just like a jet from a d-uark... Need to sum over flavours (u,d,c,s,t,b) and colours (r; ; b). R = 3 X i Q 2 i (3colours) where the sum is over all uark flavours p kinematically accessible at centre-of-mass enery, s, of collider. Enery Ratio R p s > 2ms ο1 GeV 3( ) = 2 u,d,s p s > 2mc ο4 GeV 3( ) = u,d,s,c p s > 2mb ο10 GeV 3(:: ) = u,d,s,c,b p s > 2mt ο350 GeV 3(:: ) = 5 u,d,s,c,b,t

20 20 R μ = ff(e+ e! hadrons) ff(e + e! μ+ μ ) Data: p s from 0 40 GeV R μ increases in steps with p s p s < 11 GeV reion complicated by resonances: charmonium (cμc) and bottomonium (b μ b). R μ Data exclude no colour hypothesis. STRONG EVIDENCE for COLOUR

21 Experimental Evidence for Colour 21 R μ The existence of the Ω (sss) The Ω (sss) is a (L=0) spin- 3 baryon consistin 2 of 3 strane-uarks. The wave-function ψ = s " s " s " is SYMMETRIC under particle interchane. However uarks are FERMIONS, therefore reuire an ANTI-SYMMETRIC wave-function, i.e. need another deree of freedom, namely COLOUR. ψ = (s " s " s ")ψ colour ψ colour = p 1 (rb+br+br-rb-rb-br) 6 ß 0! flfl decay rate Need colour to explain ß 0! flfl decay rate. π 0 u u (ß 0! flfl) / N 2 colour EXPT : N colour = 2:99 ± 0:12 u γ γ

22 Evidence for Gluons 22 In QED, electrons can radiate photons. In QCD uarks can radiate luons. e + e! e + e - α γ 1 2 α S Q α p ivin an extra factor of ff S in the matrix element, i.e. an extra factor of ff S in cross section. In QED we can detect the photons. In QCD we never see free luons due to confinement. Experimentally detect luons as an additional jet: 3-Jet Events. Anular distribution of luon jet depends on luon spin

23 3-Jet Events and Gluon Spin 23 JADE p s = 31 GeV Direct Evidence for Gluons (1978) OPAL p s = 91 GeV (1990) Distribution of the anle, ffi between the hihest enery jet (assumed to be one of the uarks) relative to the fliht direction of the other two (in their cms frame). ffi depends on the spin of the luon. ) GLUON is SPIN-1

24 Gluon Self-Interactions 24 Direct Evidence for the existence of the the luon selfinteractions from 4-JET events. Y 3 GLUON VERTEX 4 GLUON VERTEX Z X e + e - e + e - e + e + e - e - Anular distribution of jets is sensitive to existence triple luon vertex: ffl vertex consists of 2 spin- 1 2 uarks and a spin-1 luon. ffl vertex consists of 3 spin-1 luons, ) different anular distribution.

25 Experimentally: 25 Define the two lowest enery jets as the luons. (luon jets are more likely to be low enery than uark jets) Measure anle between the plane containin the uark jets and the plane containin the luon jets, χ BZ χ BZ χ BZ Gluon selfinteractions are reuired to describe the data. experimental Theory without self-interactions (ABELIAN) is inconsistent with observations

26 Measurin ff S 26 ff S can be measured in many ways. The cleanest is from R μ : In practice measure i.e. don t distinuish 2/3 jets. So measure R μ = ff(e+ e! hadrons) ff(e + e! μ+ μ ) not R μ = ff(e + e! ) ff(e + e! μ+ μ ) When luon radiation is included : R μ = 3 X Q ff S ß P Q2 = Therefore (1 + ff S ß ) ß 3:9 3:66 ivin ff S ( 2 = 25 2 ) ß 0:20

27 Many other ways to measure ff S... e.. 3 jet rate : e + e! e + e - α γ 1 2 α S Q α 27 ff(3 jet) ff(2 jet) = ff() ff() / ff S ff S Summary Summary of ff S measurements ff S RUNS!

28 Nucleon-Nucleon Interactions 28 Bound states (e.. Protons and Neutrons) are COLOURLESS (COLOUR SINGLETS). They can only interact via COLOURLESS intermediate states - i.e. not by sinle luons. (conservation of colour chare) Interact by exchane of PIONS One possible diaram shown below : d u d u p p u u π 0 u d Nuclear potential is YUKAWA potential with u d p p V (~r) = 2 4ßr e m ßr Short rane force : rane ο (m ß ) 1 Rane R = (0:140 GeV) 1 = 7 GeV 1 = 7~c=(1: ) m = 1:4 fm