7. QCD. Particle and Nuclear Physics. Dr. Tina Potter. Dr. Tina Potter 7. QCD 1

Transcription

1 7. QCD Particle and Nuclear Physics Dr. Tina Potter Dr. Tina Potter 7. QCD 1

2 In this section... The stron vertex Colour, luons and self-interactions QCD potential, confinement Hadronisation, jets Runnin of α s Experimental tests of QCD Dr. Tina Potter 7. QCD 2

3 QCD Quantum Electrodynamics is the uantum theory of the electromanetic interaction. mediated by massless photons photon couples to electric chare strenth of interaction: ψ f Ĥ ψ i α α = e2 4π = Quantum Chromodynamics is the uantum theory of the stron interaction. mediated by massless luons luon couples to stron chare only uarks have non-zero stron chare, therefore only uarks feel the stron interaction. strenth of interaction: ψ f Ĥ ψ i α s α s = 2 s 4π 1 Dr. Tina Potter 7. QCD 3

4 The Stron Vertex Basic QCD interaction looks like a stroner version of QED: QED γ QCD Qe αs α = e2 4π = antiuarks α s = 2 s 4π 1 + antiuarks The couplin of the luon, s, is to the stron chare. Enery, momentum, anular momentum and chare always conserved. QCD vertex never chanes uark flavour QCD vertex always conserves parity Dr. Tina Potter 7. QCD 4

5 Colour QED: QCD: Chare of QED is electric chare, a conserved uantum number Chare of QCD is called colour colour is a conserved uantum number with 3 values labelled red, reen and blue. Quarks carry colour r b Antiuarks carry anti- colour r b ḡ Colorless particles either have no color at all e.. leptons, γ, W, Z and do not interact via the stron interaction or eual parts r, b, e.. meson with 1 3 (r r + b b + ḡ), baryon with rb luons do not have eual parts r, b,, so carry color (e.. r r, see later) Dr. Tina Potter 7. QCD 5

6 QCD as a aue theory Recall QED was invariant under aue symmetry ψ ψ = e iα( r,t) ψ The euivalent symmetry for QCD is invariance under ψ ψ = e i λ. Λ( r,t) ψ an SU(3) transformation (λ are eiht 3x3 matrices). Operates on the colour state of the uark field a rotation of the colour state which can be different at each point of space and time. Invariance under SU(3) transformations eiht massless aue bosons, luons (eiht in this case). Gluon couplins are well specified. Gluons also have self-couplins, i.e. they carry colour themselves... Dr. Tina Potter 7. QCD 6

7 Gluons Gluons are massless spin-1 bosons, which carry the colour uantum number (unlike γ in QED which is chare neutral). Consider a red uark scatterin off a blue uark. Colour is exchaned, but always conserved (overall and at each vertex). αs r b b r r b Expect 9 luons (3x3): αs b r b rḡ r b bḡ b r r r b b ḡ However: Real luons are orthoonal linear combinations of the above states. The combination 1 3 (r r + b b + ḡ) is colourless and does not participate in the stron interaction. 8 coloured luons Conventionally chosen to be (all orthoonal): r b rḡ r b bḡ b r 1 2 (r r b b) r 1 6 (r r + b b 2ḡ) Dr. Tina Potter 7. QCD 7

8 Same colour flow in each case: rḡ + b r r + r b Dr. Tina Potter 7. QCD 8 Gluon Self-Interactions QCD looks like a stroner version of QED. However, there is one bi difference and that is luons carry colour chare. Gluons can interact with other luons Example: Gluon-luon scatterin

9 QCD Potential QED Potential: V QED = α r QCD Potential: V QCD = C α s r At short distances, QCD potential looks similar, apart from the colour factor C. For in a colourless state in a meson, C = 4/3 For in a colourless state in baryon, C = 2/3 Note: the colour factor C 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. Dr. Tina Potter 7. QCD 9

10 Confinement Never observe sinle free uarks or luons Quarks are always confined within hadrons This is a conseuence of the stron interaction of luons. Qualitatively, compare QCD with QED: QCD Colour field QED Electric field Self interactions of the luons sueezes the lines of force into a narrow tube or strin. The strin has a tension and as the uarks separate the strin stores potential enery. Enery stored per unit lenth in field constant V (r) r Enery reuired to separate two uarks is infinite. Quarks always come in combinations with zero net colour chare confinement. Dr. Tina Potter 7. QCD 10

11 How Stron is Stron? QCD potential between uark and antiuark has two components: Short rane, Coulomb-like term: 4 α s 3 r Lon rane, linear term: +kr V QCD = 4 α s 3 r + kr with k 1 GeV/fm at lare r F = dv dr = 4 α s 3 r 2 + k F = k N = 160, 000 N Euivalent to weiht of 150 people Dr. Tina Potter 7. QCD 11

12 Jets Consider the pair produced in e + e e + Qe γ Q e e As the uarks separate, the potential enery in the colour field ( strin ) starts to increase linearly with separation. When the enery stored exceeds 2m, new pairs can be created. As enery decreases, hadrons (mainly mesons) freeze out Dr. Tina Potter 7. QCD 12

13 Jets As uarks separate, more pairs are produced. This process is called hadronization. Start out with uarks and end up with narrowly collimated jets of hadrons. e + Qe γ Q e e Typical e + e event The hadrons in a uark(antiuark) jet follow the direction of the oriinal uark(antiuark). Conseuently, e + e is observed as a pair of back-to-back jets. Dr. Tina Potter 7. QCD 13

14 Nucleon-Nucleon Interactions Bound states (e.. protons and neutrons) are colourless (colour sinlets) They can only emit and absorb another colour sinlet state, i.e. not sinle luons (conservation of colour chare). Interact by exchane of pions. Example: pp scatterin (One possible diaram) p p π 0 p p Nuclear potential is Yukawa potential with V (r) = 2 Short rane force: 4π e m πr Rane = 1 m π = (0.140 GeV) 1 = 7 GeV 1 = 7 ( c) fm = 1.4 fm Dr. Tina Potter 7. QCD 14 r

15 Runnin of α s α s specifies the strenth of the stron interaction. But, just as in QED, α s is not a constant. It runs (i.e. depends on enery). In QED, the bare electron chare is screened by a cloud of virtual electron-positron pairs. In QCD, a similar colour screenin effect occurs. In QCD, uantum fluctuations lead to a cloud of virtual pairs. One of many (an infinite set) of 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) of such diarams. No analoy in QED, photons do not carry the chare of the interaction. Dr. Tina Potter 7. QCD 15

16 Colour Anti-Screenin Due to luon self-interactions bare colour chare is screened by both virtual uarks and luons. The cloud of virtual luons carries colour chare and the effective colour chare decreases at smaller distances (hih enery)! Hence, at low eneries, α s is lare cannot use perturbation theory. But at hih eneries, α s is small. In this reime, can treat uarks as free particles and use perturbation theory Asymptotic Freedom. s = 100 GeV, αs = 0.12 Dr. Tina Potter 7. QCD 16

17 Scatterin in QCD Example: Hih enery proton-proton scatterin. αs M 1 2 αs αs αs dσ dω (α s) 2 sin 4 θ/2 Upper points: Geier and Marsden data (1911) for the elastic scatterin of a particles from old and silver foils. Lower points: anular distribution of uark jets observed in pp scatterin at 2 = 2000 GeV 2. Both follow the Rutherford formula for elastic scatterin. Dr. Tina Potter 7. QCD 17

18 Scatterin in QCD Example: pp vs π + p scatterin Calculate ratio of σ(pp) total to σ(π + p) total QCD does not distinuish between uark flavours, only colour chare of uarks matters. At hih enery (E bindin enery of uarks within hadrons), ratio of σ(pp) total and σ(π + p) total depends on number of possible uark-uark combinations. Predict: σ(πp) σ(pp) = = 2 3 Experiment: σ(πp) σ(pp) = 24 mb 38 mb 2 3 Dr. Tina Potter 7. QCD 18

19 QCD in e + e Annihilation e + e annihilation at hih eneries provides direct experimental evidence for colour and for luons. Start by comparin the cross-sections for e + e µ + µ and e + e e + µ e + Qe γ Qe Qe γ Q e e µ + e M 1 2 α α M 1 2Q α α σ(e + e µ + µ ) = 4πα2 3s If we nelect the mass of the final state uarks/muons then the only difference is the chare of the final state particles: Q µ = 1 Q = + 2 3, 1 3 Dr. Tina Potter 7. QCD 19

20 Evidence for Colour Consider the ratio R = σ(e+ e hadrons) σ(e + e µ + µ ) For a sinle uark of a iven colour R = Q 2 However, we measure σ(e + e hadrons) not just σ(e + e uū). A jet from a u-uark looks just like a jet from a d-uark etc. Thus, we need to sum over all available flavours (u, d, c, s, t, b) and colours (r,, b): R = 3 Qi 2 (3 colours) i where the sum is over all uark flavours (i) that are kinematically accessible at centre-of-mass enery, s, of the collider. Dr. Tina Potter 7. QCD 20

21 Evidence for Colour Expect to see steps in R as enery is increased. R = 3 i Q 2 i Enery Expected ratio R s > 2ms, 1 GeV 3 ( ) = 2 uds s > 2mc, 4 GeV 3 ( ) 4 9 = udsc s > 2mb, 10 GeV 3 ( ) 1 9 = udscb s > 2mt, 350 GeV 3 ( ) 4 = 5 udscbt Dr. Tina Potter 7. QCD 21

22 Evidence for Colour R = σ(e+ e hadrons) σ(e + e µ + µ ) R increases in steps with s Stron evidence for colour s < 11 GeV reion observe bound state resonances: charmonium (c c) and bottomonium (b b) s > 50 GeV reion observe low ede of Z resonance Γ 2.5 GeV. Dr. Tina Potter 7. QCD 22

23 Experimental Evidence for Colour R = σ(e+ e hadrons) σ(e + e µ + µ ) The existence of Ω (sss) The Ω (sss) is a (L = 0) spin-3/2 baryon consistin of three s-uarks. The wavefunction: 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, whose wavefunction must be antisymmetric. ψ = (s s s )ψ colour ψ colour = 1 (rb + br + br rb rb br) 6 i.e. need to introduce a new uantum number ( colour ) to distinuish the three uarks in Ω avoids violation of Pauli s Exclusion Principle. Drell-Yan process Need colour to explain cross-section; colours of the annihilatin uarks must match to form a virtual photon. Cross-section suppressed by a factor N 2 colour. π Q ū u e Qe d d Dr. Tina Potter 7. QCD 23 p d u u γ d u µ µ +

24 Evidence for Gluons In QED, electrons can radiate photons. In QCD, uarks can radiate luons. Example: e e + e + e Qe γ αs Q e M Q 2 α α αs Givin an extra factor of α s in the matrix element, i.e. an extra factor of α s in the 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. Dr. Tina Potter 7. QCD 24

25 Evidence for Gluons JADE event s = 31 GeV First direct evidence of luons (1978) ALEPH event s = 91 GeV (1990) Distribution of the anle, φ, between the hihest enery jet (assumed to be one of the uarks) relative to the fliht direction of the other two (in their cm frame). φ distribution depends on the spin of the luon. Gluon is spin 1 Dr. Tina Potter 7. QCD 25

26 Evidence for Gluon Self-Interactions Direct evidence for the existence of the luon self-interactions comes from 4-jet events: e + e + γ γ e e e + e + γ γ e The anular distribution of jets is sensitive to existence of triple luon vertex (lower left diaram) vertex consists of two spin 1/2 uarks and one spin 1 luon vertex consists of three spin-1 luons Different anular distribution. Dr. Tina Potter 7. QCD 26 e

27 Evidence for Gluon Self-Interactions ALEPH 4-jet event Experimental method: Define the two lowest enery jets as the luons. (Gluon jets are more likely to be lower enery than uark jets). Measure anle χ between the plane containin the uark jets and the plane containin the luon jets. Gluon self-interactions are reuired to describe the experimental data. Dr. Tina Potter 7. QCD 27

28 Measurements of α s α s can be measured in many ways. The cleanest is from the ratio R = σ(e+ e hadrons) σ(e + e µ + µ ) e + e + In practise, measure Qe Q e + γ Qe γ Q e +... αs i.e. don t distinuish between 2 and 3 jets e e When luon radiation is included: R = 3 ( Q α ) s π ( Therefore, 1 + α ) s 3.9 π 3.66 α s ( 2 = 25 2 ) 0.2 Dr. Tina Potter 7. QCD 28

29 Measurements of α s Many other ways to measure α s Example: 3-jet rate e + e R 3 = σ(e+ e 3 jets) σ(e + e 2 jets) α s e + e Qe γ αs Q e α s decreases with enery α s runs! in accordance with QCD Dr. Tina Potter 7. QCD 29

30 Observed runnin of α s Dr. Tina Potter 7. QCD 30

31 Summary QCD is a aue theory, similar to QED, based on SU(3) symmetry Gluons are vector aue bosons, which couple to (three types of) colour chare (r, b, ) Gluons themselves carry colour chare hence they have self-interactions (unlike QED). Leads to runnin of α s, in the opposite sense to QED. Force is weaker at hih eneries ( asymptotic freedom ) and very stron at low eneries. Quarks and luons are confined. Seen as hadrons and jets of hadrons. Tests of QCD Evidence for colour Existence of luons, test of their spin and self-interactions Measurement of α s and observation that it runs. Up next... Section 8: Quark Model of Hadrons Dr. Tina Potter 7. QCD 31