Particle Physics. From Monday: Summary. Lecture 9: Quantum Chromodynamics (QCD) q 2 m 2 Z. !q q!q q scattering

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1 Paticle Physics Lectue 9: Quantum Chomodynamics (QCD)!Colou chage and symmety!gluons!qcd Feynman Rules!! scatteing!jets! 1 Fom Monday: Summay Weak Neutal Cuent Caied by the massive Z-boson: acts on all uaks and leptons. No flavou changes obseved. ū(e)γ µ (c e V c e Aγ 5 )u(e) At low enegies, esponsible fo neutino scatteing Electon poton scatteing, At lowe enegies: elastic scatteing e " p! e " p " poton emains intact, scatteing can be descibed by poton fom facto. At highe enegies: Deep inelastic scatteing e " p! e " X " Scatteing fom individual uaks within the poton. " Each paton caies the momentum faction x, descibed by poton stuctue functions, (x) " Only ~50% poton momentum caied by up and down uaks, emainde caied by gluons, sea uaks. g µν m Z Fiday: QCD Giffiths chapte 8.

2 Each uak caies a colou chage: ed, blue o geen. The coupling stength is the same fo all thee colous colous. To descibe a uak, need Diac spino u(p) plus a colou d.o.f. descibed by a column vecto: Colou Chage Gluons esponsible fo exchanging momentum and colou between uaks. Mathematically, symmety is descibed by an SU() symmety goup. Inteactions ae invaiant unde otations in colou space: Each gluon contains colou and anticolou. Naively expect nine gluons: b g b bb bg g gb gg Howeve gluons ae descibed by the geneatos of the SU() goup, giving eight linea colou!anticolou combinations of these Eight Gluons The geneatos of SU() ae descibed by the Gell-Mann matices They also descibe the allowed colou configuations of gluons. Each gluon is descibed by: g i = ( b g )! i ( ) b g g 1 = 1 ( b + b ) g = 1 ( b b ) g = 1 ( b b) g = 1 (ḡ + g ) g 5 = 1 (ḡ g ) g = 1 (bḡ g b) g 7 = 1 (bḡ + g b) g 8 = 1 6 ( + b b gḡ)

3 Feynman Rules fo QCD α S = g S π 5 Gluon-Gluon Inteactions Gluons also cay colou chage and can theefoe self-inteact. Two allowed possibilities: no QED analogues) k-uak scatteing, theefoe can ha Gluon inteactions ae believed to give ise to colou confinement Ty to sepaate an electon-positon pai V QED () = 1 π 0 = α Ty to sepaate an uak anti-uak pai V QCD () λ A gluon flux tube of inteacting gluons is fomed. Enegy ~1 GeV/fm. Gluon-gluon inteactions ae esponsible fo holding uaks in mesons and bayons. 6

4 ! scatteing To wite down the matix element, follow the femions aow backwads. " Fo the uak line j!i: #ji tem at vetex " Fo the antiuak line k!l: #kl tem at vetex g S g M = ū j λa jiγ µ µν v g S u i δab k λb klγ ν v l M = g λ a S ji λa kl [ū j γ µ u i ][ v k γ µ v l ] The matix element looks vey simila to electomagnetic scatteing except e!gs, and the addition of the tems λ a jiλ a kl/ In the lowest ode appoximation, the dynamics of the! scatteing is the same as electomagnetic e + e "!e + e " scatteing. Descibe in tems of Coloumb-like potential V = fα S The colou facto f = 1 λa ji λa kl = 1 a λa ji λa kl is a sum ove elements in the! matices. 7 Colou Facto fo! We want to the colou facto f = 1 λa ji λa kl = 1 a λa ji λa kl Fo the calculation we choose colous fo and. As the theoy is invaiant unde otations in colou space any choice of colous will give the same answe. Thee colou options: 1. i=1 k=1 j=1 l=1 e.g.! f 1 = 1. i=1 k= j=1 l= e.g. b! b f = 1 a λa 11λ a 11 a λa 11λ a a λa 1λ a 1. i=1 k=1 j= l= e.g.! b b f = 1 Calculate option. The only matices with non-zeo element in the eded (11) and blue-blue () elements ae " and " 8. a λa 11λ a = 1 [λ 11λ + λ 8 11λ 8 ] = 1 [(1)( 1) + ( 1 )( 1 )] = 1 6 f = 1 Similaly, f 1 = 1 f = 1 a λa 11λ a 1 = 1 [λ 11λ 11 + λ 8 11λ 8 11] = 1 [(1)(1) + ( 1 )( 1 )] = 1 a λa 1λ a 1 = 1 (λ1 1λ λ 1λ 1) = 1 [( i)(i) + (1)(1)] = 1 8

5 Colou Facto fo Mesons Mesons ae colouless states in a colou singlet : + g g + b b Calculate colou facto fo! scatteing in a meson. g g b b Two possibilities fo colou combinations:!quaks stay the same colou e.g.! f1=!!quaks change colou e.g.! b b and! g g each contibutes f=" Sum ove all possible final states fo! gives f =! + " + " = / Aveage ove all possible initial states,, g g, b b : f =! (! + g g! + b b! ) =! ( / + / + /) = / The colou facto fo the inteactions within a meson is / The potential within a meson (to lowest ode) is: V = α S 9 QCD Potential At lage distances: gluon-gluon inteactions V QCD () λ At shot distances:! scatteing V = α S Oveall potential is: V QCD () = α S + λ cc bb This model povides a good desciption of the bound states of heavy uaks: chamonium ( c c ) bottomonium ( b b ) 10

6 Jets Conside a uak and anti-uak poduced in electon positon annihilation (i) (ii) Initially Quaks sepaate at high velocity Colou flux tube foms between uaks (iii) Enegy stoed in the flux tube sufficient to poduce pais (iv) Pocess continues until uaks pai up into jets of colouless hadonsobseved as jets of paticles e + e k and anti-uak poduced in electon positon annihilation aate at ms e to alled hadonisation. It is not (yet) calculable. Michaelmas 011 This pocess is called hadonisation. It is not (yet) calculable. The main conseuence is that at collide expeiments uaks and gluons obseved as jets of paticles 11 LHC Event with Two Jets 1

7 LHC Event with Multi Jets jet jet uak scatteing, t (no QED analogue 1 QCD Summay QCD: Quantum Chomodymanics is the uantum desciption of the stong foce. Gluons ae the popagatos of the QCD and cay colou and anti-colou, descibed by 8 Gell-Mann matices,!. Fo M calculate the appopiate colou facto fom the! matices. The coupling constant $S is lage at small (confinement) and lage at high (asymptotic feedom). Mesons and hadons ae held togethe by QCD. In high enegy collisions, jets ae the signatues of uak and gluon poduction. 1