Lecture 12 Quantum chromodynamics (QCD) WS2010/11: Introduction to Nuclear and Particle Physics

Transcription

1 Lctur Quntum chromodynmics (QCD) WS/: Introduction to Nuclr nd Prticl Physics

2 QCD Quntum chromodynmics (QCD) is thory of th strong intrction - bsd on color forc, fundmntl forc dscribing th intrctions of th qurks nd gluons mking up hdrons (such s th proton, nutron or pion). QCD consquncs: Confinmnt, which mns tht th forc btwn qurks dos not dcrs s thy r sprtd. Bcus of this, it would tk n infinit mount of nrgy to sprt two qurks; thy r forvr bound into hdrons such s th proton nd th nutron. Although nlyticlly unprovn, confinmnt is widly blivd to b tru bcus it xplins th consistnt filur for fr qurk srchs, nd it is sy to dmonstrt in lttic QCD. Asymptotic frdom mns tht in vry high-nrgy rctions, qurks nd gluons intrct only vry wkly. This prdiction of QCD ws first discovrd in th rly 97 s by Dvid Politzr, Frnk Wilczk nd Dvid Gross. For this work thy wr wrdd th 4 Nobl Priz in Physics.

3 QED, QCD intrctions I. Quntum lctrodynmics (QED) lctron-lctron scttring - γ nnihiltion γ + Forc-crrirs: γ * - virtul photons, lctriclly nutrl nd msslss II. Quntum chromodynmics (QCD) qurk-qurk scttring qurk-ntiqurk nnihiltion q g q qbr g qbr q q q q g - gluons vctor gug bosons lctriclly nutrl nd msslss but color stts + gluon slfintrction 8 color combintions (color-nticolor,.g. rg ) + color singlt (color nutrl): r r+ bb+ gg 3

4 QCD Lgrngin Th dynmics of qurks is dscribd by Quntum CromoDynmics (QCD) Th dynmics of lctrons is dscribd by Quntum ElctroDynmics (QED) QCD Lgrngin: L QCD [ igt A ] ( ) ˆ ν iγ M G G 4 ν () Gluonic fild strngth tnsor: G bc b ν Aν ν A + gf A A c ν only in QCD (not in QED!) QCD QED Gnrl structur of Lgrngin (): L konjugt frmion fild q kintic nrgy coupling const g gluon photon fild Mss q frmion fild + nrgy dnsity for gluons photons 4

5 QCD Lgrngin Th structur of (): (x) - qurk fild (rstrictd to 3 flvors t low nrgy) flvor spc Dirc spc color q u, d, s,,,3 c spc r, b, g () ( x ) q c () u d s 3 r b g flvor spc Dirc spc color spc 5

6 QCD Lgrngin Α (x) - gluon fild color indx of SU(3) color group :,,3,,8 t r 8 gnrtors of th SU(3) color group: λ - 3x3 Gll-Mnn mtrics with Tr λ t λ Proprtis of gluons: vctor gug bosons lctriclly nutrl msslss color stts spin Thr r 8 indpndnt color stts: + color singlt stt: Not: thr r mny othr possibl choics, but ll r mthmticlly quivlnt, t lst qully complx, nd giv th sm physicl rsults! 6

7 SU(3) group Th Gll-Mnn mtrics (ot t mtrics) r on possibl rprsnttion of th infinitsiml gnrtors of th spcil unitry group clld SU(3). This group (which follows th Li lgbr) hs dimnsion ight nd thrfor it hs som st with ight linrly indpndnt gnrtors. Proprtis of t : t oby th commuttion rltions [ t, t ] if t whr b bc f structur constnts of th SU(3) group: c t λ (3) t hrmitin: t t + t trclss: Tr t Tr ( t t b ) δ b (4) 7

8 Stts of th SU(3) color group Rltions btwn λ nd qqbr color mtrics λ rr br gr rb bb gb rg bg gg (5) ( r b + br )/ ( gr rg) / i ( br rb )/ i ( b g + gb )/ ( rr bb )/ ( gb bg) / i ( g r + rg) / ( rr + bb gg )/ 6 8

9 Stts of th SU(3) color group Color vctors: r, b, g (6) ignvctors of SU(3) color gnrtors λ 3 nd λ 8 E.g.: λ 3 r r ignvlu (7) λ 8 r r (8) 9

10 Irrducibl stts of th SU(3) color group Color trnsformtions: b λ 8 3 ( λ ± iλ ) r + : : br rb λ 3 + : : bg gb ( λ ± iλ ) ( λ ± iλ ) 6 7 g : + : gr rg Anlogy: ( τ ± iτ ) τ 3 τ,τ,τ 3 gnrtors of th SU() group p n p, n

11 QED Lgrngin In gnrl, th structur of th Lgrngin follows from symmtry principls. QED Locl U() invrinc: locl gug trnsformtion iα ( x ) (9) α(x) rbitrry rl function of coordints nd tim Strt with Dirc qution L i γ m () Us U() trnsformtion: iα ( x ) iα ( x ) () iα ( x ) iα ( x ) + i α () This trm violts th invrinc of L undr U() locl trnsformtions

12 Modify th drivtiv such tht QED Lgrngin iα ( x ) D D To this im introduc th covrint drivtiv: D ia (3) (4) A gug vctor fild undr locl gug trnsformtions thn must follow A A + α Substitut (4),(5) into (): (5) L i γ D m ( ) iγ m + γ A (6)

13 QED Lgrngin To idntify A with photons, on hs to dd th kintic nrgy of photons to th Lgrngin (6): 4 F ν F ν ν (7) whr F ν Aν ν A - fild strngth tnsor (8) Eq. (8) should b invrint with rspct to th trnsformtion (5). thus, L QED i γ ( ) ν iγ m + γ A F F D m 4 F ν F ν 4 ν (9) QED: intrction of photons with lctrons 3

14 QCD Lgrngin. QCD Symmtry: locl SU(3) color invrinc locl gug trnsformtions with rspct to color SU(3) group iα ( x ) t () α (x) rbitrry rl functions of coordints nd tim Anlogy to QED: strt with Dirc qution (): L i γ m Introduc th covrint drivtiv: D igt A A (x) gluon vctor filds A bc b c A α f A g α Substitut (),() into (): L i γ D m nw trm (dosn t xist in QED) () () (3) 4

15 QCD Lgrngin Add th kintic nrgy of gluons to th QCD Lgrngin (3): ν G ν G 4 Gluonic fild strngth tnsor: G bc b ν Aν ν A + gf A A c ν (4) (5) Finlly obtin (): L QCD [ igt A ] gluon slf-intrction trm ( ) ˆ ν iγ M G G 4 ν QED: Photons cnnot intrct γγ sinc photons do not hv chrg! QED is blin thory QCD: gluons cn intrct gg sinc gluons hv color chrg! QCD is nonblin thory! Not: Gluons (photons) r msslss sinc mss trm in th Lgrngin would violt th SU(3) color invrinc! 5

16 . QED α 4 π 37 QED coupling constnt Th fin-structur constnt is fundmntl physicl constnt, chrctrizing th strngth of th lctromgntic intrction. α (r) Chrg scrning with dcrsing r du to th vcuum polriztion, i.. virtul dipols scrn th chrg - 6

17 QCD coupling constnt. QCD α ( 4π g Q S ( Q ) ) α S (r) prturbtiv QCD strong QCD Color chrg ntiscrning with incrsing r: α S (Q ) incrss for lrg distncs du to th gluon slf-intrction NO prturbtiv thory t smll Q (Q /r ) or lrg distncs! 7