Perturbative QCD Lecture 1

Perturbtive QCD Lecture 1 Aude Gehrmnn-De Ridder AcdemicTrining Lectures, CERN, My 2013

The Stndrd model of prticle physics Present knowledge on nture of mtter t distnces of 10-15 -10-18 m Elementry mtter prticles (spin ½) Chrged leptons: e -, μ -, τ - Neutrinos: ν e, ν μ, ν τ Qurks: u,d,c,s,t,b Force crriers (spin 1) Photon γ: electromgnetism Gluon g: strong nucler force Z 0, W ± : wek nucler force Higgs boson H (spin 0) Reltive strength of the forces t 10-15 m ( = proton rdius): Strong : Electromgnetic : Wek 1 : 1 / 100 : 1/10000 Note: Grvity left out here 1

Quntum Chromodynmics (QCD) in brief QCD: Quntum field theory of strong interctions (C.N. Yng, R. Mills; H. Fritzsch, M. Gell-Mnn, H. Leutwyler) interction crried by gluons cting on urks nd gluons QCD-chrge: colour : of three types (`colours`: red, blue, green) QCD coupling strength α s depends on energy low energy ( = long distnce or time) α s is lrge (confinement): non-perturbtive regime of QCD high energy ( = short distnce or time) α s is smll (symptotic freedom): perturbtive regime of QCD This lecture untify these sttements obtin detiled understnding of QCD pply to interprettion of collider dt 0.5! s (Q) 0.4 0.3 0.2 0.1 QCD Deep Inelstic Scttering e + e Annihiltion Hevy Qurkoni!!!""!!#!$!%&''()!*!%&%%%+ s Z 1 10 100 Q [GeV] July 2009 Prticle Dt Group 2

Expecttions t LHC LHC brings new frontiers in energy nd luminosity Production of short-lived hevy sttes (Higgs, top, SUSY...) detected through their decy products yield multi-prticle finl sttes involving: jets, leptons, γ, E/ T Jets re the signtures of urks nd gluons: built with clusters of prticles moving in common direction will be defined untittively in the second lecture g χ 0 ~ ~ g ~ g ~ g ~ Signl χ 0 g g E/ T χ 0 ~ jet jet ~ g jet ~ g ~ g jet ~ Bckground Mutli-prticle finl sttes produced in hrd scttering processes QCD effects re essentil here χ 0 E/ T g g g Z g ν ν g jet g Z jet g jet jet ν ν E/ T Exmple: SUSY signture 4j + E/ T 3

Hrd scttering processes t LHC Hrd scttering processes re rre Represent only tiny frction of ll events t hdron collider totl inelstic pp cross section: σ pp ~70 mb e.g. vector boson production: σ W = 150 nb ~ 2 10-6 σ pp Involve lrge trnsfer of momentum Q M p Q: typicl scle of the process M p : proton mss Typicl hrd scttering processes Jet production Top urk production Vector boson production Higgs boson production 4

Hrd scttering t hdron colliders Proton-proton collisions two bems of prtons (urks, gluons) initite the prton-level interction Proton-proton cross sections relted to prton-prton cross sections through fctoriztion of short nd long distnce processes Exmple: Z+jet production is computed s 5

Hrd scttering cross sections d h 1h 2!cd = Z 1 dx 1 Z 1 X dx 2 f /h1 (x 1,µ 2 F )f b/h2 (x 2,µ 2 F )dˆb!cd (Q 2,µ 2 F ) 0 0,b f /hi (x i ): prton distribution function: probbility of finding prton of type with momentum frction x i in the hdron h i process-independent but not clculble in perturbtion theory needs to be determined from dt contins ll unresolved emission below fctoriztion scle µ F σ b cd : prton-level hrd scttering cross section clculble in perturbtive QCD s series expnsion in α s contins only hrd emissions bove fctoriztion scle µ F 6

Aim of these 3 lectures Aim1: Introduce fundmentl fetures of QCD leding to the derivtion of the hrd scttering cross section formul d h 1h 2!cd = Z 1 dx 1 Z 1 X dx 2 f /h1 (x 1,µ 2 F )f b/h2 (x 2,µ 2 F )dˆb!cd (Q 2,µ 2 F ) 0 0,b Aim2: Study the phenomenology for severl high energy collider processes Jet production in e + e - nnihiltion Deep inelstic lepton-proton scttering Higgs nd guge boson production t the LHC 7

Outline of the first lecture Bsics of QCD SU(3) guge invrince Lgrngin of QCD nd Feynmn rules Properties of QCD: running of α s, symptotic freedom, confinement, perturbtive QCD Appliction of perturbtive QCD to e + e - hdrons Infrred divergences nd Kinoshit-Lee-Nuenberg theorem Scle dependence 8

Bsics of QCD QCD: Quntum Field Theory of strong interctions between urks nd gluons which both crry colour chrge three types (`colours`: red, blue, green) Theory is bsed on the symmetry group SU(3) C rottion in colour spce prt of Stndrd Model symmetry group SU(3) C SU(2) L U(1) Y Qurks re in the fundmentl representtion of SU(3) triplets of spin-1/2 spinors: ѱ = (ѱ 1, ѱ 2, ѱ 3 ) Gluons re in the djoint representtion of SU(3) octets of spin-1 vectors: A μ, =1..8 Lgrngin of QCD 9 L QCD = L clss. QCD + L guge QCD + Lghost QCD

SU(3) guge invrince L clss. QCD cn be explicitly constructed by reuiring locl SU(3) guge invrince of free fermion field Lgrngin SU(3) trnsformtion! L = (i/@ m) V(x): complex 3x3 mtrix with det(v) = 1 (specil) nd V + V = 1 (unitry) rel group prmeters: T : SU(3) genertors: hermitin nd trceless 3x3 mtrices fulfill SU(3) Lie lgebr: 0 = V (x) V (x) 2 SU(3) (x),=1...8 T =(T ) +, tr(t )=0 [T,T b ]=if bc T c V (x) =e i (x)t 10

1 g 0 bc b c 2.1. NON-ABELIAN GAUGE THEORIES AGAUGE = A + @ + f Aµ 2.1. NON-ABELIAN THEORIES µ µ µ (2.39) wherein weqcd, used tht issototlly wheres F(3) is corresponds symmetric under wo terms of which nlogous tois QED nd thesulst to in term µ Fprt Note tht Nre =fbc 3, tht ntisymmetric, triplet nd describes the rottions color $ c. The QCD Lgrngin the non-abelin nture of SU ). pce. As thehve cse we (N demnd tht L shll be invrint under We locl SU (N ) Weinnow lloftheqed, ingredients to build loclly SU (N ) invrint Lgrngin. we need kinetic term for Aµ (the nlogue of Fµ nd F µ wein QED). rnsformtion chrcterized by just hve to of use covrint derivtives insted of usul derivtives hve to mketo surethis end, Imposing invrince on guge invrint under terms like Fµ F µ. Of course, it hs to be invrint ht tht L depends! 0 (x) = V (x) (x) ei (x)t (x). (2.28) lso under globl SU (N(x) ) trnsformtions. For N = 3, the QCD-Lgrngin contining the Yng-Mills prt reds [Dclssicl,D! V (x)[d (2.40) results clssicl Lgrngin µin ] (x)yng-mills µ, D ] (x). The strtegy is the sme s in QED. We strt by constructing covrint derivtive. We 1 µ clss. define comprtor U (y, x) which is n N +covrint N (imtrix / m) trnsforming L = Fof D (2.45)strength QCD µ F ore, we cn write the commuttor two derivtives ss field 4 {z! exercise): LYM } {z LF µ } U (y, x)! V (y)u (y, x)v (x) with U (y, y) = 1 (2.29) first term: decribes the gluon field dynmics () 1 () µ () It is Yng-Mills F is guge-invrint kinetic term for A where F [D, D ] = igf T (2.41) is µ. µ µ µ 4 gluon field strength tensor uch tht (y)clss. nd U (y, x) (x) trnsform in the sme wy. clled Thethe covrint derivtive term nd LQCD is clled Yng-Mills Lgrngin. gin chrcterized nd vertices in Lclss. bc. The b derivtion c It follows listby of the propgtors of these using with Fµ = @µ A @ Aµ + gfqcd Aµ A. Aµ the functionl pproch will be sketched in section 2.3.1. Propgtors for come from 1 µ term µ µ bc yields nd+four-gluon the term (@µ A f @n A[ ):(x = lim@three "n) U (xinterctions + "n, x) (bsent (x)]. in QED) A µ )(@ µ A D (2.42) (2.30) (unlike in QED) Fµ is derivtive not"!0 guge invrint under the guge trnsformtion of " Covrint ct, itexpnd trnsforms in the djoint representtion SU (N ) (! exercise): We the comprtor s Tylor series in of terms of the Hermitin opertors ner U = 1: k D = @µ µ µ, ig µ Tb iga = k 2+i" µ, b Fµ! Fµ + Fµ = Fµ c gf bc b Fµ. µ 2 ig"n A T + bc O(" b c ) U (x + "n, x) = 1 + 11 interction Audeterms Gehrmnn-De Ridder(@µ APerturbtive Three boson rise from @ Aµ )( Aµ A1 ). The three gluon µ gf- Lecture QCD (2.43) (2.31) ordervertex to mke L invrint, we do not use Fµ directly but rther the trce, reds (! exercise) where )µ Aµ T is Lorentz vector field (the sum is over the genertors of the guge group,, which is indeed guge invrint:

The QCD Lgrngin With the covrint derivtive given by: D µ = @ µ iga µt Trnsformtion of the gluon field obtined by reuiring: D µ! Dµ 0 0 = e i (x)t D µ. results in: A 0 µ = A µ + 1 g @ µ + f bc A b µ c pple hs n belin nd pure non belin piece 12

7.4. QUANTUM CHROMODYNAMICS AND COLOR SU (3) 147 ThenNON-ABELIAN we hve 3-gluon term GAUGE gs fbc Abµ Ac THEORIES (@µ A @ Aµ ) yielding 3-gluon vertex 2.1. 2.1. NON-ABELIAN GAUGE THEORIES Feynmn7.4.rules QUANTUM CHROMODYNAMICS AND COLOR SU (3) b 7.4. QUANTUM CHROMODYNAMICS AND COLOR SUA(3) (k2 ) 147 The second prt of the Lgrngin, LF, gives rise to the well known propgtor147 for fermio (coming from (i@/ m) ). Furthermore we get guge boson-fermion interction ter µ g µ T A : F which (@ A )i yielding ij j µ 3-gluon vertex µ Arises@from b c of the Lgrngin, L The second Then we hve 3-gluon term gs fbc Aprt µ A b c, gives rise to the well known pro µ Then we hvethe 3-gluon term @ A s fbc Aµ Aclssicl Feynmn rules for Lgrngin (@µ A ) yielding 3-gluon vertex (i (coming from @/vertices: m) ).gfrom Furthermore we get guge boson-ferm Aµ b i g µ T j AA: b (k2) c Covrint derivtive term which rises Afrom (k2 ) µ A (k ) ij A (k ) NTUM CHROMODYNAMICS AND COLOR SU (3) 147 1 3 yields boson-fermion-fermion vertex µ = ig µtij 147 g (k k ) ].147 7.4. QUANTUM CHROMODYNAMICS = gs fbc [gµ AND (k1 COLOR k2 ) + g SU (k2(3) k3 )A µ+ µ µ 3 1 gs fbc Abµ Ac (@µ A Aµ ) 7.4. term QUANTUM CHROMODYNAMICS AND COLORvertex SU (3) ve 3-gluon @ yielding 3-gluon b A (k2 )term Then we hve 3-gluon gs fbc Abµ Ac (@µ A (7.30) @ Aµ ) yielding 3-gluon vertex µ b remrks µ the some form of Lclss.. A priori the ter c Then hve Finlly 3-gluon term gs fbc4-gluon AbµFinlly Ac (@ A mke @s fa ) yielding concerning 3-gluon vertex µwe we three-gluon term: yields three-gluon vertex A we hve lso term g A ( g f A A ) yielding the QCD 4-gluon bc s de µ e d b () 1 ()µ c A (k ) F F not Aisµ (k (k3 )tht one cn dd to the Lgrngin whi 4 µ 1 ) the only invrintaterm vertex 2 () c kinetic term for Aµ. We note tht Lclss. b cnaserve s of mss dime QCD contins opertors Aµ (k1 ) 2 ) (k3 ) µ (ka = ig Tij (7.30) sion 4. Since c b ) (k(k 2 )1 Aµ (k1 ) Ac (k3 ) = gs fbca [g µ k2 ) + g A(k(k + g µ (k3 k1 ) ]. 2 3 k3 )µ Z = gs fbc [gµ (k1 =g f [g (k k2 ) + g s(k2 bck3 )µ µ + g µ (k13 Aµ (k1 ) k ) +c g (7.30) (k2 k1 ) 2]. k3 )µ + g µ (k 3 k1 ) b].c S= d4 x L µ(7.30) (2.4 (k3 ) lso 4-gluon term Finlly wea hve gs fbc Aµ A ( gs fde Ad Ae ) yielding the 4-gluon µ hs mss dimension 0, L hs to hve mss dimension 4 (d4 x hs mss dimension 4 hve lso 4-gluon term gs fbc Abµ Ac ( gs fde Advertex A e ) yielding the 4-gluon = gs fbc [gµ (k1 k2 ) + g (k2 k3 )µ + g µ (k ) ]. (7.30) Other terms inµ Lclss. b3 kc1possible could be QCD clss. Finlly we hve lso 4-gluonFinlly term Agwe A A ( g f A A the 4-gluon four-gluon term: yields four-gluon vertex s de s fbc c µ some e ) yielding d mke remrks concerning the form of L. c b (k ) A (k ) 1 3 (kµ µ QCD A ) (k ) A c b 3 2 of mss dimension 4: terms like " F F. However, this term is not very usef µ A (k ) (k ) A µ b c 3 2 lso 4-gluon term vertex Finlly we hve gs fbc Aµ A ( gs fde Ad Ae ) yielding 4-gluon Aµ (kthe 1 () Ad T-invrince (k4 ) ()µ becuse nd therefore lso CPT.1 1it) violtes P- nd vertex F f F[g (k isk )not the only invrint term tht one cn dd to th 4= gµ + gof k ) + g (k k ) ]. (7.30) s bbc µ 1 2 c(k 2 3 µ µ 3 1 µ 2 2 higher thn () c (gµ g [fbe gµ ) 3+)dimension fde fbce (g gµ 4:g Possible ) + fceextr fbde (gterms g include gµ g(f )] F ). Howeve =2 ) A ig s(k Agmss cde (k µ g µ clss. µ (k3 ) Ab (k 2 ) fa cn serve s kinetic toterm Aµ. 4,We tht keep Lfor of dimension these note terms hve to bel multiplied by(7.31) couplings ofoper negti QCD contins µ mss Such re forbidden by the reuirement of renormlizbilit Finlly we hve lso 4-gluon term gs fbc Abµdimensions. Ac ( gs fde Ad terms A e ) yielding the 4-gluon dsion 4. Since Aµ (k1 ) A (k4 ) vertex If we reuire CPT-invrince nd renormlizbility, term defined bo Unlike in QED, gluons re ble to interct with themselves. This comesthen fromthe thekinetic fct tht Z d µ () clss. gµ g ) +13 fde fbce (gµ g Aude gµ g )Gehrmnn-De + fce fbde (gµ g Ridder gµ g ()] 1 F () be fcde (gµ g (kqcd Perturbtive Lecture term 1A (k F )A isµthe to4 )be included in LQCD. 1 )only- llowed µ the theory is non-belin. there is no superposition (7.31) 4As conseuence, c b 4 principle for QCD: A (k ) (k ) A 3 2 ) = L system intercting is not sum the individul Athe 4 fcdeof µ (k1 )field [fd (k (gµstrongly g gµ g ) + fdeprticles fbce (gµ g S gµ the g )d + fxceof fbde (gµ g gµ g )] = of igs2a be 2.1.3 Polristion Vectors for the Guge Fields QED, gluons re ble to interct with themselves. This comes from the fct tht fields. Thence, there is no plne wve solution to QCD problems, nd we cnnot mke(7.31) 2

Guges nd Ghosts Clssicl Lgrngin not sufficient to define the gluon propgtor {z (kinetic term not invertible) need to specify choice of guge dd guge-fixing term to the Lgrngin L guge QCD = 1 2 (@ µa µ ) 2 defines the clss of covrint guges with prmeterξ yields gluon propgtor k µ,,b i k 2 b g µ +(1 ) kµ k k 2 14

Guges nd Ghosts In non-belin guge theories (like QCD) this term must be supplemented (not discussed in this lecture) by ghost term L ghost QCD =(@ µ c)(d µ c) Ghost fields c re unphysicl (sclrs with fermionic sttistics) cncel the unphysicl polriztions of gluons in covrint guges For prcticl purposes: it is enough to know tht there re lso Feynmn rules for ghosts nd tht every Feynmn digrm with closed loop of internl gluons needs to be supplemented by corresponding digrm with gluons replced by ghosts (to obtin meningful result) 15

Feynmn rules (Summry) Ω Vertices: Propgtors: k i j i δ ij (/k + m) k 2 m 2 + i µ k - b ν i δ b g k 2 µν (1 η) k µk ν + i k 2 i µ ig s γ µ Tji ρ p r j c ν b µ g s f bc [(p ) ν g ρµ +( r) ρ g µν +(r p) µ g νρ ] k b i δ b k 2 + i, b µ p g s f bc p µ (p µ outgoing) Ω 1, Feynmn guge η fixes the guge: η = 0, Lndu guge ρ b µ σ ν c d c igsf 2 be f cde (g ρν g µσ g ρσ g µν ) igsf 2 ce f bde (g ρµ g νσ g ρσ g µν ) igsf 2 de f cbe (g ρν g µσ g ρµ g σν ) 16

Colour fctors SU(3) genertors T nd SU(3) structure constnts f bc present in Feynmn rules In cross section clcultions, pper in prticulr combintions: Tr(t A t B )=T R δ AB, T R = 1 2 A B A t A b ta bc = C F δ c, C F = N 2 C 1 2N C = 4 3 f ACD f BCD = C A δ AB, C A = N C =3 C,D A c B t A b ta cd = 1 2 δ bcδ d 1 2N C δ b δ cd (Fierz) Ech colour fctor corresponds to specific splitting _ T R : g ; C F : g; C A : g gg b c = d 1 1 2 2N 17

Perturbtive QCD Prmeters in QCD Lgrngin Qurk msses Strong coupling constnt: α s = g s2 /(4π) Perturbtive QCD: expnsion in powers of α s 1 Expnsion of observble f = f 0 + s f 1 + 2 sf 2 + 3 sf 3 +... often compute only the first one (leding order, LO) or two (next-to-leding order, NLO) terms Techniue for clcultions : using Feynmn digrms Energy dependence of α s : first ppers t NLO 18

Ultrviolet Divergencies Closed loop contributions t higher orders yield ultrviolet divergences if loop momentum k E.g.: p k =( ie) 2 µ 2" Z d d k µ i(/k + m) i (2 ) d k 2 m 2 + i" µ (p k) 2 + i" p k p Renormliztion of QCD Lgrngin: Redefine prmeters (coupling, msses, field strengths) to bsorb ultrviolet divergences Renormliztion performed t mss scle μ r Prmeters become renormliztion scle dependent: α s (μ r ), m (μ r ) 19

Running of the QCD coupling Scle dependence (running) of α s is expressed through the renormliztion group eution (RGE) dα s (µ 2 ) d ln µ 2 = β(α s(µ 2 )), β(α s )= α 2 s (b 0 + b 1 α s + b 2 α 2 s +...) b b 0 = 11C A 2n f 12π, b 1 = 17C2 A 5C An f 3C F n f 24π 2 = 153 19n f 24π 2, n f : number of light urks; C A, C F : QCD colour fctors Negtive sign of bet function: symptotic freedom coupling becomes weker t high momentum scles 20

Asymptotic freedom Keeping only the b 0 term in β(α s ), cn solve the RGE exctly: α s (µ 2 )= α s (µ 2 0 ) 1+b 0 α s (µ 2 µ2 0 )ln µ 2 0 = 1 b 0 ln µ2 Λ 2 α s (μ 2 ) cn be expressed in terms of coupling α s (μ 02 ) t reference scle μ 0 or by introducing non-perturbtive constnt Λ 200 MeV, corresponding to the divergence (Lndu pole) of α s (μ 2 ) α (Q2) s Λ QCD symptotic freedom 200 MeV Q2 21

Asymptotic freedom nd confinement Behviour of α s (μ 2 ) s function of energy Q (=inverse distnce 1/R) determines the properties of QCD nd the dynmics of urks nd gluons Lrge Q (smll distnce R): α s smll QCD wekly intercting urks nd gluons symptoticlly free regime of perturbtive QCD Smll Q Λ (lrge distnce R): α s lrge QCD strongly intercting urks nd _ gluons form colour-less bound sttes: bryons () nd mesons () nd do not exist s free prticles confinement regime of QCD α (R) s symptotic freedom confinement R 22

Asymptotic freedom nd confinement Effective QCD potentil V 1/r t smll distnce (like electromgnetism) V r t lrge distnce (like rubber bnd) JLQCD Collb. Seprting urk ntiurk bound stte: potentil energy becomes lrge enough to crete second pir form new pir of bound sttes urks cn not exist s free prticles 23

Renormlistion scle dependence of physicl observbles Let R be dimensionless physicl observble function of single scle Q After UV-renormlistion: If clculted to ll orders inα s, R does not depend on μ If only the first terms in the perturbtive series re clculted, i.e for R computed to orderα sn : uncompensted log(q/μ) dependence ppers t order α s n+1 Exmple: e + e - hdrons R = R(Q 2 /µ 2, s (µ 2 )) 24

e + e - Hdrons The hdronic R-rtio R(Q 2 )= (e+ e! hdrons) (e + e! µ + µ ) R-rtio: Clssicl precision observble (e.g. t LEP) Depends only on e + e - center-of-mss energy Q Cn be computed in perturbtion theory from Perturbtive series for R-rtio known to order α s 4 (Bikov, Chetyrkin, Kühn) (e + e! / g/...) 25

Renormlistion scle dependence: e + e - hdrons Cross section t NLO: (t orderα s ) σ NLO = σ (1 + c 1 α s (µ R )) Using n expnsion of the running coupling α s (µ R )=α s (Q) 2b 0 α 2 s (Q)ln µ R Q + O ( α 3 ) s ) ( O ( ) Rewrite the NLO cross section s σ NLO (µ R )=σ ( 1+c 1 α s (Q) 2c 1 b 0 α 2 s (Q)ln µ R Q + O ( α 3 s Observe: scle dependent logrithms enter only one order higher: t orderα s 2 ) ) 26

Renormlistion scle uncertinty Vrition of renormliztion scle μ r in NLO prediction introduces uncompensted NNLO terms Cn use these to estimte the uncertinty from unclculted NNLO contribution Sme behvior t higher orders scle vrition probes impct of the next unclculted order Scle dependence of observbles is reduced order-by-order Hdron collider observbles lso depend on fctoriztion scleμ f of prton distributions (see lecture 2) 1.1 1.08 1.06 1.04 1.02 1 0.98 0.96 scle-dep. of σ(e + e - hdrons) Q = M Z LO NLO NNLO conventionl rnge 0.5 < x µ < 2 0.1 1 10 µ R / Q 27

Infrred singulrities After UV renormliztion: higher order perturbtive QCD contributions still contin divergences from infrred configurtions rising in: rel emission of soft or colliner prton soft or colliner configurtions of moment in virtul loop Infrred divergences cncel order-by-order in perturbtion theory when dding rel nd virtul corrections s stted in: Kinoshit-Lee-Nuenberg (KLN) theorem Fctoriztion theorem (for hdron collider processes) 28

The KLN theorem nd its pplicbility Consider sufficiently inclusive observbles Stisfying infrred sfety: definition of observble is unchnged by soft emission or colliner splitting (See lecture 2) e.g. jet cross sections, event shpes, but not: finl stte multiplicities Hrd finl sttes from virtul nd rel corrections re experimentlly indistinguishble Theorem: Infrred divergences cncel mong different subprocesses yielding the sme hrd finl sttes: From rel emission of soft gluon or rel colliner splitting nd virtul corrections Theorem vlid t ny order in perturbtion theory Consider e + e - hdrons t NLO s n exmple 29

e + e - hdrons t leding order Leding order cross section: from urk - nti-urk production (prton-hdron dulity) e + e! 0 = 4 em e 2 3s N c Normlized to muon pir production e + e!µ + µ 0 = 4 em 3s yields hdronic R-Rtio e + e!hdrons X R = = N c e + e!µ + µ In good greement with experimentl observtion Sufficient to compute decy of virtul photon e 2 30

NLO corrections to e + e - hdrons Rel emission corrections: from e + e - g 3-prticle phse spce d 3 = 1 2 p s 1 (2 ) 5 Z d 4 p 1 d 4 p 2 d 4 p g, (p 2 1) (p 2 2) (p 2 g) (4) (p p 1 p 2 p g ) define x i = 2E i / s, phse spce fctorizes s Z s d 3 = d 2 16 2 dx 1 dx 2 e _ g 3-prticle mtrix element e + M g 2 =2C F g 2 s 1 s M 2 x 2 1 + x 2 2 (1 x 1 )(1 x 2 ) 31

NLO corrections to R-rtio Three-prton contribution to R-rtio R g 1 = R 0 2C F g 2 s 16 2 Z 1 0 dx 1 Z 1 x 2 1 + x 2 2 dx 2 1 x 1 (1 x 1 )(1 x 2 ) Is singulr for x 1 1 or x 2 1(correspond to s ig 0) Infrred singulrities Soft gluon: x g = 2 x 1 x 2 : x g 0 for x 1 1 nd x 2 1 Colliner urk-gluon: s g = 2E E g (1-cosΘ g ) = s(1-x 2 ) s g 0 for x 2 1 _ Colliner ntiurk-gluon: s g = 2E E g (1-cosΘ g ) = s(1-x 1 ) _ s g 0 for x 1 1 Divergent three-prton contributions re compensted by divergent virtul gluon corrections to two-prton contribution Need method to untify divergences 32

Dimensionl regulriztion Extend Lorentz nd Dirc lgebr to d = 4 2εspcetime dimensions Regultes both ultrviolet nd infrred divergences Preserves Lorentz nd guge invrince of the Lgrngin By dimensionl nlysis: coupling obtins mss dimension g s! g s µ Need to extend loop nd phse spce integrls to d dimensions Poles from ultrviolet nd infrred divergent contributions pper s inverse powers of ε 33

Renormlistion in dimensionl regulriztion Most QCD clcultions re performed in dimensionl regulriztion Dimensionlly regulrized divergences pper s poles in ε Renormliztion of coupling, msses nd fields not unmbiguous Different schemes: differ in finite prts tht re bsorbed in redefinitions of prmeters: Commonly used: modified miniml subtrction (MS) scheme Absorb only poles, multiplied with universl fctor 1 e E (4 ) 34

Rel corrections to R-rtio in d-dimensions Phse spce recomputed in d dimensions Z s 1 s d 3 = d 2 16 2 (1 ) 4 µ 2 Mtrix element recomputed in d dimensions M g 2 =2C F g 2 s Contribution to R-rtio dx 1 dx 2 [(1 x g )(1 x 1 )(1 x 2 )] 1 s M 2 (1 )(x2 1 + x 2 2)+2 (1 x g ) (1 x 1 )(1 x 2 ) R g 1 = R 2C F g 2 Z s s 1 0 16 2 (1 ) 4 µ 2 0 (1 )(x 2 1 + x 2 2)+2 (1 x g ) = R 0 (1 x 1 )(1 x 2 ) s 4 µ 2 C F s 2 (1 ) pple 2 2 + 3 + 19 2 Double pole: ssocited with soft gluon rdition Single pole: with colliner gluon 2 Z 1 dx 1 dx 2 1 x 1 2 [(1 x g )(1 x 1 )(1 x 2 )] 2 35

NLO corrections to R-rtio One-loop correction to urk-ntiurk finl stte yields R 1 = R 0 C F s 2 (1 ) s 4 µ 2 pple 2 3 2 Adding rel nd virtul corrections: R = R 0 + R 1 + R g 1 = R 0 1+ s(µ) Cncelltion of infrred singulrities explicit (KLN) Obtin finite NLO corrections to R-rtio 8 2 36

End of Lecture 1 37