QCD at hadron colliders Lecture 3: Parton Distribution Functions

Transcription

1 QCD at hadron colliders Lecture : Parton Distribution Functions Gavin Salam CERN, Princeton & LPTHE/CNRS (Paris) Maria Laach Herbtschule für Hochenenergiephysik September, Germany

2 QCD lecture (p. ) PDF introduction Factorization & parton distributions Z H Cross section for some hard process in hadron-hadron collisions p σ p σ = d f q/p (,µ ) p p d f q/ p (,µ )ˆσ( p, p,µ ), ŝ = s Total X-section is factorized into a hard part ˆσ( p, p,µ ) and normalization from parton distribution functions (PDF). Measure total cross section need to know PDFs to be able to test hard part (e.g. Higgs electroweak couplings). Picture seems intuitive, but how can we determine the PDFs? does picture really stand up to QCD corrections? NB: non-perturbative

3 QCD lecture (p. ) PDF introduction Factorization & parton distributions Z H Cross section for some hard process in hadron-hadron collisions p σ p σ = d f q/p (,µ ) p p d f q/ p (,µ )ˆσ( p, p,µ ), ŝ = s Total X-section is factorized into a hard part ˆσ( p, p,µ ) and normalization from parton distribution functions (PDF). Measure total cross section need to know PDFs to be able to test hard part (e.g. Higgs electroweak couplings). Picture seems intuitive, but how can we determine the PDFs? does picture really stand up to QCD corrections? NB: non-perturbative

4 QCD lecture (p. ) PDF introduction Factorization & parton distributions Z H Cross section for some hard process in hadron-hadron collisions p σ p σ = d f q/p (,µ ) p p d f q/ p (,µ )ˆσ( p, p,µ ), ŝ = s Total X-section is factorized into a hard part ˆσ( p, p,µ ) and normalization from parton distribution functions (PDF). Measure total cross section need to know PDFs to be able to test hard part (e.g. Higgs electroweak couplings). Picture seems intuitive, but how can we determine the PDFs? does picture really stand up to QCD corrections? NB: non-perturbative

5 QCD lecture (p. ) PDF introduction Factorization & parton distributions Z H Cross section for some hard process in hadron-hadron collisions p σ p σ = d f q/p (,µ ) p p d f q/ p (,µ )ˆσ( p, p,µ ), ŝ = s Total X-section is factorized into a hard part ˆσ( p, p,µ ) and normalization from parton distribution functions (PDF). Measure total cross section need to know PDFs to be able to test hard part (e.g. Higgs electroweak couplings). Picture seems intuitive, but how can we determine the PDFs? does picture really stand up to QCD corrections? NB: non-perturbative

6 QCD lecture (p. ) PDF introduction Factorization & parton distributions Z H Cross section for some hard process in hadron-hadron collisions p σ p σ = d f q/p (,µ ) p p d f q/ p (,µ )ˆσ( p, p,µ ), ŝ = s Total X-section is factorized into a hard part ˆσ( p, p,µ ) and normalization from parton distribution functions (PDF). Measure total cross section need to know PDFs to be able to test hard part (e.g. Higgs electroweak couplings). Picture seems intuitive, but how can we determine the PDFs? does picture really stand up to QCD corrections? NB: non-perturbative

7 QCD lecture (p. ) PDF introduction DIS kinematics Deep Inelastic Scattering: kinematics Hadron-hadron is comple because of two incoming partons so start with simpler Deep Inelastic Scattering (DIS). Kinematic relations: e + k "( y)k" q (Q = q ) p p proton = Q p.q ; y = p.q p.k ; Q = ys s = c.o.m. energy Q = photon virtuality transverse resolution at which it probes proton structure = longitudinal momentum fraction of struck parton in proton y = momentum fraction lost by electron (in proton rest frame)

8 QCD lecture (p. ) PDF introduction DIS kinematics Deep Inelastic Scattering: kinematics Hadron-hadron is comple because of two incoming partons so start with simpler Deep Inelastic Scattering (DIS). Kinematic relations: e + k "( y)k" q (Q = q ) p p proton = Q p.q ; y = p.q p.k ; Q = ys s = c.o.m. energy Q = photon virtuality transverse resolution at which it probes proton structure = longitudinal momentum fraction of struck parton in proton y = momentum fraction lost by electron (in proton rest frame)

9 QCD lecture (p. 4) PDF introduction DIS kinematics Deep Inelastic scattering (DIS): eample Q = 5 GeV ; y = :56; =.5 e + e + proton e + Q jet jet proton

10 QCD lecture (p. 5) PDF introduction DIS X-sections E.g.: etracting u & d distributions Write DIS X-section to zeroth order in α s ( quark parton model ): d σ em ( ) ddq 4πα +( y) Q 4 F em +O(α s ) F em [structure function] F = (e uu()+e d d()) = ( 4 9 u()+ 9 d() ) NB: [u(), d(): parton distribution functions (PDF)] use perturbative language for interactions of up and down quarks but distributions themselves have a non-perturbative origin.

11 QCD lecture (p. 5) PDF introduction DIS X-sections E.g.: etracting u & d distributions Write DIS X-section to zeroth order in α s ( quark parton model ): d σ em ( ) ddq 4πα +( y) Q 4 F em +O(α s ) F em [structure function] F = (e uu()+e d d()) = ( 4 9 u()+ 9 d() ) NB: [u(), d(): parton distribution functions (PDF)] use perturbative language for interactions of up and down quarks but distributions themselves have a non-perturbative origin. F gives us combination of u and d. How can we etract them separately?

12 QCD lecture (p. 6) Quark distributions u & d Etracting full flavour structure? Using neutrons and isospin F n = 4 9 u n()+ 9 d n() Using charged-current (W ± ) scattering [neutrinos instead of electrons in initial or final-state] ν interacts only with d,ū angular structure of interaction differs between d and ū

13 QCD lecture (p. 6) Quark distributions u & d Etracting full flavour structure? Using neutrons and isospin F n = 4 9 u n()+ 9 d n() 4 9 d p()+ 9 u p() Using charged-current (W ± ) scattering [neutrinos instead of electrons in initial or final-state] ν interacts only with d,ū angular structure of interaction differs between d and ū

14 QCD lecture (p. 6) Quark distributions u & d Etracting full flavour structure? Using neutrons and isospin F n = 4 9 u n()+ 9 d n() 4 9 d p()+ 9 u p() Using charged-current (W ± ) scattering [neutrinos instead of electrons in initial or final-state] ν interacts only with d,ū angular structure of interaction differs between d and ū

15 QCD lecture (p. 7) Quark distributions u & d All quarks d V d S quarks: q() Q = GeV u V s S c S u S CTEQ6D fit These&othermethods wholeset of quarks & antiquarks NB: also strange and charm quarks valence quarks (u V = u ū) are hard : q V () ( ) quark counting rules : q V ().5 Regge theory sea quarks (u S = ū,...) fairly soft (low-momentum) : q S () ( ) 7 : q S ().

16 QCD lecture (p. 7) Quark distributions u & d All quarks d V d S quarks: q() Q = GeV u V s S c S u S CTEQ6D fit These&othermethods wholeset of quarks & antiquarks NB: also strange and charm quarks valence quarks (u V = u ū) are hard : q V () ( ) quark counting rules : q V ().5 Regge theory sea quarks (u S = ū,...) fairly soft (low-momentum) : q S () ( ) 7 : q S ().

17 QCD lecture (p. 7) Quark distributions u & d All quarks d V d S quarks: q() Q = GeV u V s S c S u S CTEQ6D fit These&othermethods wholeset of quarks & antiquarks NB: also strange and charm quarks valence quarks (u V = u ū) are hard : q V () ( ) quark counting rules : q V ().5 Regge theory sea quarks (u S = ū,...) fairly soft (low-momentum) : q S () ( ) 7 : q S ().

18 QCD lecture (p. 8) Quark distributions u & d Momentum sum rule Check momentum sum-rule (sum over all species carries all momentum): d q i () = i q i momentum d V. u V.67 d S.66 u S.5 s S. c S.6 total.546 Where is missing momentum? Only parton type we ve neglected so far is the gluon Not directly probed by photon or W ±. NB: need to know it for gg H To discuss gluons we must go beyond naive leading order picture, and bring in QCD splitting...

19 QCD lecture (p. 8) Quark distributions u & d Momentum sum rule Check momentum sum-rule (sum over all species carries all momentum): d q i () = i q i momentum d V. u V.67 d S.66 u S.5 s S. c S.6 total.546 Where is missing momentum? Only parton type we ve neglected so far is the gluon Not directly probed by photon or W ±. NB: need to know it for gg H To discuss gluons we must go beyond naive leading order picture, and bring in QCD splitting...

20 QCD lecture (p. 8) Quark distributions u & d Momentum sum rule Check momentum sum-rule (sum over all species carries all momentum): d q i () = i q i momentum d V. u V.67 d S.66 u S.5 s S. c S.6 total.546 Where is missing momentum? Only parton type we ve neglected so far is the gluon Not directly probed by photon or W ±. NB: need to know it for gg H To discuss gluons we must go beyond naive leading order picture, and bring in QCD splitting...

21 QCD lecture (p. 8) Quark distributions u & d Momentum sum rule Check momentum sum-rule (sum over all species carries all momentum): d q i () = i q i momentum d V. u V.67 d S.66 u S.5 s S. c S.6 total.546 Where is missing momentum? Only parton type we ve neglected so far is the gluon Not directly probed by photon or W ±. NB: need to know it for gg H To discuss gluons we must go beyond naive leading order picture, and bring in QCD splitting...

22 QCD lecture (p. 8) Quark distributions u & d Momentum sum rule Check momentum sum-rule (sum over all species carries all momentum): d q i () = i q i momentum d V. u V.67 d S.66 u S.5 s S. c S.6 total.546 Where is missing momentum? Only parton type we ve neglected so far is the gluon Not directly probed by photon or W ±. NB: need to know it for gg H To discuss gluons we must go beyond naive leading order picture, and bring in QCD splitting...

23 QCD lecture (p. 9) Initial-state splitting st order analysis Recall final-state splitting Tuesday s lecture: calculated q qg (θ, E p) for final state of arbitrary hard process (σ h ): σ h+g σ h α s C F π de E zp p θ σ h θ ω = ( z)p dθ Rewrite with different kinematic variables σ h+g σ h α s C F π dz dkt z kt E = ( z)p k t = E sinθ Eθ If we avoid distinguishing q + g final state from q (infrared-collinear safety), then divergent real and virtual corrections cancel σ h+v σ h α s C F π dz dkt z kt σ h p p

24 QCD lecture (p. ) Initial-state splitting st order analysis Initial-state splitting For initial state splitting, hard process occurs after splitting, and momentum entering hard process is modified: p zp. σ g+h (p) σ h (zp) α sc F π dz dkt z kt p zp σ h ( z)p For virtual terms, momentum entering hard process is unchanged σ V+h (p) σ h (p) α sc F π dz dkt z kt p p σ h Total cross section gets contribution with two different hard X-sections σ g+h +σ V+h α sc F dk t dz π kt z [σ h(zp) σ h (p)] NB: We assume σ h involves momentum transfers Q k t, so ignore etra transverse momentum in σ h

25 QCD lecture (p. ) Initial-state splitting st order analysis Initial-state collinear divergence σ g+h +σ V+h α sc F π Q dkt kt }{{ } infinite dz z [σ h(zp) σ h (p)] }{{} finite In soft limit (z ), σ h (zp) σ h (p) : soft divergence cancels. For z, σ h (zp) σ h (p), so z integral is non-zero but finite. BUT: k t integral is just a factor, and is infinite This is a collinear (k t ) divergence. Cross section with incoming parton is not collinear safe! This always happens with coloured initial-state particles So how do we do QCD calculations in such cases?

26 QCD lecture (p. ) Initial-state splitting st order analysis Initial-state collinear divergence σ g+h +σ V+h α sc F π Q dkt kt }{{ } infinite dz z [σ h(zp) σ h (p)] }{{} finite In soft limit (z ), σ h (zp) σ h (p) : soft divergence cancels. For z, σ h (zp) σ h (p), so z integral is non-zero but finite. BUT: k t integral is just a factor, and is infinite This is a collinear (k t ) divergence. Cross section with incoming parton is not collinear safe! This always happens with coloured initial-state particles So how do we do QCD calculations in such cases?

27 QCD lecture (p. ) Initial-state splitting st order analysis Initial-state collinear divergence σ g+h +σ V+h α sc F π Q dkt kt }{{ } infinite dz z [σ h(zp) σ h (p)] }{{} finite In soft limit (z ), σ h (zp) σ h (p) : soft divergence cancels. For z, σ h (zp) σ h (p), so z integral is non-zero but finite. BUT: k t integral is just a factor, and is infinite This is a collinear (k t ) divergence. Cross section with incoming parton is not collinear safe! This always happens with coloured initial-state particles So how do we do QCD calculations in such cases?

28 QCD lecture (p. ) Initial-state splitting st order analysis Initial-state collinear divergence σ g+h +σ V+h α sc F π Q dkt kt }{{ } infinite dz z [σ h(zp) σ h (p)] }{{} finite In soft limit (z ), σ h (zp) σ h (p) : soft divergence cancels. For z, σ h (zp) σ h (p), so z integral is non-zero but finite. BUT: k t integral is just a factor, and is infinite This is a collinear (k t ) divergence. Cross section with incoming parton is not collinear safe! This always happens with coloured initial-state particles So how do we do QCD calculations in such cases?

29 QCD lecture (p. ) Initial-state splitting st order analysis Collinear cutoff By what right did we go to k t =? GeV p p Q σ h zp ( z)p We assumed pert. QCD to be valid for all scales, but below GeV it becomes non-perturbative. Cut out this divergent region, & instead put non-perturbative quark distribution in proton. σ = d σ h (p) q(, GeV ) σ α sc F π Q GeV dk t k t }{{} finite (large) d dz z [σ h(zp) σ h (p)] q(, GeV ) }{{} finite In general: replace GeV cutoff with arbitrary factorization scale µ.

30 QCD lecture (p. ) Initial-state splitting st order analysis Collinear cutoff By what right did we go to k t =? p µ p Q σ h zp ( z)p We assumed pert. QCD to be valid for all scales, but below GeV it becomes non-perturbative. Cut out this divergent region, & instead put non-perturbative quark distribution in proton. σ = d σ h (p) q(,µ ) σ α sc F π Q µ dk t k t }{{} finite (large) d dz z [σ h(zp) σ h (p)] q(,µ ) }{{} finite In general: replace GeV cutoff with arbitrary factorization scale µ.

31 QCD lecture (p. ) Initial-state splitting st order analysis Summary so far Collinear divergence for incoming partons not cancelled by virtuals. Real and virtual have different longitudinal momenta Situation analogous to renormalization: need to regularize (but in IR instead of UV). Technically, often done with dimensional regularization Physical sense of regularization is to separate (factorize) proton non-perturbative dynamics from perturbative hard cross section. Choice of factorization scale, µ, is arbitrary between GeV and Q In analogy with running coupling, we can vary factorization scale and get a renormalization group equation for parton distribution functions. Dokshizer Gribov Lipatov Altarelli Parisi equations (DGLAP)

32 QCD lecture (p. ) Initial-state splitting st order analysis Summary so far Collinear divergence for incoming partons not cancelled by virtuals. Real and virtual have different longitudinal momenta Situation analogous to renormalization: need to regularize (but in IR instead of UV). Technically, often done with dimensional regularization Physical sense of regularization is to separate (factorize) proton non-perturbative dynamics from perturbative hard cross section. Choice of factorization scale, µ, is arbitrary between GeV and Q In analogy with running coupling, we can vary factorization scale and get a renormalization group equation for parton distribution functions. Dokshizer Gribov Lipatov Altarelli Parisi equations (DGLAP) d u u increase Q d u g u g increase Q d g u g g u u u

33 QCD lecture (p. 4) Initial-state splitting DGLAP DGLAP equation (q q) Change convention: (a) now fi outgoing longitudinal momentum ; (b) take derivative wrt factorization scale µ (+δ)µ µ + µ (+δ)µ p /z ( z)/z p dq(,µ ) d lnµ = α s π dz p qq (z) q(/z,µ ) z α s π dz p qq (z)q(,µ ) p qq is real q q splitting kernel: p qq (z) = C F +z z Until now we approimated it in soft (z ) limit, p qq C F z

34 QCD lecture (p. 5) Initial-state splitting DGLAP DGLAP rewritten Awkward to write real and virtual parts separately. Use more compact notation: dq(,µ ) d lnµ = α s π dzp qq (z) q(/z,µ ) z } {{ } P qq q This involves the plus prescription: dz [g(z)] + f(z) = dz g(z)f(z) ( +z ), P qq = C F z dz g(z)f() z = divergences of g(z) cancelled if f(z) sufficiently smooth at z = +

35 QCD lecture (p. 6) Initial-state splitting DGLAP DGLAP flavour structure Proton contains both quarks and gluons so DGLAP is a matri in flavour space: ( ) ( ) ( ) d q Pq q P d lnq = q g q g P g q P g g g [In general, matri spanning all flavors, anti-flavors, P qq = (LO), P qg = P qg ] Splitting functions are: [ P qg (z) = T R z +( z) ] [ ] +( z), P gq (z) = C F, z [ z P gg (z) = C A + z ] +z( z) +δ( z) (C A 4n f T R ). ( z) + z 6 Have various symmetries / significant properties, e.g. P qg, P gg : symmetric z z P qq, P gg : diverge for z (ecept virtuals) soft gluon emission P gg, P gq : diverge for z Implies PDFs grow for

36 QCD lecture (p. 7) Initial-state splitting Eample evolution Effect of DGLAP (initial quarks) q(,q ), g(,q ) g(,q ) q + qbar Q =. GeV Take eample evolution starting with just quarks: lnq q = P q q q lnq g = P g q q quark is depleted at large gluon grows at small..

37 QCD lecture (p. 7) Initial-state splitting Eample evolution Effect of DGLAP (initial quarks) q(,q ), g(,q ) g(,q ) q + qbar Q = 5. GeV Take eample evolution starting with just quarks: lnq q = P q q q lnq g = P g q q quark is depleted at large gluon grows at small..

38 QCD lecture (p. 7) Initial-state splitting Eample evolution Effect of DGLAP (initial quarks) q(,q ), g(,q ) g(,q ) q + qbar Q = 7. GeV Take eample evolution starting with just quarks: lnq q = P q q q lnq g = P g q q quark is depleted at large gluon grows at small..

39 QCD lecture (p. 7) Initial-state splitting Eample evolution Effect of DGLAP (initial quarks) q(,q ), g(,q ) g(,q ) q + qbar Q = 5. GeV Take eample evolution starting with just quarks: lnq q = P q q q lnq g = P g q q quark is depleted at large gluon grows at small..

40 QCD lecture (p. 7) Initial-state splitting Eample evolution Effect of DGLAP (initial quarks) q(,q ), g(,q ) g(,q ) q + qbar Q = 46. GeV Take eample evolution starting with just quarks: lnq q = P q q q lnq g = P g q q quark is depleted at large gluon grows at small..

41 QCD lecture (p. 7) Initial-state splitting Eample evolution Effect of DGLAP (initial quarks) q(,q ), g(,q ) g(,q ) q + qbar Q = 6. GeV Take eample evolution starting with just quarks: lnq q = P q q q lnq g = P g q q quark is depleted at large gluon grows at small..

42 QCD lecture (p. 7) Initial-state splitting Eample evolution Effect of DGLAP (initial quarks) q(,q ), g(,q ) g(,q ) q + qbar Q = 9. GeV Take eample evolution starting with just quarks: lnq q = P q q q lnq g = P g q q quark is depleted at large gluon grows at small..

43 QCD lecture (p. 7) Initial-state splitting Eample evolution Effect of DGLAP (initial quarks) q(,q ), g(,q ) g(,q ) q + qbar Q = 5. GeV Take eample evolution starting with just quarks: lnq q = P q q q lnq g = P g q q quark is depleted at large gluon grows at small..

44 QCD lecture (p. 8) Initial-state splitting Eample evolution Effect of DGLAP (initial gluons) 5 4 q(,q ), g(,q ) g(,q ) q + qbar Q =. GeV nd eample: start with just gluons. lnq q = P q g g lnq g = P g g g gluon is depleted at large. high- gluon feeds growth of small gluon & quark...

45 QCD lecture (p. 8) Initial-state splitting Eample evolution Effect of DGLAP (initial gluons) 5 4 q(,q ), g(,q ) g(,q ) q + qbar Q = 5. GeV nd eample: start with just gluons. lnq q = P q g g lnq g = P g g g gluon is depleted at large. high- gluon feeds growth of small gluon & quark...

46 QCD lecture (p. 8) Initial-state splitting Eample evolution Effect of DGLAP (initial gluons) 5 4 q(,q ), g(,q ) g(,q ) q + qbar Q = 7. GeV nd eample: start with just gluons. lnq q = P q g g lnq g = P g g g gluon is depleted at large. high- gluon feeds growth of small gluon & quark...

47 QCD lecture (p. 8) Initial-state splitting Eample evolution Effect of DGLAP (initial gluons) 5 4 q(,q ), g(,q ) g(,q ) q + qbar Q = 5. GeV nd eample: start with just gluons. lnq q = P q g g lnq g = P g g g gluon is depleted at large. high- gluon feeds growth of small gluon & quark...

48 QCD lecture (p. 8) Initial-state splitting Eample evolution Effect of DGLAP (initial gluons) 5 4 q(,q ), g(,q ) g(,q ) q + qbar Q = 46. GeV nd eample: start with just gluons. lnq q = P q g g lnq g = P g g g gluon is depleted at large. high- gluon feeds growth of small gluon & quark...

49 QCD lecture (p. 8) Initial-state splitting Eample evolution Effect of DGLAP (initial gluons) 5 4 q(,q ), g(,q ) g(,q ) q + qbar Q = 6. GeV nd eample: start with just gluons. lnq q = P q g g lnq g = P g g g gluon is depleted at large. high- gluon feeds growth of small gluon & quark...

50 QCD lecture (p. 8) Initial-state splitting Eample evolution Effect of DGLAP (initial gluons) 5 4 q(,q ), g(,q ) g(,q ) q + qbar Q = 9. GeV nd eample: start with just gluons. lnq q = P q g g lnq g = P g g g gluon is depleted at large. high- gluon feeds growth of small gluon & quark...

51 QCD lecture (p. 8) Initial-state splitting Eample evolution Effect of DGLAP (initial gluons) 5 4 q(,q ), g(,q ) g(,q ) q + qbar Q = 5. GeV nd eample: start with just gluons. lnq q = P q g g lnq g = P g g g gluon is depleted at large. high- gluon feeds growth of small gluon & quark...

52 QCD lecture (p. 9) Initial-state splitting Eample evolution DGLAP evolution As Q increases, partons lose longitudinal momentum; distributions all shift to lower. gluons can be seen because they help drive the quark evolution. Now consider data

53 QCD lecture (p. ) Determining gluon Evolution versus data DGLAP with initial gluon = p F (,Q ) DGLAP: g(,q ) = ZEUS NMC Q =. GeV Fit quark distributions to F (,Q ), at initial scale Q = GeV. NB: Q often chosen lower Assume there is no gluon at Q : g(,q ) = Use DGLAP equations to evolve to higher Q ; compare with data.... Complete failure!

54 QCD lecture (p. ) Determining gluon Evolution versus data DGLAP with initial gluon = p F (,Q ) DGLAP: g(,q ) = ZEUS NMC Q = 5. GeV Fit quark distributions to F (,Q ), at initial scale Q = GeV. NB: Q often chosen lower Assume there is no gluon at Q : g(,q ) = Use DGLAP equations to evolve to higher Q ; compare with data.... Complete failure!

55 QCD lecture (p. ) Determining gluon Evolution versus data DGLAP with initial gluon = p F (,Q ) DGLAP: g(,q ) = ZEUS NMC Q = 7. GeV Fit quark distributions to F (,Q ), at initial scale Q = GeV. NB: Q often chosen lower Assume there is no gluon at Q : g(,q ) = Use DGLAP equations to evolve to higher Q ; compare with data.... Complete failure!

56 QCD lecture (p. ) Determining gluon Evolution versus data DGLAP with initial gluon = p F (,Q ) DGLAP: g(,q ) = ZEUS NMC Q = 5. GeV Fit quark distributions to F (,Q ), at initial scale Q = GeV. NB: Q often chosen lower Assume there is no gluon at Q : g(,q ) = Use DGLAP equations to evolve to higher Q ; compare with data.... Complete failure!

57 QCD lecture (p. ) Determining gluon Evolution versus data DGLAP with initial gluon = p F (,Q ) DGLAP: g(,q ) = ZEUS NMC Q = 46. GeV Fit quark distributions to F (,Q ), at initial scale Q = GeV. NB: Q often chosen lower Assume there is no gluon at Q : g(,q ) = Use DGLAP equations to evolve to higher Q ; compare with data.... Complete failure!

58 QCD lecture (p. ) Determining gluon Evolution versus data DGLAP with initial gluon = p F (,Q ) DGLAP: g(,q ) = ZEUS NMC Q = 6. GeV Fit quark distributions to F (,Q ), at initial scale Q = GeV. NB: Q often chosen lower Assume there is no gluon at Q : g(,q ) = Use DGLAP equations to evolve to higher Q ; compare with data.... Complete failure!

59 QCD lecture (p. ) Determining gluon Evolution versus data DGLAP with initial gluon = F p (,Q ) DGLAP: g(,q ) = ZEUS Fit quark distributions to F (,Q ), at initial scale Q = GeV. Q = 9. GeV NB: Q often chosen lower Assume there is no gluon at Q : g(,q ) = Use DGLAP equations to evolve to higher Q ; compare with data.... Complete failure!

60 QCD lecture (p. ) Determining gluon Evolution versus data DGLAP with initial gluon = F p (,Q ) DGLAP: g(,q ) = ZEUS Fit quark distributions to F (,Q ), at initial scale Q = GeV. Q = 5. GeV NB: Q often chosen lower Assume there is no gluon at Q : g(,q ) = Use DGLAP equations to evolve to higher Q ; compare with data.... Complete failure!

61 QCD lecture (p. ) Determining gluon Evolution versus data DGLAP with initial gluon F p (,Q ).6 DGLAP (CTEQ6D) ZEUS NMC. Q =. GeV If gluon, splitting g q q generates etra quarks at large Q. faster rise of F Find a gluon distribution that leads to correct evolution in Q. Done for us by CTEQ, MRST,... PDF fitting collaborations. Success!

62 QCD lecture (p. ) Determining gluon Evolution versus data DGLAP with initial gluon F p (,Q ).6 DGLAP (CTEQ6D) ZEUS NMC. Q = 5. GeV If gluon, splitting g q q generates etra quarks at large Q. faster rise of F Find a gluon distribution that leads to correct evolution in Q. Done for us by CTEQ, MRST,... PDF fitting collaborations. Success!

63 QCD lecture (p. ) Determining gluon Evolution versus data DGLAP with initial gluon F p (,Q ).6 DGLAP (CTEQ6D) ZEUS NMC. Q = 7. GeV If gluon, splitting g q q generates etra quarks at large Q. faster rise of F Find a gluon distribution that leads to correct evolution in Q. Done for us by CTEQ, MRST,... PDF fitting collaborations. Success!

64 QCD lecture (p. ) Determining gluon Evolution versus data DGLAP with initial gluon F p (,Q ).6 DGLAP (CTEQ6D) ZEUS NMC. Q = 5. GeV If gluon, splitting g q q generates etra quarks at large Q. faster rise of F Find a gluon distribution that leads to correct evolution in Q. Done for us by CTEQ, MRST,... PDF fitting collaborations. Success!

65 QCD lecture (p. ) Determining gluon Evolution versus data DGLAP with initial gluon F p (,Q ).6 DGLAP (CTEQ6D) ZEUS NMC. Q = 46. GeV If gluon, splitting g q q generates etra quarks at large Q. faster rise of F Find a gluon distribution that leads to correct evolution in Q. Done for us by CTEQ, MRST,... PDF fitting collaborations. Success!

66 QCD lecture (p. ) Determining gluon Evolution versus data DGLAP with initial gluon F p (,Q ).6 DGLAP (CTEQ6D) ZEUS NMC. Q = 6. GeV If gluon, splitting g q q generates etra quarks at large Q. faster rise of F Find a gluon distribution that leads to correct evolution in Q. Done for us by CTEQ, MRST,... PDF fitting collaborations. Success!

67 QCD lecture (p. ) Determining gluon Evolution versus data DGLAP with initial gluon F p (,Q ).6 DGLAP (CTEQ6D) ZEUS. Q = 9. GeV If gluon, splitting g q q generates etra quarks at large Q. faster rise of F Find a gluon distribution that leads to correct evolution in Q. Done for us by CTEQ, MRST,... PDF fitting collaborations. Success!

68 QCD lecture (p. ) Determining gluon Evolution versus data DGLAP with initial gluon F p (,Q ).6 DGLAP (CTEQ6D) ZEUS. Q = 5. GeV If gluon, splitting g q q generates etra quarks at large Q. faster rise of F Find a gluon distribution that leads to correct evolution in Q. Done for us by CTEQ, MRST,... PDF fitting collaborations. Success!

69 QCD lecture (p. ) Determining full PDFs Global fits Gluon distribution q(), g() Q = GeV CTEQ6D fit gluon Gluon distribution is HUGE! Can we really trust it? d S, u S u V.. Consistency: momentum sum-rule is now satisfied. NB: gluon mostly at small Agrees with vast range of data

70 QCD lecture (p. ) Determining full PDFs Global fits DIS data and global fits HERA F Q / GeV 4 HERA Data: H 994- ZEUS ZEUS BPT 997 ZEUS SVX 995 H SVX 995, H QEDC 997 y = -log () em F 5 4 =6.E-5 =. =.6 =.5 =.4 =.5 =.6 =.8 =. =. =. =.5 ZEUS NLO QCD fit H PDF fit H 94- H (prel.) 99/ ZEUS 96/97 BCDMS E665 NMC Fied Target Eperiments: NMC BCDMS E665 SLAC =.8 =. =. =. =.5 =.8 - y = =. =.8 =.5 =.4 = Q (GeV )

71 QCD lecture (p. ) Determining full PDFs Global fits DIS data and global fits HERA F Q (GeV ) 5 4 H ZEUS CDF/D Inclusive jets η<.7 D Inclusive jets η< Fied Target Eperiments: CCFR, NMC, BCDMS, E665, SLAC -log () em F 5 4 =6.E-5 =. =.6 =.5 =.4 =.5 =.6 =.8 =. =. =. =.5 ZEUS NLO QCD fit H PDF fit H 94- H (prel.) 99/ ZEUS 96/97 BCDMS E665 NMC y= (HERA s= GeV) =.8 =. =. =. =.5 =.8 - =. =.8 = =.4 = Q (GeV )

72 QCD lecture (p. 4) Determining full PDFs Back to factorization Crucial check: other processes Factorization of QCD cross-sections into convolution of: hard (perturbative) process-dependent partonic subprocess non-perturbative, process-independent parton distribution functions e + q > e + + jet qg > jets e + Q q(, Q ) q (, Q ) g (, Q ) proton proton proton σ ep = σ eq q σ pp jets = σ qg jets q g +

73 QCD lecture (p. 4) Determining full PDFs Back to factorization Crucial check: other processes Factorization of QCD cross-sections into convolution of: hard (perturbative) process-dependent partonic subprocess non-perturbative, process-independent parton distribution functions e + q > e + + jet qg > jets e + Q q(, Q ) q (, Q ) g (, Q ) proton proton proton σ ep = σ eq q σ pp jets = σ qg jets q g +

74 QCD lecture (p. 5) Determining full PDFs Jet production Jet production in p p dσ / dp T [pb / (GeV/c)] DO Run II preliminary DO data, Cone R=.7 y <.5.5 < y <.. < y <.4 NLO (JETRAD) CTEQ6M ma =.5 p T = µ R R sep =., µ F s=.96tev - = 4 pb L int p T [GeV/c] Jet production in proton-antiproton collisions is good test of large gluon distribution, since there are large direct contributions from gg gg, qg qg NB: more complicated to interpret than DIS, since many channels, and, dependence. p T s jet transverse mom. Q y log y = logtan θ jet angle wrt p p beams Good agreement confirms factorization

75 QCD lecture (p. 5) Determining full PDFs Jet production Jet production in p p Jet production in proton-antiproton collisions is good test of large gluon distribution, since there are large direct contributions from gg gg, qg qg NB: more complicated to interpret than DIS, since many channels, and, dependence. p T s jet transverse mom. Q Good agreement confirms factorization y log y = logtan θ jet angle wrt p p beams

76 QCD lecture (p. 5) Determining full PDFs Jet production Jet production in p p Jet production in proton-antiproton collisions is good test of large gluon distribution, since there are large direct contributions from gg gg, qg qg NB: more complicated to interpret than DIS, since many channels, and, dependence. p T s jet transverse mom. Q Good agreement confirms factorization y log y = logtan θ jet angle wrt p p beams

77 QCD lecture (p. 5) Determining full PDFs Jet production Jet production in p p Jet production in proton-antiproton collisions is good test of large gluon distribution, since there are large direct contributions from gg gg, qg qg NB: more complicated to interpret than DIS, since many channels, and, dependence. p T s jet transverse mom. Q Good agreement confirms factorization y log y = logtan θ jet angle wrt p p beams

78 QCD lecture (p. 6) Determining full PDFs Jet production Fractional contributions Which PDF channels contribute? Inclusive jet cross sections with MSTW 8 NLO PDFs Tevatron, s =.96 TeV. < y <.7.9. gg jets. gq jets qq jets p T (GeV) Fractional contributions LHC, gg jets s = 4 TeV. < y <.8 qq jets gq jets fastnlo with µ = µ = p R F T k algorithm with D =.7 T p (GeV) A large fraction of jets are gluon-induced T

79 QCD lecture (p. 7) Determining full PDFs Jet production Uncertainties on predictions Ratio to MSTW 8 NNLO Gluon at Q = M H = ( GeV) MSTW 8 NNLO (68% C.L.) Fi α S Fi α S at +68% C.L. limit at - 68% C.L. limit y = at LHC y = at Tevatron General message Data-related errors on PDFs are such that uncertainties are just a few % for many key Tevatron and LHC observables

80 QCD lecture (p. 8) Closing remarks Conclusions on PDFs Eperiments tell us that proton really is what we epected (uud) Plus lots more: large number of sea quarks (q q), gluons (5% of momentum) Factorization is key to usefulness of PDFs Non-trivial beyond lowest order PDFs depend on factorization scale, evolve with DGLAP equation Pattern of evolution gives us info on gluon (otherwise hard to measure) PDFs really are universal! Precision of data & QCD calculations is striking. Crucial for understanding future signals of new particles, e.g. Higgs Boson production at LHC.

81 QCD lecture (p. 9) Etras EXTRAS

82 QCD lecture (p. ) Etras Sea and valence NMC proton & deuteron data.8 F p 6 5 Fd u() F p + 4 p & Fd data, Combine F 5 Fd d() deduce u(), d(): Definitely more up than down ( ) f().6.4 Q = 7 GeV NMC data How much u and d? Total U = d u() F = ( 4 9 u + 9 d) u() d().5. non-integrable divergence So why do we say proton = uud?

83 QCD lecture (p. ) Etras Sea and valence NMC proton & deuteron data.8 F p 6 5 Fd u() F p + 4 p & Fd data, Combine F 5 Fd d() deduce u(), d(): Definitely more up than down ( ) f().6.4 Q = 7 GeV NMC data CTEQ6D fit How much u and d? Total U = d u() F = ( 4 9 u + 9 d) u() d().5. non-integrable divergence So why do we say proton = uud?

84 QCD lecture (p. ) Etras Sea and valence NMC proton & deuteron data.8 F p 6 5 Fd u() F p + 4 p & Fd data, Combine F 5 Fd d() deduce u(), d(): Definitely more up than down ( ) f().6.4 Q = 7 GeV NMC data CTEQ6D fit How much u and d? Total U = d u() F = ( 4 9 u + 9 d) u() d().5. non-integrable divergence So why do we say proton = uud?

85 QCD lecture (p. ) Etras Sea and valence NMC proton & deuteron data.8 F p 6 5 Fd u() F p + 4 p & Fd data, Combine F 5 Fd d() deduce u(), d(): Definitely more up than down ( ) f().6.4 Q = 7 GeV NMC data CTEQ6D fit How much u and d? Total U = d u() F = ( 4 9 u + 9 d) u() d().5. non-integrable divergence So why do we say proton = uud?

86 QCD lecture (p. ) Etras Sea and valence NMC proton & deuteron data 8 F p,d / u() F p,d / d() Combine F p & Fd data, deduce u(), d(): Definitely more up than down ( ) f() 6 4 Q = 7 GeV NMC data CTEQ6D fit How much u and d? Total U = d u() F = ( 4 9 u + 9 d) u() d().5 non-integrable divergence So why do we say proton = uud?

87 QCD lecture (p. ) Etras Sea and valence NMC proton & deuteron data 8 F p,d / u() F p,d / d() Combine F p & Fd data, deduce u(), d(): Definitely more up than down ( ) f() 6 4 Q = 7 GeV NMC data CTEQ6D fit How much u and d? Total U = d u() F = ( 4 9 u + 9 d) u() d().5 non-integrable divergence So why do we say proton = uud?

88 QCD lecture (p. ) Etras Sea & valence Anti-quarks in proton u u d u u u d u u u u d d u d How can there be infinite number of quarks in proton? Proton wavefunction fluctuates etra uū, d d pairs (sea quarks) can appear: Antiquarks also have distributions, ū(), d() F = 4 9 (u()+ū())+ 9 (d()+ d()) NB: photon interaction square of charge +ve Previous transparency: we were actually looking at u +ū, d + d Number of etra quark-antiquark pairs can be infinite, so d (u()+ū()) = as long as they carry little momentum (mostly at low )

89 QCD lecture (p. ) Etras Sea & valence Anti-quarks in proton u u d u u u d u u u u d d u d How can there be infinite number of quarks in proton? Proton wavefunction fluctuates etra uū, d d pairs (sea quarks) can appear: Antiquarks also have distributions, ū(), d() F = 4 9 (u()+ū())+ 9 (d()+ d()) NB: photon interaction square of charge +ve Previous transparency: we were actually looking at u +ū, d + d Number of etra quark-antiquark pairs can be infinite, so d (u()+ū()) = as long as they carry little momentum (mostly at low )

90 QCD lecture (p. ) Etras Sea & valence Anti-quarks in proton u u d u u u d u u u u d d u d How can there be infinite number of quarks in proton? Proton wavefunction fluctuates etra uū, d d pairs (sea quarks) can appear: Antiquarks also have distributions, ū(), d() F = 4 9 (u()+ū())+ 9 (d()+ d()) NB: photon interaction square of charge +ve Previous transparency: we were actually looking at u +ū, d + d Number of etra quark-antiquark pairs can be infinite, so d (u()+ū()) = as long as they carry little momentum (mostly at low )

91 QCD lecture (p. ) Etras Sea & valence Valence quarks When we say proton has up quarks & down quark we mean d (u() ū()) =, d (d() d()) = u ū = u V is known as a valence distribution. How do we measure difference between u and ū? Photon interacts identically with both no good... Question: what interacts differently with particle & antiparticle? Answer: W + or W

92 QCD lecture (p. ) Etras Sea & valence Valence quarks When we say proton has up quarks & down quark we mean d (u() ū()) =, d (d() d()) = u ū = u V is known as a valence distribution. How do we measure difference between u and ū? Photon interacts identically with both no good... Question: what interacts differently with particle & antiparticle? Answer: W + or W

93 QCD lecture (p. ) Etras Sea & valence Valence quarks When we say proton has up quarks & down quark we mean d (u() ū()) =, d (d() d()) = u ū = u V is known as a valence distribution. How do we measure difference between u and ū? Photon interacts identically with both no good... Question: what interacts differently with particle & antiparticle? Answer: W + or W

94 QCD lecture (p. ) Etras PDF evolution matters Taking PDFs from HERA to LHC Suppose we produce a system of mass M at LHC from partons with momentum fractions, : M = s rapidity y = ln pseudorapidity η lntan θ = rapidity for massless objects 5 at LHC Are PDFs being used in region where measured? Only partial kinematic overlap DGLAP evolution is essential for the prediction of PDFs in the LHC domain.

95 QCD lecture (p. ) Etras PDF evolution matters Taking PDFs from HERA to LHC Suppose we produce a system of mass M at LHC from partons with momentum fractions, : M = s = DGLAP = DGLAP rapidity y = ln pseudorapidity η lntan θ = rapidity for massless objects 5 at LHC Are PDFs being used in region where measured? Only partial kinematic overlap DGLAP evolution is essential for the prediction of PDFs in the LHC domain.

96 QCD lecture (p. 4) Etras PDF evolution matters By how much do PDFs evolve? g(, Q = GeV) / g(, Q = GeV). Gluon evolution from to GeV LO evolution Input: CTEQ6 at Q = GeV Evolution: HOPPET Illustrate for the gluon distribution Here using fied Q scales But for HERA LHC relevant Q range is -dependent See factors. Remember: LHC involves product of two parton densities It s crucial to get this right! Without DGLAP evolution, you couldn t predict anything at LHC

97 QCD lecture (p. 5) Etras Higher orders It s not enough for data-related errors to be small. DGLAP evolution must also be well constrained. So evolution must be done with more than just leading-order DGLAP splitting functions

98 QCD lecture (p. 6) Etras Higher orders Higher-order calculations Earlier, we saw leading order (LO) DGLAP splitting functions, P ab = αs π P() ab : [ P qq () + () =C F + ] ( ) + δ( ), P () [ qg () =T R +( ) ], [ ] P gq () +( ) () =C F, [ P gg () () =C A + ( ) + +δ( ) (C A 4n f T R ) 6 ] +( ).

99 QCD lecture (p. 6) Etras Higher orders ( [ P () ps () = 4C F n 8 f + 6 4H + 9 H 56 9 ] [ ]) + ( + ) 5H H, ( [ ] P () qg () = 4C A n f pqg( )H, pqg()h, H 9 [ ] ) ( [ +4( ) H, H + H 4ζ 6H, + 9H + 4C F n f pqg() H, + H, + H ] [ ζ + 4 H + H, + 5 ] [ + ( ) H + H, H + 9 ] 5 4 H, ) H ( P () gq () = 4C A C F [H, + pgq() + H, + H 6 ] [ H 8 H 44 ] + 4ζ 9 [ 7H + H, H + ( + ) H, 5H + 7 ] ) ( pgq( )H, 4C F n f 9 [ pgq() H ]) ( [ ] + 4C F pgq() H H, + ( + ) [H, ] H H, + ) H + H Higher-order calculations NLO: P ab = α s π P() + α s 6π P() Curci, Furmanski & Petronzio 8 ( P () gg () = 4C A n f 9 pgg() ( 9 [ +( + ) H + 8H, 7 ) ( + )H ) ( δ( ) + 4C A 7 ] 67 ( ) 9 H ] [ ]) ( 8 + δ( ) + 4C F n f H ] [ + pgg( ) H, H, ζ 44 [ 67 H + pgg() ζ + H, + H, + H 8 + ζ + + [ ] + ( + ) 4 5H H, ) δ( ).

100 QCD lecture (p. 7) Etras Higher orders NNLO splitting functions Divergences for are understood in the sense of -distributions. The third-order pure-singlet contribution to the quark-quark splitting function (.4), corresponding to the anomalous dimension (.), is given by P ps 6C A C F n f 4 H 4 9 H H ζ H H H 6 ζ H 9ζ 9 4 H H H Hζ 6 H H H H 8 9 H H H H 6 H H H H ζ H H H ζ Hζ 9 4 H 4 H H H 7 Hζ 6 ζ 9 8 H 7 H 8 ζ 86 7 H 5 H 6 H 6H 6 H 7 6 H 7 ζ 9Hζ 5 H ζ H H H Hζ 7 H H H H H4 5H H H ζ 4H 4Hζ 9 H 9 H 5 ζ 5 97 H 4 H H Hζ ζ H H 8 4 H 4 4 H ζ H 4 H 4ζ Hζ H H 6H 6C F n f 7 H H ζ H ζ 9 6 H 9 H 5 H 6 H 7 6 H H 5 7 H H 4 ζ ζ H H Hζ 9 6 H H 6C F n f 85 H 5 4 H H 58 H ζ 7 4 H H 5H H Hζ H H H 8 9 ζ 8 9 H 6 Hζ 6 H 4H 4 H 6 ζ H 4 H 5 44 H ζ 55 6 H H H H H 7 H 74 7 H 5 H 5 H 5 4 ζ 5 4 H 8 H H Hζ H H H H 4H 7H ζ 7 ζ 7Hζ 6H ζ 4H H H H 4H H 6H4 (4.) Due to Eqs. (.) and (.) the three-loop gluon-quark and quark-gluon splitting functions read P qg 6C AC Fn f pqg 9 Hζ 4H H 5 4 H 9 4 H H Hζ H 4Hζ 7 Hζ 55 7 H 64 ζ ζ 49 4 H H H H H H 49 4 H 5 Hζ 79 6 H 7 H H 6H H 9H ζ 6H ζ H H 4H H 6H 6H 49 4 ζ pqg 7 H ζ 5 H 5 H 9 H 5 H H H H4 6H 6H 6H H ζ 9H ζ H H 6H 6H 6H 9H ζ 9H ζ H H 6H 55 4ζ 9 H 4 H H 7 8 H 9 H H 6H H 5 6 H 7H H 9Hζ 4Hζ 6 ζ H 54 Hζ H ζ 67 6 H 5 Hζ 65 6 H 9 H 8 H 89 8 H 67 H 9H ζ 67 H 4 H H H H H ζ 6H H ζ 9H 7H 9H H 4H 4H4 4H 4H 7 H 5 H ζ 4H H ζ Hζ H H H 9H 9 H H 9 H 9 H 9 Hζ 8H ζ 5 H 5 H 9 H 9 H 47 Hζ H 7 ζ 59 4 ζ 69 8 H 4 Hζ H 67 4 H 9 8 H 8 8 H 85 H 75 4 H 7 9 ζ 85 4 H 45 H H H 4H 4H 6C An f 6 pqg H Hζ H H H 9 8 H 4 H 6 H 5 8 H 7 6 H ζ 8 ζ 9 8 pqg H H 7 6 H H 7 H 7 9 H 7 4 H 9 54 H H 5 9 H 5 9 H C A n f pqg H 6 H 7 H 7 5 ζ 55 H H Hζ 5 H 6 8 H H 55 6 ζ 5 4 H 57 7 H H H4 H 8 7 H 5 H 8 ζ 7 4 Hζ H ζ Hζ 6 H 5 H 56 7 H H H H 5H ζ H H ζ H 4H H H H pqg H ζ H 6H H H ζ H 77 6 H H ζ H 5 H ζ H H H H 7 6H H H H 9 ζ H 4 H 9 H 8 H H 6ζ 6 6 H H 8 9 H H H Hζ H ζ H H 5H 5Hζ 65 6 ζ 6 H H4 5 H ζ H 6 H 7 H 55 ζ 9 4 H 4 H H H H 9H H H ζ 9 6 Hζ 4 H 6 H 9 Hζ 7 H 4 6 H 6 H 8H 4H 7H 5 6 H 5H ζ H H 5 H ζ 8H H 5Hζ 4H H 6Hζ 5Hζ H 6H 6H H H 5H 5 4 H H ζ 7 H H Hζ 7 4 H H 7 H 4 H4 4 H ζ H H 79 H 67 8 H 6 8 ζ 9 ζ H 5 H 4Hζ H ζ 75 7 H H H 79 4 H 7 4 H 7 H 79 7 ζ 4 Hζ 7 H 7 Hζ 8 H H 45 H 55 4 H 6C Fn f 7 6 H 6 H H 7 4 H H 5 9 H 5 9 H 5 8 H 5 9 ζ 6 pqg H 9 5 H H H 6H ζ H 7 9 H 77 8 H H 6 H 7 9 H 7 9 H 8 9 H 7 9 ζ H H 6C F n f pqg 7H 7H4 H 7Hζ 5H 6H 6H H 4H H H 5 H 6 8 H 6 8 ζ 87 8 H H 6 8 H 7 H 7H ζ 5 H 5 H 9 ζ 8 H Hζ 7 Hζ 5 H 87 8 H 5 ζ H 5Hζ 7Hζ H H 7H ζ H 5H ζ 4H H H 5H 6H 6H 4pqg H H H H H 5 8 H 5 4 H H H ζ H 4 H H H H H H H ζ H H H Hζ ζ 4 8 H 49 8 ζ 8 H 8 6 H 5 H 7 H ζ 4 ζ H H 4 H H Hζ Hζ 7 H4 Hζ 9 H 9 6 H 45 H 8 H H H 4 H 9 4 H 4H 5H 5H H H 4H 8 H 7 8 ζ 5 4 H H 8 H 7 H 7 H ζ 5 H 5 H 5 ζ H 9 4 H 5 4 H H 4 Hζ Hζ 5 Hζ 9 5 ζ 7 H4 7 Hζ 49 4 H 9 6 H 4 6 H H H H H H 6H ζ H H ζ 4H (4.) and P gq 6C AC Fn f H 4 ζ H H H 5 6 H H 5 6 pgq H H H 9 H ζ 5 Hζ 5 Hζ 4 H H 5 H H 5 H pgq H ζ 7 4 ζ 4 H 5 7 H H 5 H H H H H ζ H 79 8 H 5 9 ζ 5 9 H 5 6 H 67 6 H H 4 Hζ H 9 H 4H 4 H 7 8 H 5 H H Hζ H ζ H H 4 ζ H 7 8 H 5 6 H 6C A C F H ζ H 6 9 H 6H H 6ζ 5 54 H 8 9 H 8 H pgq Hζ 6 5ζ 7 4 ζ 65 4 H 9 H 5 H 4H 9 H 4H H Hζ 4 H H 9 4 H H H H ζ 59 Hζ 5H H ζ 5 H H ζ H 5H H 4H H H H pgq H H ζ H 7 ζ H H ζ H H H 5H ζ 4H H 6H H Hζ H H 76 4 H H 8 H H H 44 4 H H H 5 ζ 7 H Hζ 4Hζ 49 6 H H 55 6 Hζ H 59 6 ζ ζ 85 8 H 6 H 95 9 H 9 6 H 7 4 H H 7H ζ 6H 5 H H 4H 5H 79 7 H 4 44 H 9 6 H H Hζ H H 5 H ζ 8ζ 9 4 H 55 H H H 7 4 H H 8H 4ζ H H 5 Hζ 8H 4H H H ζ 7Hζ 6H ζ H 6H H H ζ 9 H 5 8 H H4 H H 6C A C F Hζ H 6H 6 ζ H 4 H H ζ 4 H 4 9 H 4 H 7 9 H 4 H 4 9 ζ 8 H 45 8 H 4 H H 9 7 H 8 H 6Hζ 4H H pgq 7 Hζ H 4 H ζ H H 4H 4 6 H 9 ζ 7 H 7 4 H 6 H ζ H H H H ζ H ζ 55 H H 4H H 4H 55 H 6H 4H 4H H H pgq H ζ 5H ζ H 9 H Hζ 7 5 ζ 6 Hζ Hζ 65 4 H 9 H 4H H 7H H 79 6 H 4H 49 6 H H H ζ 8H 6H H H H ζ 5H H H 6 Hζ H H ζ 8 9 ζ 4H 6 ζ 5 H 7Hζ 97 H H 45 H 8H 4H H 9 6 H 7 6 H H H H Hζ 6 6 H 4Hζ H ζ 46 H 5 4 H4 9 4 Hζ 55 9 H 7 8 H 49 8 H H ζ 47 4 ζ 6 59 H ζ 5H 9 H H 9 4 H 5 H H 5H 7 6 H 8Hζ 67 4 ζ 9 6 H H 8H 5Hζ 4 9 H 98 9 H 4 H4 65H 8H 9H 7 H 7 H H 4 H ζ H 4H H 4 6 ζ Hζ 7 7 H H 5 4 ζ 97 4 H 4 Hζ 85 H ζ H 5 H 9 4 H H 6 H 7 4 H 4H 4H 6C F n f H 6 pgq H 5 H 6C F n f 4 9 H 6 H 7 H pgq H H Hζ 9ζ 8 H H 7 6 H Hζ H H 5 H 8 pgq 95 9 H ζ H 6H H 5 88 ζ 4H H H H H 65 4 H Hζ 87 6 H 8 9 H 85 8 H 8 ζ 8 7 H 8 ζ H 5 4 H 9 H 7 H 8 H 5 54 Hζ H H H 6C F pgq H ζ Hζ 7 ζ 8 H 8Hζ 6H H ζ H H H H H H H 9 H H H 5 ζ pgq H 6H H ζ 7 4 H 6 5 ζ 6H 7 H 4H H ζ H 9H H Hζ Hζ H Hζ H 5H 4H H H ζ H H4 6 H 49 ζ ζ 4 H 4 H 9 6 H 6H 8H 4H 7H ζ H 4Hζ H H 5H H H H H ζ 4 ζ 9 4 H 9 ζ 87 6 H 4H 6H 4H ζ 8H 5Hζ 9 4 H H 5 8 H 9Hζ 5H 6H 58 ζ 7 Hζ 4H H 5 H H 9 ζ H 4H H ζ H H4 H H H 5 6 H H 7 6 Hζ H 9 6 H 85 8 H (4.4) Finally the Mellin inversion of Eq. (.) yields the NNLO gluon-gluon splitting function P gg 6C A C F n f 4 9 H H 97 H 8 H Hζ 7 H 6 ζ H 6H H 7 8 H 5 pgg ζ H 4 8 H 5 44 H 69 4 H Hζ Hζ H H H H 75 H 6H 8Hζ 6H 5 6 Hζ 49 H 85 4 ζ 5 H H 4H 67 H 4 ζ H 97 H 4ζ 9 H 8H H 4 H H H H 9 6 ζ ζ H ζ 4H H H 9Hζ H Hζ 7 H H 9Hζ 6H 4H 8H ζ H 4 H H H 6 H 95 H 49 6 H 45 8 H 7H 9 H 9 6 H 99 6 ζ 6 6 ζ 5 H 7 6 H 4 ζ H ζ 8H H 6H4 4H 6H ζ 8Hζ 7H H H 4H 9H 4 88 δ 6C An f 9 54 H 4 H 7 pgg 54 5 H 7 H ζ H H H 9 88 δ 6C A n f ζ 9 ζ 9 H H Hζ 69 8 H H pgg ζ 9 6 8ζ H H H H H H H 9 pgg ζ H Hζ H H Hζ H Hζ 5 6 H H H 5 54 H 8 ζ H ζ ζ H H 7 9 H H 5 6 H H H 69 6 ζ H ζ 6H H 7 Hζ H H 7 4 H 9 6 H H ζ 9 ζ 7 H 5 9 H 7 Hζ 5 H ζ 5ζ 7H 77 6 H 9 H H Hζ H H H4 9 ζ 7H H H H ζ ζ 4H H 8 Hζ 7 H H 7 6 H 94 6 H 6Hζ δ 88 6 ζ ζ 5 ζ 6C A H Hζ 49 8 H 44H H 44 H 85 6 ζ H pgg ζ ζ ζ 4H 6H ζ 4H H 4H 4H 6 H 7Hζ 67 9 H 8H ζ 4H 6Hζ 4H H 6H ζ 8H 8H 8H4 4 9 H 6 H 8H 8H 4 9 H 4Hζ 8H 8H 6 H H 8H pgg ζ 6 Hζ 4H 6H ζ H 4 9 H Hζ 8H H ζ 8H 8H 6H ζ 4H 6H 8H ζ 6H 4H 6H 5Hζ 8H ζ 4H H 67 9 ζ 67 9 H 8H H 55 ζ Hζ 67 9 H 67 9 H 4 8 H Hζ H 7 54 H 6 H ζ H H ζ H 5 98 H 8 H H 6 H 7 H 5 H 4H 6 H 9 H 9 H 7 8 4H ζ 8H H Hζ 7 Hζ 4 6 H 9 H H 9 8 ζ ζ 4 5 ζ 5 6 H H ζ 8Hζ H 8H 5H H 4Hζ 6H 6H H 5 H ζ 9H 4 H 7H 4 Hζ H 4H 5 6 ζ 6 H 4ζ ζ 7H 4H ζ 6Hζ 6H 8H δ 79 ζζ 6 ζ 4 ζ 67 6 ζ 5ζ5 6C Fn f 9 6 H H ζ H 9 H ζ H H 9 8 H H H 77 8 H 6 H H H 68 9 H 4 H 4 ζ 9 6 H ζ Hζ H H H H 44 δ 6C F n f H 7 H H ζ 9 4 H H H 85 6 H H H ζ 4 6 H H H Hζ H H H ζ 4 H ζ H H 5 H H 6 H 5 ζ H H 85 H 4 H 8H ζ Hζ H 6Hζ 8H 4H 7 H 5 6 H 4ζ Hζ 8H 4H ζ 4H 4Hζ 7 H 7 H H H 5 4 H H 8H 54 4 H 4H H ζ 9 ζ 5Hζ 5H 4H ζ 8H 67 H 4H H ζ H 4Hζ H H H H H4 6 δ (4.5) The large- behaviour of the gluon-gluon splitting function P gg is given by P gg A g B g δ C g ln O (4.6) NNLO, P () ab : Moch, Vermaseren & Vogt 4

101 QCD lecture (p. 8) Etras Higher orders Evolution uncertainty uncertainty on g(, Q = GeV) Uncert. on gluon ev. from to GeV LO evolution Input: CTEQ6 at Q = GeV Evolution: HOPPET Estimate uncertainties on evolution by changing the scale used for α s inside the splitting functions Talk more about such tricks in net lecture with LO evolution, uncertainty is % NLO brings it down to 5% NNLO % Commensurate with data uncertainties

102 QCD lecture (p. 8) Etras Higher orders Evolution uncertainty uncertainty on g(, Q = GeV) Uncert. on gluon ev. from to GeV LO evolution NLO evolution Input: CTEQ6 at Q = GeV Evolution: HOPPET Estimate uncertainties on evolution by changing the scale used for α s inside the splitting functions Talk more about such tricks in net lecture with LO evolution, uncertainty is % NLO brings it down to 5% NNLO % Commensurate with data uncertainties

103 QCD lecture (p. 8) Etras Higher orders Evolution uncertainty uncertainty on g(, Q = GeV) Uncert. on gluon ev. from to GeV LO evolution NLO evolution NNLO evolution Input: CTEQ6 at Q = GeV Evolution: HOPPET Estimate uncertainties on evolution by changing the scale used for α s inside the splitting functions Talk more about such tricks in net lecture with LO evolution, uncertainty is % NLO brings it down to 5% NNLO % Commensurate with data uncertainties