Initial-state splitting

Transcription

1 QCD lecture (p. 14) 1st order analysis 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 1 z kt p zp! h (1 z)p For virtual terms, momentum entering hard process is unchanged σ V +h (p) σ h (p) α sc F π dz dkt 1 z kt p p! 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 1 z [σ h(zp) σ h (p)] NB: We assume σ h involves momentum transfers Q k t,soignoreetra transverse momentum in σ h

2 QCD lecture (p. 15) 1st order analysis Initial-state collinear divergence σ g+h + σ V +h α sc F π Q dkt kt }{{ } infinite dz 1 z [σ h(zp) σ h (p)] }{{} finite In soft limit (z 1), σ h (zp) σ h (p) : soft divergence cancels. For 1 z,σ h (zp) σ h (p),soz 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?

3 QCD lecture (p. 15) 1st order analysis Initial-state collinear divergence σ g+h + σ V +h α sc F π Q dkt kt }{{ } infinite dz 1 z [σ h(zp) σ h (p)] }{{} finite In soft limit (z 1), σ h (zp) σ h (p) : soft divergence cancels. For 1 z,σ h (zp) σ h (p),soz 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?

4 QCD lecture (p. 15) 1st order analysis Initial-state collinear divergence σ g+h + σ V +h α sc F π Q dkt kt }{{ } infinite dz 1 z [σ h(zp) σ h (p)] }{{} finite In soft limit (z 1), σ h (zp) σ h (p) : soft divergence cancels. For 1 z,σ h (zp) σ h (p),soz 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?

5 QCD lecture (p. 15) 1st order analysis Initial-state collinear divergence σ g+h + σ V +h α sc F π Q dkt kt }{{ } infinite dz 1 z [σ h(zp) σ h (p)] }{{} finite In soft limit (z 1), σ h (zp) σ h (p) : soft divergence cancels. For 1 z,σ h (zp) σ h (p),soz 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?

6 QCD lecture (p. 16) 1st order analysis Collinear cutoff By what right did we go to k t =? 1 GeV p p Q! h zp (1 z)p We assumed pert. QCD to be valid for all scales, but below 1GeVit becomes non-perturbative. Cut out this divergent region, & instead put non-perturbative quark distribution in proton. σ = d σ h (p) q(, 1GeV ) σ 1 α sc F π Q 1GeV dk t k t }{{} finite (large) d dz 1 z [σ h(zp) σ h (p)] q(, 1GeV ) }{{} finite In general: replace 1 GeV cutoff with arbitrary factorization scale µ.

7 QCD lecture (p. 16) 1st order analysis Collinear cutoff By what right did we go to k t =? p µ p Q! h zp (1 z)p We assumed pert. QCD to be valid for all scales, but below 1GeVit becomes non-perturbative. Cut out this divergent region, & instead put non-perturbative quark distribution in proton. σ = d σ h (p) q(, µ ) σ 1 α sc F π Q µ dk t k t }{{} finite (large) d dz 1 z [σ h(zp) σ h (p)] q(, µ ) }{{} finite In general: replace 1 GeV cutoff with arbitrary factorization scale µ.

8 QCD lecture (p. 17) 1st 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,isarbitrarybetween1gev 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)

9 QCD lecture (p. 17) 1st 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,isarbitrarybetween1gev 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

10 QCD lecture (p. 18) DGLAP DGLAP equation (q q) Change convention: (a) now fi outgoing longitudinal momentum ; (b) take derivative wrt factorization scale µ (1+$)µ µ + µ (1+$)µ p /z (1 z)/z p dq(,µ ) d ln µ = α s π 1 dz p qq (z) q(/z,µ ) z α s π 1 dz p qq (z) q(,µ ) p qq is real q q splitting kernel: p qq (z) =C F 1+z 1 z Until now we approimated it in soft (z 1) limit, p qq C F 1 z

11 QCD lecture (p. 19) DGLAP DGLAP rewritten Awkward to write real and virtual parts separately. Use more compact notation: dq(,µ ) d ln µ = α s π 1 dz P qq (z) q(/z,µ ) z } {{ } P qq q ( 1+z ), P qq = C F 1 z + This involves the plus prescription: 1 dz [g(z)] + f (z) = 1 dz g(z) f (z) 1 dz g(z) f (1) z =1divergencesofg(z) cancellediff (z) sufficiently smooth at z =1

12 QCD lecture (p. ) DGLAP DGLAP flavour structure Proton contains both quarks and gluons so DGLAP is a matri in flavour space: ( ) ( ) ( ) d q Pq q P d ln Q = 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 +(1 z) ] [ ] 1+(1 z), P gq (z) =C F, z [ z P gg (z) =C A + 1 z ] + z(1 z) + δ(1 z) (11C A 4n f T R ). (1 z) + z 6 Have various symmetries / significant properties, e.g. P qg, P gg : symmetric z 1 z P qq, P gg : diverge for z 1 (ecept virtuals) soft gluon emission P gg, P gq : diverge for z Implies PDFs grow for

13 QCD lecture (p. 1) Eample evolution Effect of DGLAP (initial quarks) q(,q ), g(,q ) g(,q ) q + qbar Q = 1. GeV Take eample evolution starting with just quarks: ln Q q = P q q q ln Q g = P g q q quark is depleted at large gluon grows at small.1.1 1

14 QCD lecture (p. 1) Eample evolution Effect of DGLAP (initial quarks) q(,q ), g(,q ) g(,q ) q + qbar Q = 15. GeV Take eample evolution starting with just quarks: ln Q q = P q q q ln Q g = P g q q quark is depleted at large gluon grows at small.1.1 1

15 QCD lecture (p. 1) 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: ln Q q = P q q q ln Q g = P g q q quark is depleted at large gluon grows at small.1.1 1

16 QCD lecture (p. 1) Eample evolution Effect of DGLAP (initial quarks) q(,q ), g(,q ) g(,q ) q + qbar Q = 35. GeV Take eample evolution starting with just quarks: ln Q q = P q q q ln Q g = P g q q quark is depleted at large gluon grows at small.1.1 1

17 QCD lecture (p. 1) 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: ln Q q = P q q q ln Q g = P g q q quark is depleted at large gluon grows at small.1.1 1

18 QCD lecture (p. 1) 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: ln Q q = P q q q ln Q g = P g q q quark is depleted at large gluon grows at small.1.1 1

19 QCD lecture (p. 1) 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: ln Q q = P q q q ln Q g = P g q q quark is depleted at large gluon grows at small.1.1 1

20 QCD lecture (p. 1) Eample evolution Effect of DGLAP (initial quarks) q(,q ), g(,q ) g(,q ) q + qbar Q = 15. GeV Take eample evolution starting with just quarks: ln Q q = P q q q ln Q g = P g q q quark is depleted at large gluon grows at small.1.1 1

21 QCD lecture (p. ) Eample evolution Effect of DGLAP (initial gluons) q(,q ), g(,q ) g(,q ) q + qbar Q = 1. GeV nd eample: start with just gluons. ln Q q = P q g g ln Q g = P g g g 1 gluon is depleted at large. high- gluon feeds growth of small gluon & quark

22 QCD lecture (p. ) Eample evolution Effect of DGLAP (initial gluons) q(,q ), g(,q ) g(,q ) q + qbar Q = 15. GeV nd eample: start with just gluons. ln Q q = P q g g ln Q g = P g g g 1 gluon is depleted at large. high- gluon feeds growth of small gluon & quark

23 QCD lecture (p. ) Eample evolution Effect of DGLAP (initial gluons) q(,q ), g(,q ) g(,q ) q + qbar Q = 7. GeV nd eample: start with just gluons. ln Q q = P q g g ln Q g = P g g g 1 gluon is depleted at large. high- gluon feeds growth of small gluon & quark

24 QCD lecture (p. ) Eample evolution Effect of DGLAP (initial gluons) q(,q ), g(,q ) g(,q ) q + qbar Q = 35. GeV nd eample: start with just gluons. ln Q q = P q g g ln Q g = P g g g 1 gluon is depleted at large. high- gluon feeds growth of small gluon & quark

25 QCD lecture (p. ) Eample evolution Effect of DGLAP (initial gluons) q(,q ), g(,q ) g(,q ) q + qbar Q = 46. GeV nd eample: start with just gluons. ln Q q = P q g g ln Q g = P g g g 1 gluon is depleted at large. high- gluon feeds growth of small gluon & quark

26 QCD lecture (p. ) Eample evolution Effect of DGLAP (initial gluons) q(,q ), g(,q ) g(,q ) q + qbar Q = 6. GeV nd eample: start with just gluons. ln Q q = P q g g ln Q g = P g g g 1 gluon is depleted at large. high- gluon feeds growth of small gluon & quark

27 QCD lecture (p. ) Eample evolution Effect of DGLAP (initial gluons) q(,q ), g(,q ) g(,q ) q + qbar Q = 9. GeV nd eample: start with just gluons. ln Q q = P q g g ln Q g = P g g g 1 gluon is depleted at large. high- gluon feeds growth of small gluon & quark

28 QCD lecture (p. ) Eample evolution Effect of DGLAP (initial gluons) q(,q ), g(,q ) g(,q ) q + qbar Q = 15. GeV nd eample: start with just gluons. ln Q q = P q g g ln Q g = P g g g 1 gluon is depleted at large. high- gluon feeds growth of small gluon & quark

29 QCD lecture (p. 3) 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

30 QCD lecture (p. 4) Determining gluon Evolution versus data DGLAP with initial gluon = p F (,Q ) DGLAP: g(,q ) = ZEUS NMC Q = 1. GeV Fit quark distributions to F (, Q ), at initial scale Q =1GeV. NB: Q often chosen lower Assume there is no gluon at Q : g(, Q )= Use DGLAP equations to evolve to higher Q ;comparewithdata Complete failure!

31 QCD lecture (p. 4) Determining gluon Evolution versus data DGLAP with initial gluon = p F (,Q ) DGLAP: g(,q ) = ZEUS NMC Q = 15. GeV Fit quark distributions to F (, Q ), at initial scale Q =1GeV. NB: Q often chosen lower Assume there is no gluon at Q : g(, Q )= Use DGLAP equations to evolve to higher Q ;comparewithdata Complete failure!

32 QCD lecture (p. 4) 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 =1GeV. NB: Q often chosen lower Assume there is no gluon at Q : g(, Q )= Use DGLAP equations to evolve to higher Q ;comparewithdata Complete failure!

33 QCD lecture (p. 4) Determining gluon Evolution versus data DGLAP with initial gluon = p F (,Q ) DGLAP: g(,q ) = ZEUS NMC Q = 35. GeV Fit quark distributions to F (, Q ), at initial scale Q =1GeV. NB: Q often chosen lower Assume there is no gluon at Q : g(, Q )= Use DGLAP equations to evolve to higher Q ;comparewithdata Complete failure!

34 QCD lecture (p. 4) 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 =1GeV. NB: Q often chosen lower Assume there is no gluon at Q : g(, Q )= Use DGLAP equations to evolve to higher Q ;comparewithdata Complete failure!

35 QCD lecture (p. 4) 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 =1GeV. NB: Q often chosen lower Assume there is no gluon at Q : g(, Q )= Use DGLAP equations to evolve to higher Q ;comparewithdata Complete failure!

36 QCD lecture (p. 4) 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 =1GeV. 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 ;comparewithdata Complete failure!

37 QCD lecture (p. 4) 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 =1GeV. Q = 15. GeV NB: Q often chosen lower Assume there is no gluon at Q : g(, Q )= Use DGLAP equations to evolve to higher Q ;comparewithdata Complete failure!

38 QCD lecture (p. 5) Determining gluon Evolution versus data DGLAP with initial gluon F p (,Q ) 1.6 DGLAP (CTEQ6D) ZEUS NMC 1. Q = 1. GeV If gluon,splittingg 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!

39 QCD lecture (p. 5) Determining gluon Evolution versus data DGLAP with initial gluon F p (,Q ) 1.6 DGLAP (CTEQ6D) ZEUS NMC 1. Q = 15. GeV If gluon,splittingg 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!

40 QCD lecture (p. 5) Determining gluon Evolution versus data DGLAP with initial gluon F p (,Q ) 1.6 DGLAP (CTEQ6D) ZEUS NMC 1. Q = 7. GeV If gluon,splittingg 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!

41 QCD lecture (p. 5) Determining gluon Evolution versus data DGLAP with initial gluon F p (,Q ) 1.6 DGLAP (CTEQ6D) ZEUS NMC 1. Q = 35. GeV If gluon,splittingg 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!

42 QCD lecture (p. 5) Determining gluon Evolution versus data DGLAP with initial gluon F p (,Q ) 1.6 DGLAP (CTEQ6D) ZEUS NMC 1. Q = 46. GeV If gluon,splittingg 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!

43 QCD lecture (p. 5) Determining gluon Evolution versus data DGLAP with initial gluon F p (,Q ) 1.6 DGLAP (CTEQ6D) ZEUS NMC 1. Q = 6. GeV If gluon,splittingg 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!

44 QCD lecture (p. 5) Determining gluon Evolution versus data DGLAP with initial gluon F p (,Q ) 1.6 DGLAP (CTEQ6D) ZEUS 1. Q = 9. GeV If gluon,splittingg 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!

45 QCD lecture (p. 5) Determining gluon Evolution versus data DGLAP with initial gluon F p (,Q ) 1.6 DGLAP (CTEQ6D) ZEUS 1. Q = 15. GeV If gluon,splittingg 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!

46 QCD lecture (p. 6) Determining full PDFs Global fits Gluon distribution q(), g() Q = 1 GeV CTEQ6D fit gluon Gluon distribution is HUGE! Can we really trust it? 3 1 d S, u S u V Consistency: momentum sum-rule is now satisfied. NB: gluon mostly at small Agrees with vast range of data

47 QCD lecture (p. 7) Determining full PDFs Global fits DIS data and global fits HERA F Q / GeV HERA Data: H ZEUS ZEUS BPT 1997 ZEUS SVX 1995 H1 SVX 1995, H1 QEDC 1997 y = 1 -log 1 () em F 5 4 =6.3E-5 =.1 =.161 =.53 =.4 =.5 =.63 =.8 =.13 =.1 =.3 =.5 ZEUS NLO QCD fit H1 PDF fit H1 94- H1 (prel.) 99/ ZEUS 96/97 BCDMS E665 NMC Fied Target Eperiments: NMC BCDMS E665 SLAC 3 =.8 =.13 =.1 =.3 =.5 = y = =.13 1 =.18 =.5 =.4 = Q (GeV )

48 QCD lecture (p. 7) Determining full PDFs Global fits DIS data and global fits HERA F Q (GeV ) H1 ZEUS CDF/D Inclusive jets &<.7 D Inclusive jets &<3 Fied Target Eperiments: CCFR, NMC, BCDMS, E665, SLAC -log 1 () em F 5 4 =6.3E-5 =.1 =.161 =.53 =.4 =.5 =.63 =.8 =.13 =.1 =.3 =.5 ZEUS NLO QCD fit H1 PDF fit H1 94- H1 (prel.) 99/ ZEUS 96/97 BCDMS E665 NMC y=1 (HERA %s=3 GeV) 3 =.8 =.13 =.1 =.3 =.5 = =.13 =.18 = =.4 = Q (GeV )

49 QCD lecture (p. 8) 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 1 q(, Q ) q 1 ( 1, Q ) g (, Q ) proton proton 1 proton σ ep = σ eq q σ pp jets = σ qg jets q 1 g +

50 QCD lecture (p. 8) 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 1 q(, Q ) q 1 ( 1, Q ) g (, Q ) proton proton 1 proton σ ep = σ eq q σ pp jets = σ qg jets q 1 g +