Regge theory as a tool for global QCD fits Gregory Soyez 16-18 - PowerPoint PPT Presentation

Regge theory as a tool for global QCD fits Gregory Soyez 16-18 December 2004, Spa, Belgium

2 Outline Introduction: usual technique of Global QCD fits Problem 1: the initial condition problem Digression: Regge theory Problem Consequences on Global fits Problem 2 : the essential singularity problem Problem One possible solution form Regge theory Conclusions : new constraints from “soft” physics

3 Global QCD Fits initial PDF at Q 2 = Q 0 2 Q 2 DGLAP evolution DGLAP pQCD Adjust initial PDF to 2 Q 0 reproduce all data: ??? p , F 2 n , F 2 d , F 2 , F 3 F 2 x 0 1 Obtain a parton set: u V , d V , u, d, s, c, b, g ( x, Q 2 )

4 Global QCD Fits Applied many times with many updates: http://durpdg.dur.ac.uk/hepdata/pdf3.html Alekhin 2002 LO, 2002 NLO, 2002 NNLO ZEUS 2002 TR, 2002 FF, 2002 ZM, H1 2000 CTEQ 6.1M, 6M, 6D, 6L, 6L1, 5M, 5D, 5L, 5HJ, 5HQ, 5F3, 5F4, 5M1, 5HQ1, 4M, 4L, 4D, 4A1, 4A2, 4A3, 4A4, 4A5, 4HJ, 4LQ, 3M, 3D, 3L GRV 98 LO, 98 NLD, 98 NLM, 94 L0, 94 HO, 94 D1 MRST 2004 NLO, 2004 NNLO, 2003c NLO, 2003c NNLO, 2002 NLO, 2002 NNLO, 2001 LO, 2001-NLO1, 2001-NLO2, 2001-NLO3, 2001-NLO4, 2001- NNLO1, 2001-NNLO2, 2001-NNLO3, 2001-NNLO4, MRST 99-1, ..., MRST 99- 12, MRST 99-dis1, ..., MRST 99-dis11 MRS*, MRSCHM*, MRSR*, MRSALPHAS*, MRSG*, KMRS*, HMRS*, ... (67 fits) 145 PDF sets !!! 20 “commonly used” !!!

5 Global QCD Fits Gluon distribution ( Q 2 = 100 GeV 2 )

6 Digression: Regge theory Analyticity of the S matrix Partial-wave expansion + Sommerfeld-Watson: A(s,t) A(j,t) j = complex angular momentum Leading Singularities high-energy behaviour in A(j,t) of A(s,t) Example: s a simple pole at j= 1 +a log 2 ( s ) triple pole at j= 1

7 Digression: Regge theory Applies at large energy (small x ) We know that it may be used to reproduce all soft hadronic data with the SAME singularities (cross sections and DIS) it is is consistent with t -channel unitarity Examples: F 2 = A(Q 2 ) s 0.4 + B(Q 2 ) s 0.08 + D(Q 2 ) s -0.4 F 2 = A(Q 2 ) [ log (s) - B(Q 2 ) ] 2 + C(Q 2 ) + D(Q 2 ) s -0.47

8 Problem 1: The Initial Condition Description: The initial PDF is a soft input not given by DGLAP Example: MRST 2002 xq(x) x -0.12 xg(x) x -0.27 vs. Not the sam e in quarks and gluons Not present in soft amplitudes Can we use tools from soft physics as a constraint ?

9 The initial PDF problem revisited Use Regge theory to constrain initial PDF Q 2 DGLAP 2 Q 0 Same high-energy behaviour: x as in soft data 0 1 G.S. hep-ph/0407098 for all PDF + to be written

10 The initial PDF problem revisited PDF at Q 2 = Q 0 2 = 5 GeV 2 : p , F 2 n , F 2 d , F 2 , F 3 And reproduce F 2

11 The initial PDF problem revisited Pomeron : same singularity, flavour independent Reggeon : same singularity, different couplings Common large-x behaviour Mass effect: normalisation

12 The initial PDF problem revisited Domain: Q 2 > 5 GeV 2 (pQCD) W 2 > 12.5 GeV 2 (Higher-twists) Results:

13 The initial PDF problem revisited Distributions: quarks ( Q 2 = Q 0 2 ) valence sea

14 The initial PDF problem revisited Distributions: gluons Q 2 = 5 GeV 2 Q 2 = 100 GeV 2

15 The initial PDF problem revisited Predictions for F L

16 The initial PDF problem revisited c Predictions for F 2

17 The initial PDF problem revisited Advantages: Consistent with S-matrix theory Q 2 Regge theory describes the “soft” data DGLAP 2 Q 0 Regge Extend the PDF down to x 0 1 Q 2 =0

18 Description of the soft data xq = [ A ( Q 2 ) log 2 (1/ x ) + B ( Q 2 ) log(1/ x ) + C ( Q 2 ) + D ( Q 2 ) x n ] (1- x ) b( Q 2 ) Q 2 -dependent Basically: impose matching at Q 2 = Q 0 2 With a few simplifications: 10 parameters for the soft data

19 Description of the soft data Regg DGLAP Regge DGLAP e NMC data

20 Description of the soft data Final result W 2 > 12.5 GeV 2

21 Problem 2: The Essential Singularity Description: At small x , the DGLAP-evolved gluon is given by Essential singularity at j =1 Idea: Only a full resummation should give a correct singularity DGLAP = “numerical” approximation

22 Problem 2: The Essential Singularity 2 1) Select Q 0 Q 2 G.S. Phys.Rev.D69 (2004) 096005 Q 2 2 2) Regge PDF at Q 0 max forward Standard set 3) DGLAP PDF at Q 2 DGLAP 4) Compare with data 2 Q 0 backward 5) Repeat 2, 3, 4 DGLAP 2 residues at Q 0 Q 2 min 2 6) Repeat for all Q 0 x x max 1 0 2 - Physics is Q 0 independent

23 Problem 2: The Essential Singularity Quarks ang Pomeron: 5 S = 2 ( xu S + xd S + xs S + xc S ) F 2 = S 18 S = A S log 2 ( x ) + B S log( x ) + C S + D S x r Result:

24 Problem 2: The Essential Singularity Stability: quarks DGLAP is a numerically close to the Regge description

25 Problem 2: The Essential Singularity Stability: gluons All produce correct fit error on gluon distribution

26 Conclusions Global QCD fit with Regge-compatible initial condition Description for ALL Q 2 : p , F 2 n , F 2 d , F 2 , F 3 DGLAP F 2 Regge c , F L Prediction for F 2 Forward+Backward DGLAP: High-Q2 PDF compatible with Regge theory gluon errors estimation

27 Perspectives Additional applications: Small- Q 2 description for the NLO fit Other Regge models NNLO ? Combined small/large Q 2 fit (determination of 2 ) Q 0

29 Various problems Good knowledge required for the LHC 2 of a few GeV 2 Q 0 x larger than 10 -4