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

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

2

Global QCD Fits

initial PDF at Q2 = Q0

2

DGLAP evolution Adjust initial PDF to reproduce all data: F2

p, F2 n, F2 d, F2 , F3

Obtain a parton set: u V , d V , u, d, s, c, b, g(x, Q2)

3 1 Q0

2

DGLAP

Q2 x pQCD ???

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” !!!

4

Global QCD Fits 5

Gluon distribution (Q2 = 100 GeV2)

Digression: Regge theory

6

Analyticity of the S matrix Partial-wave expansion + Sommerfeld-Watson:

j = complex angular momentum Example: sa simple pole at j=1+a

log2(s) triple pole at j=1

A(s,t) A(j,t)

high-energy behaviour

• f A(s,t)

Leading Singularities in A(j,t)

Digression: Regge theory 7

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: F2 = A(Q2) s0.4 + B(Q2) s0.08 + D(Q2) s-0.4 F2 = A(Q2) [log(s) - B(Q2)]2 + C(Q2) + D(Q2) s-0.47

Problem 1: The Initial Condition

8

Description: The initial PDF is a soft input not given by DGLAP

Example: MRST 2002

xq(x) x-0.12 vs. xg(x) x-0.27 Not the sam e in quarks and gluons Not present in soft amplitudes Can we use tools from soft physics as a constraint ?

The initial PDF problem revisited

9

Use Regge theory to constrain initial PDF

1 Q0

2

DGLAP

Q2 x

Same high-energy behaviour: as in soft data for all PDF

G.S. hep-ph/0407098 + to be written

The initial PDF problem revisited 10

PDF at Q2 = Q0

2 = 5 GeV2 :

And reproduce F2

p, F2 n, F2 d, F2 , F3

The initial PDF problem revisited 11

Pomeron: same singularity, flavour independent Reggeon: same singularity, different couplings

Common large-x behaviour Mass effect: normalisation

The initial PDF problem revisited 12

Domain: Q 2 > 5 GeV2 (pQCD) W 2 > 12.5 GeV2 (Higher-twists) Results:

The initial PDF problem revisited 13

Distributions: quarks (Q2 = Q0

2)

valence sea

The initial PDF problem revisited 14

Distributions: gluons Q2 = 5 GeV2 Q2 = 100 GeV2

The initial PDF problem revisited 15

Predictions for FL

The initial PDF problem revisited 16

Predictions for F2

c

The initial PDF problem revisited 17

Regge theory describes the “soft” data

1 Q0

2

DGLAP

Q2 x

Regge

Extend the PDF down to

Q2=0 Advantages: Consistent with S-matrix theory

18 Description of the soft data

xq = [A(Q2) log2(1/x) + B(Q2) log(1/x) + C(Q2) + D(Q2) xn] (1-x)b(Q2)

Q2-dependent Basically: impose matching at Q2 = Q0

2

With a few simplifications: 10 parameters for the soft data

19 Description of the soft data

NMC data

Regg e Regge DGLAP DGLAP

20 Description of the soft data

Final result W 2 > 12.5 GeV2

Problem 2: The Essential Singularity

21

Description: At small x, the DGLAP-evolved gluon is given by Idea:

Only a full resummation should give a correct singularity DGLAP = “numerical” approximation

Essential singularity at

j=1

Problem 2: The Essential Singularity 22

1) Select Q0

2

2) Regge PDF at Q0

2

3) DGLAP PDF at Q2 4) Compare with data 5) Repeat 2, 3, 4 residues at Q0

2

6) Repeat for all Q0

2

Q0

2

xmax 1 x Q2 Q2

min

Q2

max

backward DGLAP forward DGLAP Standard set

Physics is Q 0

2-

independent

G.S. Phys.Rev.D69 (2004) 096005

23

Quarks ang Pomeron:

S = 2 (xu S + xd S + xsS + xcS ) S = A S log2(x) + B S log(x) + CS + DS xr

Result:

Problem 2: The Essential Singularity

F2 = S 5 18

24

Stability: quarks DGLAP is a numerically close to the Regge description

Problem 2: The Essential Singularity

25

Stability: gluons All produce correct fit error on gluon distribution

Problem 2: The Essential Singularity

Global QCD fit with Regge-compatible initial condition Description for ALL Q2 :

F2

p, F2 n, F2 d, F2 , F3

Prediction for F2

c , FL

Forward+Backward DGLAP: High-Q2 PDF compatible with Regge theory gluon errors estimation

26

Conclusions

DGLAP Regge

Additional applications: Small-Q2 description for the NLO fit Other Regge models NNLO ? Combined small/large Q2 fit (determination of Q0

2 )

27

Perspectives

Various problems 29

Good knowledge required for the LHC

Q0

2 of a few GeV2

x larger than 10-4