Parton Distributions for the LHC and small-x physics

Robert Thorne

March 13, 2009

University College London

Soton 2009

Hadrons are particles made up of the more fundamental constituents, quarks andgluons (partons). These are bound together by the strong force, described by thequantum field theory QCD.

Most important particle colliders use hadrons – HERA was an ep collider, the Tevatronis a pp collider, the LHC (large hadron collider) at CERN is a pp collider.

The strong coupling constant αS(µ2) runs with the energy scale µ2 of a process,decreasing as µ2 increases (asymptotic freedom).

αs(µ2) ≈

4π

(11 − 2/3Nf) ln(µ2/Λ2QCD)

αs(µ2) is very large if µ2 ∼ Λ2

QCD (∼ 0.3GeV), the scale of nonperturbative physics,

but αs(µ2) ¿ 1 if µ2 À Λ2

QCD, and perturbation theory can be used.

Because of the strong force it is difficult to perform analytic calculations of scatteringprocesses involving hadronic particles from first principles. However, the weakeningof αS(µ2) at higher scales → the Factorization Theorem – separates processes intononperturbative parton distributions which describe the composition of the protonand which can be determined from experiment, and perturbative parts associated withhigher scales which are calculated as a power-series in αS(µ2).

Soton 2009 1

e e

γ? Q2

x

P

perturbativecalculable

coefficient function

CPi (x, αs(Q

2))

nonperturbativeincalculable

parton distribution

fi(x, Q2, αs(Q2))

Hadron scattering with anelectron factorises.

Q2 – Scale of scattering

x = Q2

2P ·q – Momentum fraction ofParton in light-cone frame.

Soton 2009 2

The cross-section for this process can be written in the factorised form

σ(ep → eX) =∑

i

CPi (x, αs(Q

2)) ⊗ fi(x, Q2, αs(Q2))

where fi(x,Q2, αs(Q2)) represent the probability to find a parton of type i carrying a

fraction x of the momentum of the hadron.

Corrections to above formula of size Λ2QCD/Q2.

The partons are intrinsically nonperturbative. However, once Q2 is large enough theydo evolve with Q2 in a perturbative manner.

dfi(x,Q2, αs(Q2))

d ln Q2=

∑

j

Pij(x, αs(Q2)) ⊗ fj(x,Q2, αs(Q

2))

where the splitting functions Pij(x,Q2, αs(Q2)) describing how a parton splits into

more partons are calculable order by order in perturbation theory.

Partons parametrized at one low scale Q20, evolved to higher Q2.

Soton 2009 3

P

P

fi(xi, Q2, αs(Q

2))

CPij(xi, xj, αs(Q

2))

fj(xj, Q2, αs(Q

2))

The coefficient functionsCP

i (x, αs(Q2)) are process

dependent (new physics) butare calculable as a power-seriesin αs(Q

2).

CPi (x, αs(Q

2)) =∑

k

CP,ki (x)αk

s(Q2).

The parton distributionsfi(x,Q2, αs(Q

2)) are process-independent, i.e. universal,and evolve with Q2 in aperturbative manner. Oncethey have been measuredat one experiment, one canpredict many other scatteringprocesses.

Soton 2009 4

I am a member of MSTW (previously MRST) group. Use all available data –largely ep → eX (Structure Functions), and the most up-to-date QCD calculations todetermine parton distributions and their consequences.

Many reasons for this

→ Understanding structure of hadrons.

→ One of the best tests of QCD.

→ Good determination of strong coupling constant (important in unification).

→ Necessary to look for new physics (LHC, Tevatron).

Currently use LO, NLO and NNLO–in–αs(Q2), i.e.

CPi (x, αs(Q

2)) = αPS (Q2)(CP,0

i (x) + αS(Q2)CP,1i (x) + α2

S(Q2)CP,2i (x)).

Pij(x, αs(Q2)) = αS(Q2)P 0

ij(x) + α2S(Q2)P 1

ij(x) + α3S(Q2)P 2

ij(x).

Perturbation theory valid if αs(Q2) < 0.3, i.e. Q2 > 2GeV2.

Soton 2009 5

Full range of data sets used. Largely DIS data but an increasing amount of hadroncollider data.

Soton 2009 6

Use all available data – more than 30 different types of data set, and most up-to-dateQCD calculations to determine parton distributions and their consequences. A large-scale, ongoing project. Vital input for hadron collider physics – used by experimentsand theorists worldwide.

x-410 -310 -210 -110 1

)2xf

(x,Q

0

0.2

0.4

0.6

0.8

1

1.2

g/10

d

d

u

uss,

cc,

2 = 10 GeV2Q

x-410 -310 -210 -110 1

)2xf

(x,Q

0

0.2

0.4

0.6

0.8

1

1.2

x-410 -310 -210 -110 1

)2xf

(x,Q

0

0.2

0.4

0.6

0.8

1

1.2

g/10

d

d

u

u

ss,

cc,

bb,

2 GeV4 = 102Q

x-410 -310 -210 -110 1

)2xf

(x,Q

0

0.2

0.4

0.6

0.8

1

1.2

MSTW 2008 NLO PDFs (68% C.L.)

Soton 2009 7

|y|0 0.5 1 1.5 2 2.5 3

/dy

σ dσ

1/

0

0.05

0.1

0.15

0.2

0.25

0.3

-1 Run II with 0.4 fb∅D

MSTW 2008 NLO PDF fit

MSTW 2008 NNLO PDF fit

|y|0 0.5 1 1.5 2 2.5 3

Dat

a / T

heo

ry

0.8

1

1.2

1.4

1.6 = 19 for 28 points)2χ(

MSTW 2008 NLO PDF fit

= 17 for 28 points)2χ(MSTW 2008 NNLO PDF fit

∅* rapidity shape distribution from DγZ/

Leads to very accurate andprecise predictions.

Comparison of MSTW predictionfor Z rapidity distribution withdata.

Rapidity y measures how farforward (or back) particle isproduced.

y = 1/2 ln((E +pz)/(E−pz))

New data (D0, CDF) been fit– affects partons.

Important at LHC.

Soton 2009 8

Predictions at the LHC

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100100

101

102

103

104

105

106

107

108

109

fixedtargetHERA

x1,2

= (M/14 TeV) exp(±y)Q = M

LHC parton kinematics

M = 10 GeV

M = 100 GeV

M = 1 TeV

M = 10 TeV

66y = 40 224

Q2

(GeV

2 )

x

LHCb LHCb

New kinematic regime.

High scale and small-x partondistributions are vital forunderstanding processes at theLHC.

Problems at small x due toextra powers of ln(1/x).

Some predictions potentiallyunstable, e.g. high-rapidity (allLHCb physics) and/or lowishmass final states at LHC.

(Small-x partons also necessaryfor very high-energy neutrinoscattering at neutrino telescopes.)

Soton 2009 9

Uncertainty at NLO

0 1 2 3 4 5

1

10

100

24 GeV

pdf uncertainty ondσ(W+)/dy

W, dσ(W-)/dy

W,

dσ(Z)/dyZ, dσ(DY)/dMdy

at LHC using MSTW2007NLO

% p

df u

ncer

tain

ty

y

8 GeV

Uncertainty on σ(Z) and σ(W+)grows at high rapidity.

Uncertainty on σ(W−) grows morequickly at very high y – depends onless well-known down quark.

Uncertainty on σ(γ?) is greatest asy increases. Depends on partons atvery small x.

Soton 2009 10

Interplay of LHC/Tevatron and pdfs/QCD

Make predictions for all processes, both SM and BSM, as accurately as possible givencurrent experimental input and theoretical accuracy.

Check against well-understood processes, e.g. central rapidity W, Z production(luminosity monitor), lowish-ET jets, .....

Compare with predictions with more uncertainty and lower confidence, e.g. high-ET

jets, high rapidity bosons or heavy quarks .....

Improve uncertainty on parton distributions by improved constraints, and checkunderstanding of theoretical uncertainties, and determine where NNLO, electroweakcorrections, resummations etc. needed.

Make improved predictions for both background and signals with improved partonsand surrounding theory.

Spot new physics from deviations in these predictions. As a nice by-product improveour understanding of the strong sector of the Standard Model considerably.

Some outstanding issues still in QCD/Standard Model physics, particularly for smallx,

Soton 2009 11

Small x Theory

It is known that at each order in αS each splitting function and coefficient functionobtains an extra power of ln(1/x) (some accidental zeros in Pgg), i.e.

Pij(x, αs(Q2)), CP

i (x, αs(Q2)) ∼ αm

s (Q2) lnm−1(1/x).

→ no guarantee of convergence at small x!

x < 0.01, ln(1/x) > 5, → αS ln(1/x) > 1.

The global fits usually assume that this turns out to be unimportant in practice, andproceed regardless.

Fits work fairly well at small x, but could be better.

Some predictions unstable from LO → NLO → NNLO.

Soton 2009 12

0

0.5

1

1.5

2

1 10 10 2 10 3

σ~(x,Q2)HERA (F2(x,Q2)FixedTarget) + c

x=0.00016 (c=0.4)

x=0.0005 (c=0.35)

x=0.0013 (c=0.3)

x=0.0032 (c=0.25)

x=0.008 (c=0.2)

x=0.013 (c=0.15)

x=0.05 (c=0.1)

x=0.18 (c=0.05)

x=0.35 (c=0.0)

Q2(GeV2)

H1ZEUSNMCBCDMSSLAC

NNLO NLO LO

MSTW 2008

Fit to F2(x,Q2) data.

Slope poor (too flat) at LO, ok atNLO and better at NNLO.

Even at NNLO a little bit too flat– χ2 for HERA data can improve.

Within global fit HERA dataprefer αS(M2

Z) = 0.124 at NLO(best fit (αS(M2

Z) = 0.120), andαS(M2

Z) = 0.122 at NNLO (bestfit (αS(M2

Z) = 0.117)

Some room for improvements.

Soton 2009 13

x

-510 -410

-310 -210 -110 1

)2xg

(x,Q

0

5

10

2 = 2 GeV2Q

x

-510 -410

-310 -210 -110 1

)2xg

(x,Q

0

5

10

MSTW 2008 LO

MSTW 2008 NLO

MSTW 2008 NNLO

x

-510 -410

-310 -210 -110 1

)2xg

(x,Q

0

5

10

15

2 = 5 GeV2Q

x

-510 -410

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)2xg

(x,Q

0

5

10

15

x

-510 -410

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(x,Q

0

10

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30

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x

-510 -410

-310 -210 -110 1

)2xg

(x,Q

0

10

20

30

x

-510 -410

-310 -210 -110 1

)2xg

(x,Q

0

10

20

30

40

50

2 = 100 GeV2Q

x

-510 -410

-310 -210 -110 1

)2xg

(x,Q

0

10

20

30

40

50

Stability order-by-order.

Start by looking at fixed orderQCD.

The gluon extracted from theglobal fit at LO, NLO and NNLO.

Additional and positive small-xcontributions in Pqg at each orderlead to smaller small-x gluon ateach order.

Clearly poor stability.

Similar for FL(x,Q2)

Soton 2009 14

Consequences for LHC

y0 1 2 3 4 5 6 7 8 9 10

/dM

/dy

(p

b/G

eV)

σ2 d

-2

0

2

4

6

8

10

12

14

163

10×

M = 2 GeVM = 2 GeV

y0 1 2 3 4 5 6 7 8 9 10

/dM

/dy

(p

b/G

eV)

σ2 d

0

0.5

1

1.5

23

10×

M = 4 GeVM = 4 GeV

y0 1 2 3 4 5 6 7 8 9 10

/dM

/dy

(p

b/G

eV)

σ2 d

-50

0

50

100

150

200

M = 10 GeV

(2001 LO, 2004 NLO, 2006 NNLO)Using Vrap with MRST PDFs

M = 10 GeV

y0 1 2 3 4 5 6 7 8 9 10

/dM

/dy

(p

b/G

eV)

σ2 d

-20

0

20

40

60

80

100

ZM = M

LO

NLO

qNLO q

NLO qg

NNLO

qNNLO q

NNLO qg

NNLO gg

NNLO qq

ZM = M

*/Z rapidity distributions at LHCγNow have QCD calculations atLO, NLO and NNLO in thecoupling constant αS for Z,W andγ? production Anastasiou, Dixon,Melnikov, Petriello).

Good stability in predictions for e.g.Z and γ? cross-sections for very highvirtuality.

Becomes worse at lower scales whereαS larger and large ln(s/M2) termsappear in expansion (equivalent toln(1/x)).

Soton 2009 15

Small-x corrections.

For a long time a dominant area of research for theorists working on HERA physics.

Look at a brief history. Development as long as any experiment.

Late 1970s, LO BFKL equation for high energylimit in QCD - related to gluon ladders

f(k2, x) = fI(Q20)+

∫ 1

xdx′

x′ αS

∫

∞

0

dq2

q2 K0(q2, k2)f(q2, x),

where f(k2, x) is the unintegrated gluon

distribution g(x,Q2) =∫ Q2

0(dk2/k2)f(x, k2), and

K0(q2, k2) is a calculated kernel.

Physical structure functions obtained fromconvolution

σ(Q2, x) =∫

(dk2/k2)h(k2/Q2)f(k2, x)

where h(k2/Q2) is a calculable impact factor.

Soton 2009 16

Led to predictions that

σ(x,Q2) ∼ x−λ, λ = 12/π ln 2 αS ≈ 2.7αS

Later also shown that

xPij(x), Ci,j(x) ∼ x−λ.

Early-mid 1990s analysis of structure function impact by Ellis-Hautmann-Webber, Ball-Forte, Forshaw-Roberts-Thorne, Martin-Sutton-· · · , Forshaw-Ross and many others.

Also look for signs in jets, Orr-Stirling, + · · · , onium-onium Salam, i.e. where bothends of ladder a hard scale, Q2

0 ∼ k2.

Showed matching onto normal perturbative expansion possible. Growth really toosteep, ∼ x−0.5 asymptotically, but approaches this from below at finite x.

OK, in some respects good, but became harder to fit to data as it improved.

Soton 2009 17

However, in early 1998 data pressure overtaken by theory disaster – NLO calculationFadin and Lipatov, Camici and Ciafaloni.

f(k2, x) = fI(Q20) +

∫ 1

x

dx′

x′

∫

∞

0

dq2

q2αS(µ2)K0(q

2, k2)f(q2, x)

+

∫ 1

x

dx′

x′

∫

∞

0

dq2

q2

(

α2S(µ2)K1(q

2, k2)−β0α2S(µ2) ln(k2/µ2)K0(q

2, k2)

)

f(q2, x)

Proceeding as if at LO and ignoring running of coupling one obtains

λ = 2.7αS ∗ (1 − 6.4αS + · · · ), i.e. unstable.

Extreme lack of convergence.

But many new complications at NLO.

Almost immediately noticed (Ross) that form of solutions has different structure withconsequence of oscillatory behaviour.

Soton 2009 18

At NLO fundamentally more complicated. Cannot ignore running coupling (Preliminarywork by Collins, Kwiecinski and Martin)

Also dependence on scales at each end of gluon ladder, i.e. is variable s/k2, s/Q20 or

s/(kQ0).

Durham HERA Workshop, Sept. 1998 essentials of importance of running couplingand change of scales (collinear resummation) presented (report Forshaw, Salam, RT).

Large part of NLO correction to kernel due to large logs inherent in scale change. Canbe resummmed to all orders.(Salam).

Factorization → evolution with Q2 in BFKL equation sensitive to UV diffusion –coupling weaker more convergent expansion. Normalization/input sensitive to IRdiffusion – coupling stronger (too strong?).

2000 also duality of splitting functions between high Q2 and small x Altarelli, Ball,Forte. Not sure of importance after first two.

(BFKL physics in jets etc. very messy theoretically due to infrared contaminationunless scales large. Work by Andersen, Sabio-Vera · · · . Inputs similar story, howeversee (Ellis, Kowalski, Ross)).

Soton 2009 19

Try alternative perturbative organisation – resummation of leading ln(1/x) terms andrunning coupling effects (resummation of β0 terms). Obtained by solving running-coupling BFKL equation for unintegrated gluon f(k2).

f(k2, x) = fI(Q20) +

∫ 1

x

dx′

x′αS(k2)

∫

∞

0

dq2

q2(K0(q

2, k2) + αS(k2)K1(q2, k2))f(q2, x)

Solve by performing double Mellin transformation

f(γ,N) =

∫ 1

0

xNdx

∫

∞

0

dk2(k2)−γ−1f(k2, x)

BFKL equation becomes

d2f(γ,N)

dγ2=

d2fI(γ,N)

dγ2−

1

β0N

d(χ0(γ)f(γ,N))

dγ+

1

β20N

χ1γf(γ,N)

where χ(γ) are the Mellin transformations of the kernels.

Solved using ansatz (Ciafaloni and Colferai)

f(γ,N) = exp

(

−X1(γ)

β0

)∫

∞

γ

A(γ) exp

(

+X1(γ)

β0

)

Soton 2009 20

Used as general basis for various approaches – Altarelli, Ball, Forte (ABF), Ciafaloni,Colferai, Salam, Stasto(CCSS), Thorne, White (TW).

In TW recognise that it is impossible to obtain normalizations and inputs withoutregularisation, but letting lower limit of integral go from γ → 0, do obtain factorisedsolution for gluon.

G(Q2, Q20, N) = GE(Q2, N)GI(Q

20, N) + O(Q2

0/Q2)

GI(Q20, N) input at low scale Q0 ∼ ΛQCD. Contaminated by infrared physics.

GE(Q2, N) completely controls evolution with Q2.

GE(Q2, N) =1

2πi

∫ 1/2+i∞

1/2−i∞

dγ

γexp

(

γt −X1γ

β0

)

where t = ln(Q2/Λ2QCD) and

X1(γ,N) =

∫

∞

1/2

[

χ0(γ) + Nχ1(γ)

χ0(γ)

]

.

Structure functions given by

FE(Q2, N) =1

2πi

∫ 1/2+i∞

1/2−i∞

h(γ)

γexp

(

γt −X1γ

β0

)

dγ

Soton 2009 21

GE(Q2, N) and GE(Q2, N) calculable so it is possible to obtain splitting functions andcoefficient functions unambiguously. Integrate Mellin space expression by expandingabout γ = 0.

GE(Q2, N) ∝

∞∑

n=0

AnαnS(Q2)

(

1

Nn− a

β0

Nn−1+ · · · + z

βn−10

N

)

.

Results in splitting functions of the form

xPgg(x,Q2) =

∞∑

n=1

n−1∑

m=0

anmαnS(Q2) lnn−1−m(1/x)βm

0 .

For fixed coupling ln(1/x)-resummation xP (x) ∼ x−12 ln 2/παS.

Qualitatively running coupling leads to change in coupling scale

αS(Q2) ∼ 1/(ln(Q2/Λ2QCD) + 3.5(αS(Q2) ln(1/x))1/2).

Full calculation much more detailed. The series is asymptotic but well defined(∼ (−1)nn!). Interface with nonperturbative QCD (renormalons).

Both conventional αS expansion and fixed coupling ln(1/x) expansion show instability.The doubly resummed calculation stabilises the perturbative expansion.

Soton 2009 22

Splitting functions dip below fixed order at intermediate x before eventually rising atvery small x.

0

0.1

0.2

0.3

10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 1

P +

x0

0.02

0.04

0.06

0.08

0.1

10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 1

P qg

x

Momentum conservation implies slight increase at highest x.

Soton 2009 23

In 2006 enough finally worked out (heavy quarks a complication) for a fit with NLOplus NLO resummation (White, RT).

0

0.5

1

1.5

2

2.5

3

3.5

4

1 10 10 2 10 3

x=5×10-4

x=6.32×10-4

x=8×10-4

x=1.3×10-3

x=1.61×10-3

x=2×10-3

x=3.2×10-3

x=5×10-3

x=8×10-3

H1ZEUSNMC

NLL+NLL(2)+NLO+

Q2(GeV2)

F 2p (x,Q

2 ) + 0

.25(

9-i)

-3

-2

-1

0

1

2

10 -5 10 -4 10 -3 10 -2 10 -1 1x

Q2=1GeV2

xg(x

)

0

20

40

60

80

10 -5 10 -4 10 -3 10 -2 10 -1 1x

Q2=100GeV2

NLL+NLL(2)+NLO+

→ moderate improvement in fit to HERA data within global fit, and change inextracted gluon (more like quarks at low Q2).

Soton 2009 24

By 2008 very similar results coming from the White-RT, Ciafaloni-Colferai-Salam-Stasto and Altarelli-Ball-Forte procedures, despite some differences in technique.

TW believe some complications unnecessary for splitting functions and coefficientfunctions and have produced phenomenologically useful results, particularly fulltreatment of heavy flavours, i.e. full General-Mass Variable-Flavour Number Scheme.

Soton 2009 25

Full set of coefficient functions stillto come in many cases, but splittingfunctions comparable.

Note, in all cases NLO correctionslead to dip in functions below fixedorder values until slower growth(running coupling effect) at verysmall x.

May possibly be significant to small xdetails, and spoil 3 − 4% theoreticalaccuracy.

Soton 2009 26

All groups produce similar splitting functions.

Coefficient functions from Altarelli et al (blue, red different schemes for resummation)very similar in form.

0

0.05

0.1

10 -6 10 -510 -410 -310 -2 10 -1 1

C Lg

x

Presumably predictions will be fairly similar.

Soton 2009 27

Comparisons in simplified framework – set inputs, no heavy flavours, attempt to usesame schemes – imply qualitative agreement but different details.

TW ABF

1 10 100 1000Q / GeV

0.8

0.85

0.9

0.95

1

1.05

1.1

K2

x = 0.01x = 0.001x = 0.0001x = 0.00001x = 0.000001

1 10 100 1000Q / GeV

0.8

0.85

0.9

0.95

1

1.05

1.1

K2

x = 0.01x = 0.001x = 0.0001x = 0.00001x = 0.000001

1 10 100 1000Q / GeV

0.8

0.9

1

1.1

1.2

1.3

KL

x = 0.01x = 0.001x = 0.0001x = 0.00001x = 0.000001

1 10 100 1000Q / GeV

0.8

0.9

1

1.1

1.2

1.3

KL

x = 0.01x = 0.001x = 0.0001x = 0.00001x = 0.000001

Soton 2009 28

Dipole Cross-section.

1−z, ~p

z,−~p

x,~k x,~k

1−z, ~p

z,−~p

x,~k x,~k

Within LO kT -factorization theory can write γ?p cross-section (Bialas, Navelet,Peschanski) as

σL ∝

∫ 1

0

dz[z(1 − z)]2∫

d2k

k4

∫

d2p

(

1

Q2 + p2−

1

Q2 + (p + k)2

)2

f(x, k2)

where f(x, k2) is the unintegrated gluon distribution, Q2 = z(1 − z)Q2.

Includes some of the resummation effects, and also higher twist – different contributionsto renormalon calculation.

Misses quark and higher-x contributions. Systematic expansion difficult, and universalfactorisation less clear.

Soton 2009 29

x

-510 -410

-310 -210 -110 1

)2(x

,QL

F

-0.1

0

0.1

0.2

0.3

0.4

0.5

2 = 2 GeV2Q

x

-510 -410

-310 -210 -110 1

)2(x

,QL

F

-0.1

0

0.1

0.2

0.3

0.4

0.5

MSTW 2008 LO

MSTW 2008 NLO

MSTW 2008 NNLO

WT NLO+NLL(1/x)

Dipole (b-Sat)

x

-510 -410

-310 -210 -110 1

)2(x

,QL

F

0

0.1

0.2

0.3

0.4

0.5

2 = 5 GeV2Q

x

-510 -410

-310 -210 -110 1

)2(x

,QL

F

0

0.1

0.2

0.3

0.4

0.5

x

-510 -410

-310 -210 -110 1

)2(x

,QL

F

0

0.1

0.2

0.3

0.4

0.5

2 = 20 GeV2Q

x

-510 -410

-310 -210 -110 1

)2(x

,QL

F

0

0.1

0.2

0.3

0.4

0.5

x

-510 -410

-310 -210 -110 1

)2(x

,QL

F

0

0.1

0.2

0.3

0.4

0.5

2 = 100 GeV2Q

x

-510 -410

-310 -210 -110 1

)2(x

,QL

F

0

0.1

0.2

0.3

0.4

0.5

Overall predicted dipole FL(x,Q2)steeper at small x than fixed order,and automatically stable at lowestQ2.

Similar to fixed order predictions athigh Q2 and lower Q2 in probed xrange at HERA, but bigger at low xand Q2.

Would be good to make predictionsfor other quantities. Developmentsin these directions, e.g. calculationof impact factor for Drell-Yanproduction (Marzani, Ball).

Soton 2009 30

Look at variations in additional predictions for HERA range of measurement. Usex = Q2/35420.

)2 (GeV2Q10 210

)2(x

,QL

F

-0.2

0

0.2

0.4

0.6

0.8

1 H1 dataMSTW 2008 LOMSTW 2008 NLOMSTW 2008 NNLOWT NLO+NLL(1/x)Dipole (b-Sat)

x =

0.00

028

x =

0.00

037

x =

0.00

049

x =

0.00

062

x =

0.00

093

x =

0.00

14

x =

0.00

22

x =

0.00

36

1.28)2/GeV2(Q

-510×x = 1.1

Extrapolate using

)2 (GeV2Q10 210

)2(x

,QL

F

-0.2

0

0.2

0.4

0.6

0.8

1

Resummations and dipole model different shape to NLO and NNLO. Possible todistinguish former at lower Q2 perhaps. Is measurement accurate enough at higherQ2?

Soton 2009 31

Conclusions

One can extract parton distribution functions and make predictions for hadron collidersat LO, NLO and NNLO in αS and the framework appears to work well (best at NNLO).

Some indications of problems. dF2/d ln Q2 struggles a little to be large enough forx ∼ 0.001 and poor stability of gluon, and strange shape at low x and Q2.

Some predictions, particularly when sensitive to low x and lower scales are unstable.

Can perform resummation of ln(1/x) logarithms for structure function based on NLOBFKL equation – running coupling effects vital. Calculate splitting functions andcoefficient functions – inputs more challenging problem.

Quantitatively successful results in fit quality and form of gluon. Possibility of somereasonable test from HERA measurement of FL(x, Q2).

More detailed work required for predictions with these corrections at LHC. Changesnot likely to be huge, but may be bigger than quoted accuracy at fixed order. Couldaffect truly quantitative understanding of LHC physics.

Soton 2009 32