parton distribution functions at the lhc · 2013-02-27 · obtaining pdf sets – general...

Parton Distribution Functions at the LHC

Robert Thorne

February 15th, 2013

University College London

IPPP Research Associate

With contributions from Alan Martin, James Stirling, Graeme Watt

and Arnold Mathijssen and Ben Watt

UCL – February 2013

UCL – February 2013 1

Obtaining PDF sets – General procedure.

Start parton evolution at low scale Q20 ∼ 1GeV2. In principle 11 different

partons to consider.

u, u, d, d, s, s, c, c, b, b, g

mc,mb � ΛQCD so heavy parton distributions determinedperturbatively. Leaves 7 independent combinations, or 6 if we assumes = s (just started not to).

uV = u− u, dV = d− d, sea = 2 ∗ (u + d + s), s + s d− u, g.

Input partons parameterised as, e.g. MSTW,

xf(x,Q20) = (1− x)η(1 + εx0.5 + γx)xδ.

Evolve partons upwards using LO, NLO (or increasingly NNLO) DGLAPequations.

dfi(x,Q2,αs(Q2))

d ln Q2 =∑

j Pij(x, αs(Q2))⊗ fj(x,Q2, αs(Q2))


Fit data above ∼ 2GeV2. Need many types for full determination.

- Lepton-proton collider HERA – (DIS) → small-x quarks (best belowx ∼ 0.05). Also gluons from evolution (same x), and now FL(x,Q2).Also, jets → moderate-x gluon.Charged current data some limited infoon flavour separation. Heavy flavour structure functions – gluon andcharm, bottom distributions and masses.

- Fixed target DIS – higher x – leptons (BCDMS, NMC, . . .) → up quark(proton) or down quark (deuterium) and neutrinos (CHORUS, NuTeV,CCFR) → valence or singlet combinations.

- Di-muon production in neutrino DIS – strange quarks and neutrino-antineutrino comparison → asymmetry . Only for x > 0.01.

- Drell-Yan production of dileptons – quark-antiquark annihilation (E605,E866) – high-x sea quarks. Deuterium target – u/d asymmetry.

- High-pT jets at colliders (Tevatron) – high-x gluon distribution – x >0.01.

- W and Z production at colliders (Tevatron/LHC) – different quarkcontributions to DIS.


This procedure is generally successful and is part of a large-scale,ongoing project. Results in partons of the form shown.

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.)

Various choices of PDF – MSTW, CTEQ, NNPDF, AB(K)M, HERA,Jimenez-Delgado et al etc.. All LHC cross-sections rely on ourunderstanding of these partons.


|y|0 0.5 1 1.5 2 2.5 3

/dy

σ dσ

1/

0

0.1

0.2

0.3

Run II∅* rapidity shape distribution from DγZ/

MRST2006 NNLO PDFs

LO Vrap

NLO Vrap

NNLO Vrap

Run II∅* rapidity shape distribution from DγZ/Leads to very accurateand precise predictions.

Comparison of MSTWprediction for Z rapiditydistribution with data.


Parton Fits and Uncertainties. Two main approaches.

Parton parameterization and Hessian (Error Matrix) approach first usedby H1 and ZEUS, and extended by CTEQ.

χ2 − χ2min ≡ ∆χ2 =

∑i,j Hij(ai − a

(0)i )(aj − a

(0)j )

The Hessian matrix H is related to the covariance matrix of theparameters by

Cij(a) = ∆χ2(H−1)ij.

We can then use the standard formula for linear error propagation.

(∆F )2 = ∆χ2∑

i,j∂F∂ai

(H)−1ij

∂F∂aj

,

This is the most common approach.


Can find and rescale eigenvectors of H leading to diagonal form

∆χ2 =∑

i z2i

Implemented by CTEQ, then MRST/MSTW, HERAPDF. Uncertainty onphysical quantity then given by

(∆F )2 =∑

i

(F (S(+)

i )− F (S(−)i )

)2,

where S(+)i and S

(−)i are PDF “error sets” displaced along eigenvector

direction.

Must choose “correct” ∆χ2 given complication of errors in full fit andsometimes conflicting data sets.


Neural Network group (Ball et al.) limit parameterization dependence.Leads to alternative approach to “best fit” and uncertainties.

First part of approach, no longer perturb about best fit.

Where r(k)p are random numbers following Gaussian distribution. Hence,

include information about measurements and errors in distribution ofO

art,(k)i,p .

Fit to the data replicas obtaining PDF replicas q(net)(k)i (follows Giele et

al.)

Mean µO and deviation σO of observable O then given by

µO = 1Nrep

∑Nrep1 O[q(net)(k)

i ], σ2O = 1

Nrep

∑Nrep1 (O[q(net)(k)

i ]− µO)2.

Eliminates parameterisation dependence by using a neural net whichundergoes a series of (mutations via genetic algorithm) to find the bestfit. In effect is a much larger sets of parameters – ∼ 37 per distribution.


x-510 -410 -310 -210 -110

Rat

io t

o M

ST

W 2

008

NL

O

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

2 GeV4 = 102Up valence distribution at Q

MSTW 2008 NLO (68% C.L.)Random params. (40)Random params. (40)Random PDFs (1000)

x-510 -410 -310 -210 -110

Rat

io t

o M

ST

W 2

008

NL

O

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

x-510 -410 -310 -210 -110

Rat

io t

o M

ST

W 2

008

NL

O

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

2 GeV4 = 102Down valence distribution at Q


x-510 -410 -310 -210 -110

Rat

io t

o M

ST

W 2

008

NL

O

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

Showed equivalence of replicaapproach to Hessian study witheigenvectors for given ∆χ2.

However, can generate “random”PDF sets directly from parametersand variation from eigenvectors.

x-510 -410 -310 -210 -110

Rat

io t

o M

ST

W 2

008

NL

O

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

2 GeV4 = 102Up antiquark distribution at Q


x-510 -410 -310 -210 -110

Rat

io t

o M

ST

W 2

008

NL

O

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

x-510 -410 -310 -210 -110

Rat

io t

o M

ST

W 2

008

NL

O

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

2 GeV4 = 102Down antiquark distribution at Q


x-510 -410 -310 -210 -110

Rat

io t

o M

ST

W 2

008

NL

O

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

ai(Sk)=a0i +∑n

j=1eij

(±t±j

)|Rjk|

(k = 1, . . . , Npdf). Or fromeigenvectors directly (see LHCbstudy and De Lorenzi thesis).Far quicker.

x-510 -410 -310 -210 -110

Rat

io t

o M

ST

W 2

008

NL

O

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

2 GeV4 = 102Strange quark distribution at Q


x-510 -410 -310 -210 -110

Rat

io t

o M

ST

W 2

008

NL

O

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

x-510 -410 -310 -210 -110

Rat

io t

o M

ST

W 2

008

NL

O

0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

2 GeV4 = 102Gluon distribution at Q


x-510 -410 -310 -210 -110

Rat

io t

o M

ST

W 2

008

NL

O

0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

F (Sk)=F (S0)+∑

j

[F (S±j )−F (S0)

]|Rjk|

Can use in reweighting studiesas NNPDF, i.e. weight “random”set by χ2 of fit to new data set(s)to get new distribution.


Speed of convergence of prediction for Z cross section.

pdfNumber of random predictions, N

10 210 310

Rat

io to

bes

t-fit

pre

dict

ion

0.94

0.96

0.98

1

1.02

1.04

1.06


predictionspdf

Average and s.d. over N

= 7 TeV)s cross section at the LHC (0NLO Z

pdfSame random numbers for each value of N


10 210 310

Rat

io to

bes

t-fit

pre

dict

ion

0.94

0.96

0.98

1

1.02

1.04

1.06


predictionspdf


= 7 TeV)s cross section at the LHC (0NLO Z

pdfDifferent random numbers for each value of N

Left, add a new random set to existing ones sequentially. Right,increasing numbers of independent random sets.

Very good with 40 sets. Excellent with 100 sets.


Speed of convergence of prediction for W+/W− cross section ratio.


10 210 310

Rat

io to

bes

t-fit

pre

dict

ion

0.97

0.98

0.99

1

1.01

1.02

1.03


predictionspdf


= 7 TeV)s cross-section ratio at the LHC (-/W+NLO W



10 210 310

Rat

io to

bes

t-fit

pre

dict

ion

0.97

0.98

0.99

1

1.01

1.02

1.03


predictionspdf


= 7 TeV)s cross-section ratio at the LHC (-/W+NLO W





Speed of convergence of prediction for tt cross section.


10 210 310

Rat

io to

bes

t-fit

pre

dict

ion

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1


predictionspdf


= 7 TeV)s cross section at the LHC (tNLO t



10 210 310

Rat

io to

bes

t-fit

pre

dict

ion

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1


predictionspdf


= 7 TeV)s cross section at the LHC (tNLO t





Speed of convergence of prediction for H cross section.


10 210 310

Rat

io to

bes

t-fit

pre

dict

ion

0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05


predictionspdf


= 120 GeVH

= 7 TeV) for MsH at the LHC (→NLO gg



10 210 310

Rat

io to

bes

t-fit

pre

dict

ion

0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05


predictionspdf


= 120 GeVH

= 7 TeV) for MsH at the LHC (→NLO gg





Can combine PDF sets, e.g. comparison to PDF4LHC prescription.

) (

nb

)- l+ l

→ 0 B

(Z⋅ 0

Zσ

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1 = 7 TeV)s at the LHC (-l+ l→ 0NNLO Z

MSTW08) = 0.11712

Z(MSα

CT10) = 0.11802

Z(MSα

NNPDF2.3) = 0.11902

Z(MSα

Closed markers: average and s.d. over random predictions.Open markers: usual best-fit and 68% C.L. Hessian uncertainty.

Solid lines: envelope and midpoint.Dashed lines: statistical combination.

G. W

att

(

No

vem

ber

201

2)

-Wσ

/ +Wσ ≡ ±

R

1.4

1.41

1.42

1.43

1.44

1.45

1.46

1.47

1.48

1.49

1.5 = 7 TeV)s ratio at the LHC (-/W+NNLO W

MSTW08) = 0.11712

Z(MSα

CT10) = 0.11802

Z(MSα

NNPDF2.3) = 0.11902

Z(MSα



G. W

att

(

No

vem

ber

201

2)

(p

b)

ttσ

145

150

155

160

165

= 7 TeV)s cross sections at the LHC (tNNLO+NNLL t

MSTW08) = 0.11712

Z(MSα

CT10) = 0.11802

Z(MSα

NNPDF2.3) = 0.11902

Z(MSα



= 173.18 GeVpolet = m

Fµ =

RµTop++ (v1.4),

G. W

att

(

No

vem

ber

201

2)

(p

b)

Hσ

18.5

19

19.5

20

20.5

21

21.5

22

= 126 GeVH

= 8 TeV) for MsH at the LHC (→NNLO gg

MSTW08) = 0.11712

Z(MSα

CT10) = 0.11802

Z(MSα

NNPDF2.3) = 0.11902

Z(MSα



/ 2H = MF

µ = R

µggh@nnlo (v1.4.1),

G. W

att

(

No

vem

ber

201

2)

Smaller uncertainty and shifted central value if disagreement betweenindividual predictions. (Plots by G. Watt at http://mstwpdf.hepforge.org/random/).


Comparison to LHC data. (arXiv:1211.1215, A.D. Martin et. al.)

Start with ATLAS jets. Comparison clearly fine.


Being more quantitative Using APPLGrid or FastNLO (B. Watt).

MSTW fit very good, see χ2 per point above. Always close to, andsometimes the best of any PDF set, particularly for R = 0.6. Not a hugevariation in PDFs across groups though.

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

0.0001 0.001 0.01 0.1

Rat

io to

MS

TW

Cen

tral

Val

ue

x

g(x) at q2=10000 (GeV)2

Before Reweighting (Npdf=1000)After Reweighting (Neff=774)Can see effect of data

on the gluon by usingthe reweighting technique,R = 0.6 (R = 0.4similar). Clearly little pullwith present data.


Comparison of MSTW2008 to total W,Z excellent.

Ratio (CMS/Theory)0.6 0.8 1 1.2 1.4

= 7 TeVs at -136 pbCMS

B ( W )× σ th. 0.028± exp. 0.009±0.987

)+ B ( W× σ th. 0.030± exp. 0.009±0.982

)- B ( W× σ th. 0.029± exp. 0.010±0.993

B ( Z )× σ th. 0.032± exp. 0.010±1.002

W/ZR th. 0.015± exp. 0.010±0.981

+/-R th. 0.023± exp. 0.011±0.990

Ratio (CMS/Theory)0.6 0.8 1 1.2 1.4

lumi. uncertainty: 4%±


Comparisons between different sets for Inclusive predictions.

) (nb)ν+ l→ + B(W⋅ +Wσ

5.6 5.8 6 6.2 6.4 6.6

) (

nb

)ν- l

→ - B

(W⋅ -

Wσ

3.9

4

4.1

4.2

4.3

4.4

4.5

= 7 TeV)s cross sections at the LHC (- and W+NNLO W

uncertaintiesSαPDF+

-1CMS, L = 36 pb

-1ATLAS, L = 33-36 pb

) = -/W+R(W 1.40

1.45

1.50

G. W

att

(

No

vem

ber

201

2)

68% C.L. PDF

MSTW08

CT10

NNPDF2.3 noLHC

NNPDF2.3

HERAPDF1.5

ABM11

JR09

) (nb)ν± l→ ± B(W⋅ ±Wσ

9.8 10 10.2 10.4 10.6 10.8 11

) (

nb

)- l+ l

→ 0 B

(Z⋅ 0 Zσ

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

= 7 TeV)sNNLO W and Z cross sections at the LHC (

uncertaintiesSαPDF+

-1CMS, L = 36 pb

-1ATLAS, L = 33-36 pb

R(W/Z) =

10.810.9

11.0

G. W

att

(

No

vem

ber

201

2)

68% C.L. PDF

MSTW08

CT10

NNPDF2.3 noLHC

NNPDF2.3

HERAPDF1.5

ABM11

JR09

(Plots by G. Watt at http://mstwpdf.hepforge.org/pdf4lhc/AR/).


)2Z

(MSα0.112 0.113 0.114 0.115 0.116 0.117 0.118 0.119 0.12

(p

b)

ttσ

100

120

140

160

180

200

220

2012PDG

(prel., Sept. 2012)

-1L = 0.7-1.1 fbLHC combined,

= 7 TeV)s cross sections at the LHC (tNNLO+NNLL t

= 173.18 GeVpolet = m

Fµ =

RµTop++ (v1.4),

G. W

att

(

No

vem

ber

201

2)

68% C.L. PDF

MSTW08

CT10

NNPDF2.3 noLHC

NNPDF2.3

HERAPDF1.5

ABM11

JR09

)2Z

(MSα0.112 0.113 0.114 0.115 0.116 0.117 0.118 0.119 0.12

(p

b)

Hσ

17

17.5

18

18.5

19

19.5

20

20.5

21

21.5

22

2012PDG

= 126 GeVH

= 8 TeV) for MsH at the LHC (→NNLO gg

/ 2H = MF

µ = R

µggh@nnlo (v1.4.1),

G. W

att

(

No

vem

ber

201

2)

68% C.L. PDF

MSTW08

CT10

NNPDF2.3 noLHC

NNPDF2.3

HERAPDF1.5

ABM11

JR09

Note variety of values of αS(M2Z) used by different groups.


Correlates with PDF luminosity plots

(GeV)s10 100 1000

Rat

io t

o M

ST

W 2

008

(68%

C.L

.)

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2 = 8 TeV)s) luminosity at LHC (q(qqΣNNLO

WM ZM

MSTW 2008

CT10

NNPDF2.3 noLHC

NNPDF2.3

G. W

att

(

No

vem

ber

201

2)

(GeV)s10 100 1000

Rat

io t

o M

ST

W 2

008

(68%

C.L

.)

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2 = 8 TeV)s) luminosity at LHC (q(qqΣNNLO

WM ZM

MSTW 2008

HERAPDF1.5

ABM11

JR09

G. W

att

(

No

vem

ber

201

2)


(GeV)s10 100 1000

Rat

io t

o M

ST

W 2

008

(68%

C.L

.)

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2 = 8 TeV)sNNLO gg luminosity at LHC (

top2mHiggsM

MSTW 2008

CT10

NNPDF2.3 noLHC

NNPDF2.3

G. W

att

(

No

vem

ber

201

2)

(GeV)s10 100 1000

Rat

io t

o M

ST

W 2

008

(68%

C.L

.)

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2 = 8 TeV)sNNLO gg luminosity at LHC (

top2mHiggsM

MSTW 2008

HERAPDF1.5

ABM11

JR09

G. W

att

(

No

vem

ber

201

2)

Some pretty clear difference between some of the groups. More later.


MSTW also pretty good for inclusive distributions. Except someproblems with asymmetry. Comparison gives χ2/Npts = 60/30

|Z

|y0 0.5 1 1.5 2 2.5 3 3.5

The

ory/

Dat

a

0.91

1.10 0.5 1 1.5 2 2.5 3 3.5

| [pb

]Z

/d|y

σd

20

40

60

80

100

120

140

160

= 7 TeV)sData 2010 (

MSTW08

HERAPDF1.5

ABKM09

JR09

-1 L dt = 33-36 pb∫ -l+ l→Z

Uncorr. uncertainty

Total uncertainty

ATLAS

|l

η|0 0.5 1 1.5 2 2.5

The

ory/

Dat

a

0.91

1.10 0.5 1 1.5 2 2.5

| [pb

]lη

/d|

σd

300

400

500

600

700

800


MSTW08

HERAPDF1.5ABKM09

JR09

-1 L dt = 33-36 pb∫ lν+ l→+W

Uncorr. uncertainty

Total uncertainty

ATLAS

|l

η|0 0.5 1 1.5 2 2.5

The

ory/

Dat

a

0.91

1.10 0.5 1 1.5 2 2.5

| [pb

]lη

/d|

σd

100

200

300

400

500

600


MSTW08

HERAPDF1.5ABKM09

JR09

-1 L dt = 33-36 pb∫ lν- l→-W

Uncorr. uncertainty

Total uncertainty

ATLAS

|l

η|0 0.5 1 1.5 2 2.5

lA

0.1

0.15

0.2

0.25

0.3

0.35 = 7 TeV)sData 2010 (

MSTW08HERAPDF1.5ABKM09JR09

-1 L dt = 33-36 pb∫

Stat. uncertainty

Total uncertainty

ATLAS


MSTW pretty good for ATLAS W,Z distributions, except some problemswith asymmetry.

|ηElectron Pseudorapidity |0 1 2

Ele

ctro

n C

harg

e A

sym

met

ry

0.05

0.1

0.15

0.2

0.25

CMS = 7 TeVs at -1840 pb

ν e→W

MCFM:

theory bands: 68% CL

CT10 HERAPDF1.5 MSTW2008NLO NNPDF2.2 (NLO)

(e) > 35 GeVT

p

Similar for CMS data (will return to this later), though depends on pT

cut. Generally very good for LHCb.

µη0 1 2 3 4

µA

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

= 7 TeVsLHCb,

statData MSTW08

totData ABKM09

JR09

NNPDF21

HERA15

CTEQ6M (NLO)

> 20 GeV/cµT

p


Asymmetry used (G. Watt, R.T.) in reweighting (JHEP 1208 (2012) 052),and moves uV −dV up near x = 0.01 - where parameterisation perhapsunderestimates uncertainty. (ATLAS left, CMS pT > 25GeV right).

|l

ηLepton pseudorapidity, |0 0.5 1 1.5 2 2.5 3

Lep

ton

ch

arg

e as

ymm

etry

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3-1), L = 33-36 pbµATLAS (l = e+

MSTW08 NLO PDFs (68% C.L.)

/11 = 2.712χHessian,

/11 = 2.902χ = 1000, pdfN

/11 = 1.272χ = 264, effN

|l


Lep

ton

ch

arg

e as

ymm

etry

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3 > 40 GeV

T > 25 GeV, M

νTE > 20 GeV, l

T = 7 TeV) with ps at the LHC (ν± l→ ±W

|l


Lep

ton

ch

arg

e as

ymm

etry

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

-1CMS (l = e), L = 36 pb-1), L = 36 pbµCMS (l =

MSTW08 NLO PDFs (68% C.L.)

/12 = 1.742χHessian,

/12 = 2.012χ = 1000, pdfN

/12 = 1.202χ = 485, effN

|l


Lep

ton

ch

arg

e as

ymm

etry

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

> 25 GeVl


x-410 -310 -210 -110

Rat

io t

o M

ST

W 2

008

NL

O

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

2 GeV4 = 102 distribution at Qv - dvu

MSTW08 (68% C.L.)

= 1000)pdf

MSTW08 (N

= 264)eff

MSTW08 (N

x-410 -310 -210 -110

Rat

io t

o M

ST

W 2

008

NL

O

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

x-410 -310 -210 -110

Rat

io t

o M

ST

W 2

008

NL

O

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

2 GeV4 = 102 distribution at Qv - dvu

MSTW08 (68% C.L.)

= 1000)pdf

MSTW08 (N

= 485)eff

MSTW08 (N

x-410 -310 -210 -110

Rat

io t

o M

ST

W 2

008

NL

O

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2


Investigation of Parameterisation Issues

In the light of Monte Carlo studies investigate parameterisationdependence, initially concentrating on valence quarks.

Decide to use Chebyshev polynomials (looked at other possibilities)

xf(x,Q20) = A(1− x)ηxδ(1 +

∑n

anTn(y))

i.e. keep high and low x limits. Choose y = 1− 2√

x.

0.001 0.01 0.1 1x

-0.5

0.0

0.5

1.0

TiHyHxL � 1 - 2 xL

0.001 0.01 0.1 1x

-0.5

0.0

0.5

1.0

TiJyHxL � 1 - 2 x N

0.001 0.01 0.1 1x

-0.5

0.0

0.5

1.0

TiIyHxL � 1 - 2 x0.25M

0.001 0.01 0.1 1x

-0.5

0.0

0.5

1.0

TiHyHxL � cosHΠ xLL

Same choice as in Pumplin study. Slightly different to Glazov, Moch andRadescu.


It is an open questionhow many parametersare required to accuratelyfit a simple shape suchas that for xuV (x,Q2) orxdV (x,Q2).

Investigate by generatingpseudo-data for varioustypes of parton distributionusing a function withvery many parameters,then fitting with smallernumbers.

Fairly similar conclusionindependent of type ofparameterisation.


Example, fit to pseudo-data for valence quark generated between x =0.01 and x = 0.7. For precise match to pseudo-data need 4 polynomials.

0.001 0.005 0.010 0.050 0.100 0.500 1.000x

0.001

0.01

0.1

Fractional deviation

0.001 0.005 0.010 0.050 0.100 0.500 1.000x

0.001

0.01

0.1

Fractional deviation

Look at χ2 distribution with increasing terms in polynomial.

0.02 0.05 0.10 0.20 0.50 1.00

0.001

0.1

10

0.02 0.05 0.10 0.20 0.50 1.00

0.01

0.1

1

10

0.02 0.05 0.10 0.20 0.50 1.00

10-4

0.01

1

0.02 0.05 0.10 0.20 0.50 1.00

10-5

0.001

0.1

10

Very good fits with 4 parameters. More tends to give over-fitting andpeculiarities outside of range of x fit.


0.8

1

1.2

10-4

10-3

10-2

10-1

MSTWMSTWCPvMSTWCP

x

ratio of uV(x,Q2=1GeV2)

0.8

1

1.2

10-4

10-3

10-2

10-1

MSTWMSTWCPvMSTWCP

x


0.8

1

1.2

10-4

10-3

10-2

10-1

MSTWMSTWCPvMSTWCP

x

ratio of dV(x,Q2=1GeV2)

0.8

1

1.2

10-4

10-3

10-2

10-1

MSTWMSTWCPvMSTWCP

x


Fits to same data as MSTW2008with 2 extra parameters forvalence quarks – 28 → 32params.

Also with 2 extra parametersfor valence quarks and sea –28 → 34 params. No extraparameters in the gluon foundto be necessary.

0.9

1

1.1

10-4

10-3

10-2

10-1

MSTWMSTWCPvMSTWCP

x

ratio of sea(x,Q2=1GeV2)

0.9

1

1.1

10-4

10-3

10-2

10-1

MSTWMSTWCPvMSTWCP

x

ratio of sea(x,Q2=10000GeV2)

0.9

1

1.1

10-4

10-3

10-2

10-1

MSTWMSTWCPvMSTWCP

x

ratio of g(x,Q2=5GeV2)

0.9

1

1.1

10-4

10-3

10-2

10-1

MSTWMSTWCPvMSTWCP

x

ratio of g(x,Q2=10000GeV2)

∆χ2 = −29, MSTW2008CPpdfs. Only real change in small-x uV , and sea at low Q2.


0.8

1

1.2

10-4

10-3

10-2

10-1

MSTWMSTWCPMSTWCPdeut

x


0.8

1

1.2

10-4

10-3

10-2

10-1


x


0.8

1

1.2

10-4

10-3

10-2

10-1


x


0.8

1

1.2

10-4

10-3

10-2

10-1


x


Also do the same fits witha freedom in the deuteriumcorrections – more stablethan previous MSTW studies.MSTW2008CPdeut PDFs.

Now also get variation in dV (x) for higher x due to deuterium correction(seen before) and x ≤ 0.03 due to parameterization and corrections.

Fit to ATLAS W,Z rapidity data at NLO improves to 49/30 for MSTWCPand 46/30 for MSTWCPdeut.


0

0.1

0.2

0.3

10-4

10-3

10-2

10-1

MSTW

MSTWCP

MSTWCPdeut

x

x(uV-dV)(x,Q2=10000GeV2)

-50

0

50

10-4

10-3

10-2

10-1


x

percentage difference at Q2=10000GeV2

Shown is change in central valueand uncertainty for uV (x)−dV (x)at Q2 = 10, 000 GeV2.

Biggest effect at lower x thanprobed at the LHC (yet).

Uncertainty sets have 23 eigenvectors(20 in MSTW2008).

Main effect in uncertainty anincrease in dV (x,Q2) due todeuterium correction uncertainties,and minor valence uncertaintyincrease from extra parameter.


0 1 2 30.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

35

lepton asymmetry, variable p

Tl(min)

MSTW2008 MSTW2008CP MSTW2008CPdeut

0

20

10

30

A(y

l)

yl

LHC 7 TeVMSTW2008 NLO

Increases lepton asymmetry,but very preferentially forhigh pT cut. (Curves madehere with LO calculations).

Most of the effect alreadyobtained for parameterisationextension, but some fromdeuterium study.


|l


Lept

on c

harg

e as

ymm

etry

0

0.05

0.1

0.15

0.2

0.25

0.3

-1CMS (l = e), L = 840 pb

NLO PDFs (with NLO K-factors)

/11 = 5.342χMSTW08 (68% C.L.),

/11 = 1.532χMSTW08CP,

/11 = 0.952χMSTW08CPdeut,

> 35 GeVl


Prediction for pT > 35GeV CMS asymmetry data.

Note no change to data fit, just parameterisation and some fromdeuterium corrections.

Main deuterium effect absence of shadowing used in default fit.


|l


Lept

on c

harg

e as

ymm

etry

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3-1), L = 33-36 pbµATLAS (l = e+

NLO PDFs (with NLO K-factors)

/11 = 2.712χMSTW08 (68% C.L.),

/11 = 1.382χMSTW08CP,

/11 = 0.862χMSTW08CPdeut,

> 40 GeVT

> 25 GeV, MνTE > 20 GeV, l


Prediction for ATLAS asymmetry data.

Note no change to data fit, just parameterisation and some fromdeuterium corrections.


(GeV)s10 100 1000

Rat

io t

o M

ST

W 2

008

(68%

C.L

.)

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1 = 8 TeV)s) luminosity at LHC (q(qqΣNLO

WM ZM

MSTW08

MSTW08CP

MSTW08CPdeut

Big change in high-pT cut asymmetry,but this is very specifically sensitive touV (x,Q2)− dV (x,Q2).

What about other quantities?

(GeV)s10 100 1000

Rat

io t

o M

ST

W 2

008

(68%

C.L

.)

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1 = 8 TeV)sNLO gg luminosity at LHC (

top2mHiggsM

MSTW08

MSTW08CP

MSTW08CPdeut

Other PDFs changed little. αS free buttiny change. Expect little variation.

Seen clearly on luminosity plots (G.Watt).

(GeV)s10 100 1000

Rat

io t

o M

ST

W 2

008

(68%

C.L

.)

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

)q(q+q

Σ g + 4/9 ≡ = 8 TeV), G sNLO GG luminosity at LHC (

MSTW08

MSTW08CP

MSTW08CPdeut

A not totally insignificant change inthe high-x gluon luminosity for theMSTWCPdeut set . Due to softer dV


The percentage change ofvarious cross sections dueto the modifications of theMSTW2008 PDFs. In orderto demonstrate the smallchanges in the cross sections,we also show, in the finalcolumn, the symmetrizedPDF + αS(M2

Z) percentageuncertainties for MSTW2008PDFs.


6 7 8 9 10 11 12 13 14 15-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

_t t

gg→H(125)

Z0

W −

W +

δσ/σ

(

%)

√s (TeV)

% change in total (NLO) LHC cross sectionsfrom MSTW2008 to MSTW2008CPdeut

Can see variation of crosssections with energy.


Choices for Heavy Flavours in DIS. (Extension of work in by RT inPhys.Rev. D86 (2012) 074017.)

Near threshold Q2 ∼ m2H massive quarks not partons. Created in final

state.

Described using Fixed Flavour Number Scheme (FFNS).

F (x,Q2) = CFF,nf

k (Q2/m2H)⊗ f

nf

k (Q2)

Does not sum αnS lnn Q2/m2

H terms in perturbative expansion. Usuallyachieved by definition of heavy flavour parton distributions and solutionof evolution equations.

Additional problem FFNS known up to NLO (Laenen et al.), but are notfully known at NNLO – α3

SCFF,32,Hi unknown.

Approximations based on some or all of threshold, low-x and high-Q2

limits can be derived, see Kawamura, et al.,, and are sometimes usedin fits, e.g. ABM11 and MSTW (at low Q2). Generally not large exceptat threshold and very low x.


Variable Flavour - at high scales Q2 � m2H heavy quarks behave like

massless partons. Sum ln(Q2/m2H) terms via evolution. Zero Mass

Variable Flavour Number Scheme (ZM-VFNS). Ignores O(m2H/Q2)

corrections.

F (x,Q2) = CZM,nf

j ⊗ fnf

j (Q2).

Partons in different number regions related to each other perturbatively.

fnf+1

j (Q2) = Ajk(Q2/m2H)⊗ f

nf

k (Q2),

Perturbative matrix elements Ajk(Q2/m2H) (Buza et al.) containing

ln(Q2/m2H) terms relate f

nf

i (Q2) and fnf+1

i (Q2) → correct evolution forboth.

Want a General-Mass Variable Flavour Number Scheme (VFNS)taking one from the two well-defined limits of Q2 ≤ m2

H and Q2 � m2H.


The GM-VFNS can be defined by demanding equivalence of the nf

light flavour and nf + 1 light flavour descriptions at all orders – abovetransition point nf → nf + 1

F (x,Q2)=CFF,nf

k (Q2/m2H)⊗f

nf

k (Q2)=CV F,nf+1

j (Q2/m2H)⊗f

nf+1

j (Q2)

≡ CV F,nf+1

j (Q2/m2H)⊗Ajk(Q2/m2

H)⊗ fnf

k (Q2).

Hence, the VFNS coefficient functions satisfy

CFF,nf

k (Q2/m2H) = C

V F,nf+1


H),

which at O(αS) gives

CFF,nf ,(1)

2,Hg ( Q2

m2H

) = CV F,nf+1,(0)

2,HH ( Q2

m2H

)⊗P 0qg ln(Q2/m2

H)+CV F,nf+1,(1)

2,Hg ( Q2

m2H

),

The VFNS coefficient functions tend to massless limits as Q2/m2H →∞.

However, CV Fj (Q2/m2

H) only uniquely defined in this limit.

Can swapO(m2H/Q2) terms between CV F,0

2,HH(Q2/m2H) and CV F,1

2,g (Q2/m2H),

i.e. get scheme variations.


Difference between FFNS and GM-VFNS

0

1

1 10 102

103

MSTW08MSTW08FFNS

NLO

x=0.0005

Q2

Fc 2(x,

Q2 )

0

0.25

0.5

1 10 102

103

MSTW08

MSTW08FFNS

x=0.005

Q2

Fc 2(x,

Q2 )

0

0.05

1 10 102

103

MSTW08

MSTW08FFNS

x=0.05

Q2

Fc 2(x,

Q2 )

0

1

1 10 102

103

MSTW08MSTW08FFNS

NNLO

x=0.0005

Q2

Fc 2(x,

Q2 )

0

0.25

0.5

1 10 102

103

MSTW08

MSTW08FFNS

x=0.005

Q2

Fc 2(x,

Q2 )

0

0.05

1 10 102

103

MSTW08

MSTW08FFNS

x=0.05

Q2Fc 2(

x,Q

2 )

As seen at higher Q2 charm structure function for FFNS nearly alwayslower than any GM-VFNS. NNLO uses O(α2

S) coefficient functions forF c

2 (x,Q2). Approx. O(α3S) raises slightly F c

2 (x,Q2) at higher x andlowers it at small x (except very small x at low Q2).


µ=100 GeV

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

10-4

10-3

10-2

10-1

x

evol

ved/

fixe

d-or

der

c(2)(Nf=5)

b(2)(Nf=5)

Σ(2)(Nf=5)

G(2)(Nf=5)

Results for F c2 (x,Q2) in GM-VFNS compared to those for FFNS similar

to results for PDFs by Alekhin et al. in Phys.Rev. D81 (2010) 014032comparing NNLO evolution to the fixed order result up to O(α2

S). Detailsdepend on PDF set and αS(M2

Z) value used.


0.96

1

1.04

10 102

103

104

MSTW08MSTW08FFNS

NNLO x=0.0005

Q2

F 2(x,

Q2 )/

MST

W08

0.955

1

1.045

10 102

103

104

x=0.005

Q2

F 2(x,

Q2 )/

MST

W08

0.96

1

1.04

10 102

103

104

x=0.05

Q2

F 2(x,

Q2 )/

MST

W08

Can lead to over 4% changes inthe total F2(x,Q2) if the sameinput PDFs are used in twoschemes.

At higher x mainly due toF c

2 (x,Q2).

At lower x there is a largecontribution from light quarksevolving slightly more slowly inFFNS.

At much higher x differencedies away. Charm componentbecomes very small and lightquark evolution not much different.(Light quarks slightly bigger atthe highest x.)


0

0.5

1

1.5

2

1 10 102

103

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 2008This is certainly important giventhe precision and range of thedata on F2(x,Q2) included inPDF fits.

Mainly from HERA, but NMCdata also relevant.


Performed a series of NLO fits using the FFNS scheme and NNLO withup to O(α2

S) heavy flavour coefficient functions.(Approximations to theO(α3

S) expressions change results very little).

Fit to only DIS and Drell-Yan data but also effectively fit to Tevatron Drell-Yan or Tevatron jet data, if necessary, in 5-flavour scheme as FFNScalculations do not exist.

Fits to DIS and Drell-Yan data usually at least a few tens of unitsworse than MSTW08 to same data (even without refitting MSTW08 torestricted data sets). Often slightly better for F c

2 (x,Q2), but flatter in Q2

for x ∼ 0.01 for inclusive structure function.

As well as (usually) a worse fit to DIS and Drell-Yan data only, in FFNSthe fit quality for the DIS and low-energy Drell Yan data deteriorates byin general ∼ 50 units when all jet data is included as opposed to < 10units when using a GM-VFNS.


PDFs evolved up to Q2 = 10, 000GeV2 (using variable flavour evolutionfor consistent comparison) different in form to MSTW08.

0.9

0.95

1

1.05

1.1

10-4

10-3

10-2

10-1

MSTW08NLO-0.1202FFNSDIS-0.1187FFNS+DY-0.1185FFNSjet-0.1222FFNSjetZ-0.1225FF

NS/

2008

at N

LO

for

g(x

,Q2 )

0.9

0.95

1

1.05

1.1

10-4

10-3

10-2

10-1

MSTW08NLOFFNSDISFFNS+DYFFNSjetFFNSjetZFF

NS/

2008

at N

LO

for

u(x

,Q2 )

0.9

0.95

1

1.05

1.1

10-4

10-3

10-2

10-1

MSTW08NNLO-0.1171FFNSDIS-0.1144FFNS+DY-0.1152FFNSjet-0.1181FFNSjetZ-0.1184FF

NS/

2008

at N

NL

O f

or g

(x,Q

2 )

0.9

0.95

1

1.05

1.1

10-4

10-3

10-2

10-1

MSTW08NNLOFFNSDISFFNS+DYFFNSjetFFNSjetZFF

NS/

2008

at N

NL

O f

or u

(x,Q

2 )

In contrast in standard MSTW2008 fit PDFs usually within uncertaintiesif Tevatron jet data left out. Main effect loss of tight constraint onαS(M2

Z).


0.9

0.95

1

1.05

1.1

10-5

10-4

10-3

10-2

10-1

MSTW08GMVFNS1GMVFNS2GMVFNS3

GMVFNS4GMVFNS5GMVFNS6ZMVFNS

GMVFNSopt

GM

VFN

Sa/2

008

at N

LO

for

g(x

,Q2 )

0.9

0.95

1

1.05

1.1

10-5

10-4

10-3

10-2

10-1

MSTW08GMVFNS1GMVFNS2GMVFNS3

GMVFNS4GMVFNS5GMVFNS6ZMVFNS

GMVFNSopt

GM

VFN

Sa/2

008

at N

LO

for

u(x

,Q2 )

0.9

0.95

1

1.05

1.1

10-5

10-4

10-3

10-2

10-1

MSTW08NNLOGMVFNS1GMVFNS2GMVFNS3

GMVFNS4GMVFNS5GMVFNS6GMVFNSopt

GM

VFN

Sa/2

008

at N

NL

O f

or g

(x,Q

2 )

0.9

0.95

1

1.05

1.1

10-5

10-4

10-3

10-2

10-1

MSTW08NNLOGMVFNS1GMVFNS2GMVFNS3

GMVFNS4GMVFNS5GMVFNS6GMVFNSopt

GM

VFN

Sa/2

008

at N

NL

O f

or u

(x,Q

2 )

Using FFNS leads to much larger changes than any choice of GM-VFNS mainly due to fitting high-Q2 DIS data.


Low Q2 – Higher Twist.

Potentially large corrections at low Q2 and particularly low W 2. UsualMSTW cuts for

Q2cut - 2GeV2

W 2cut - 15GeV2

Have tried raising Q2 cut to 5GeV2 and 10GeV2 and W 2 to 20GeV2. Notmuch effect on PDFs or αS.

Can also lower W 2cut to 5GeV2 and try parameterising higher twist

contributions by

FHTi (x,Q2) = FLT

i (x,Q2)(

1 +Di(x)Q2

)

where i spans bins of x from x = 0.8− 0.9 down to x = 0− 0.0005.

Previously no evidence for much higher twist except at low W 2.


0.9

0.95

1

1.05

1.1

10-4

10-3

10-2

10-1

MSTW08NLO-0.1202

FFNSDIS-0.1187MSTW08HT-0.1189

FFNSHT-0.1188

PDF/

MST

W08

at N

LO

for

g(x

,Q2 )

0.9

0.95

1

1.05

1.1

10-4

10-3

10-2

10-1

MSTW08NLO

FFNSDISMSTW08HT

FFNSHT

PDF/

MST

W08

at N

LO

for

u(x

,Q2 )

0.9

0.95

1

1.05

1.1

10-4

10-3

10-2

10-1

MSTW08NNLO-0.1171

FFNSDIS-0.1144MSTW08HT-0.1175

FFNSHT-0.1152

PDF/

MST

W08

at N

NL

O f

or g

(x,Q

2 )

0.9

0.95

1

1.05

1.1

10-4

10-3

10-2

10-1

MSTW08NNLO

FFNSDISMSTW08HT

FFNSHTPD

F/M

STW

08 a

t NN

LO

for

u(x

,Q2 )

Now more evidence for positive contribution also at very low x. Leads tolower input quarks, more gluon for evolution. Largely washes out quicklywith Q2. Similar effect using FFNS as for GM-VFNS.


0.9

0.95

1

1.05

1.1

10-4

10-3

10-2

10-1

MSTW08NLO-0.1202

FFNS+DYHT*-0.1165

FFNS+DY-0.1185

PDF/

MST

W08

at N

LO

for

g(x

,Q2 )

0.9

0.95

1

1.05

1.1

10-4

10-3

10-2

10-1

MSTW08NLO

FFNS+DYHT*

FFNS+DY

PDF/

MST

W20

08 a

t NL

O f

or u

(x,Q

2 )

0.9

0.95

1

1.05

1.1

10-4

10-3

10-2

10-1

MSTW08NNLO-0.1171

FFNS+DYHT*-0.1136

FFNS+DY-0.1152

PDF/

MST

W08

at N

LO

for

g(x

,Q2 )

0.9

0.95

1

1.05

1.1

10-4

10-3

10-2

10-1

MSTW08NNLO

FFNS+DYHT*

FFNS+DY

PDF/

MST

W20

08 a

t NN

LO

for

u(x

,Q2 )

Restricting higher twist from lowest x value and omitting nuclear targetdata (except dimuon for strangeness) tends to keep values of αS lowerby ∼ 0.02.


Explains some PDF differences? MSTW FFNS ratios and ABKM ratios.

0.9

0.95

1

1.05

1.1

10-4

10-3

10-2

10-1

MSTW08NLO-0.1202

ABKM09-0.1179

PDF/

MST

W08

at N

LO

for

g(x

,Q2 )

0.9

0.95

1

1.05

1.1

10-4

10-3

10-2

10-1

MSTW08NLO

ABKM09

PDF/

MST

W08

at N

LO

for

u(x

,Q2 )

0.9

0.95

1

1.05

1.1

10-4

10-3

10-2

10-1

MSTW08NNLO-0.1171

ABKM09-0.1135

PDF/

MST

W08

at N

NL

O f

or g

(x,Q

2 )0.9

0.95

1

1.05

1.1

10-4

10-3

10-2

10-1

MSTW08NNLO

ABKM09

PDF/

MST

W08

at N

NL

O f

or u

(x,Q

2 )

General trend is very similar to fits on previous page.


Conclusions

There is a reasonable amount of variation in comparisons todata/predictions at the LHC using different PDF sets.

The MSTWCP and MSTWCPdeut PDFs improve dramatically thepre(post)diction for lepton asymmetries from W bosons at the LHC.

This is due to a very localised effect in small-x valence quarkdifferences. There is extremely little change in parton luminosities orpredictions for a wide variety of other cross sections.

Lowering W 2cut and allowing for higher twist terms in structure functions

makes very little change to MSTW PDFs except at extremely high x

Performing fits using an FFNS leads to worse fits to DIS and low energyDrell-Yan data than GM-VFNS, and much more tension with jet data.

Light quarks in variable flavour number scheme are automatically largerin most regions for FFNS than for GM-VFNS. The gluon is smaller athigh x and larger at small x in FFNS, and αS(M2

Z) smaller. Seems tobe the likely explanation of some major differences in PDFs and LHCpredictions. (Similar conclusions in NNPDF studies.)


Back-up


Calculate χ2/N = 60/30 for ATLAS W,Z data again at NLO usingAPPLGrid. Not best, but fairly close to any other set except CT10 whichis best. Again look at eigenvectors.

Fit improves markedly in one direction with eigenvector 9, gluon, whichalters common shape and normalisation, and 14 and 18 which alterdV and uV , i.e. affect asymmetry. Not much variation in strangenormalisation.

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 5 10 15 20

Χ2 p

er p

oint

Dev

iatio

n F

rom

Cen

tral

Val

ue

Eigenvector Number

MSTW Eigenvectors for WZ Fit


Similar but slightly smaller effect on uV − dV than asymmetry alone.

Can also see the effect on the gluon. Slight raise near x = 0.01preferred. Improves overall shape of rapidity distribution. Afterreweighting χ2 = 48/60.

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.0001 0.001 0.01 0.1

Rat

io to

MS

TW

Cen

tral

Val

ue

x

g(x) at q2=10000 (GeV)2

Before Reweighting (Npdf=1000)After Reweighting (Neff=190)


x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Rat

io t

o M

ST

W 2

008

NL

O (

90%

C.L

.)

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

2 GeV4 = 102Gluon distribution at Q

No Tevatron jetsFit only Run I jets

II jets∅Fit CDFII(kT) + D/2JET

T = p

Fµ =

RµSame with

II jets∅Fit CDFII(Midpoint) + DII jets∅Fit Run I + CDFII(kT) + D

x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Rat

io t

o M

ST

W 2

008

NL

O (

90%

C.L

.)

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

In contrast in MSTW2008 fit central gluon hardly changed if Tevatron jetdata left out, and only slight further rearrangement of quark flavours ifDrell-Yan data left out.

Main effect loss of tight constraint on αS(M2Z). Much the same at NNLO.

Similar results from various other groups.


0.94

0.96

0.98

1

1.02

1.04

10-2

10-1

MSTWCP - 2 deuteron parameters



corr

ectio

n fa

ctor

Given previous relationship betweenTevatron asymmetry and deuteriumcorrections where partial successwas noted revisit with extendedparameterisation.

Default for MSTW some shadowingfor x < 0.01.

Previously big improvement in fit,but “unusual” corrections.

Now improvement again but muchmore stable, and sensible fordeuterium corrections. (Noshadowing favoured though.)


The GM-VFNS can be defined by demanding equivalence of the nf

light flavour and nf + 1 light flavour descriptions at all orders – abovetransition point nf → nf + 1

F (x,Q2)=CFF,nf

k (Q2/m2H)⊗f

nf

k (Q2)=CV F,nf+1

j (Q2/m2H)⊗f

nf+1

j (Q2)

≡ CV F,nf+1


H)⊗fnf

k (Q2).

Hence, the VFNS coefficient functions satisfy

CFF,nf

k (Q2/m2H) = C

V F,nf+1


H),

which at O(αS) gives

CFF,nf ,(1)

2,Hg (Q2

m2H

) = CV F,nf+1,(0)

2,HH (Q2

m2H

)⊗P 0qg ln(Q2/m2

H)+CV F,nf+1,(1)

2,Hg (Q2

m2H

),

The VFNS coefficient functions tend to the m=0 limits as Q2/m2H →∞.

However, CV Fj (Q2/m2

H) only uniquely defined in this limit.

Can swapO(m2H/Q2) terms between CV F,0

2,HH(Q2/m2H) and CV F,1

2,g (Q2/m2H).


0.96

1

1.04

10 102

103

104

MSTW08MSTW08nojet

FFNSDISFFNSjetZ

NNLO x=0.0005

Q2

F 2(x,

Q2 )/

MST

W08

0.96

1

1.04

10 102

103

104

x=0.005

Q2

F 2(x,

Q2 )/

MST

W08

0.96

1

1.04

10 102

103

104

x=0.05

Q2

F 2(x,

Q2 )/

MST

W08

The results for F2(x,Q2) whenrefits are performed.

As seen very little change whenusing GM-VFNS with no jets.

Much more tension and worsefits for FFNS.


0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

10-3

10-2

10-1

x

F c , Q2 = 20 GeV22

NNLO / NLO

NNLO approx.

x

F c , Q2 = 100 GeV22

NLO scale

NNLO scale

ABM11, m2 = 2 GeV2c

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

10-3

10-2

10-1

Approximate O(α3S) corrections to F c

2 (x,Q2) by Kawamura et al. inNucl.Phys. B864 (2012) 399-468.

Similar results for O(α3S) approximation used by MSTW at low Q2

extended to higher Q2.


0.9

0.95

1

1.05

1.1

10-5

10-4

10-3

10-2

10-1

MSTW08 NLOQ2=5GeV2, W2=20GeV2

Q2=10GeV2, W2=20GeV2

Preliminary

Q2=10,000GeV2

part

ons/

MST

W20

08 a

t NL

O f

or g

(x,Q

2 )

0.9

0.95

1

1.05

1.1

10-5

10-4

10-3

10-2

10-1

MSTW08 NLOQ2=5GeV2, W2=20GeV2


Preliminary

Q2=10,000GeV2

part

ons/

MST

W20

08 a

t NL

O f

or u

(x,Q

2 )

0.9

0.95

1

1.05

1.1

10-5

10-4

10-3

10-2

10-1

MSTW08 NNLOQ2=5GeV2, W2=20GeV2


Preliminary

part

ons/

MST

W20

08 a

t NN

LO

for

g(x

,Q2 )

0.9

0.95

1

1.05

1.1

10-5

10-4

10-3

10-2

10-1

MSTW08 NNLOQ2=5GeV2, W2=20GeV2

Q2=5GeV2, W2=20GeV2

Preliminarypa

rton

s/M

STW

2008

at N

NL

O f

or u

(x,Q

2 )

Similar to effect of higher twist, particularly at NNLO. Remember losedata at lowest x.


0.9

0.95

1

1.05

1.1

10-4

10-3

10-2

10-1

MSTW08NLO-0.1202FFNSHT*DIS-0.1162FFNS+DYHT*-0.1165FFNSjetHT*-0.1199FFNSjetZHT*-0.1215PD

F/M

STW

2008

at N

LO

for

g(x

,Q2 )

0.9

0.95

1

1.05

1.1

10-4

10-3

10-2

10-1

MSTW08NLOFFNSDISYHT* 2 LFFNS+DYHT*FFNSjetHT*FFNSjetZHT*PD

F/M

STW

08 a

t NL

O f

or u

(x,Q

2 )

0.9

0.95

1

1.05

1.1

10-4

10-3

10-2

10-1

MSTW08NNLO-0.1171FFNSDISHT*-0.1132FFNS+DYHT*-0.1136FFNSjetHT*-0.1155FFNSjetZHT*-0.1174PD

F/M

STW

08 a

t NN

LO

for

g(x

,Q2 )

0.9

0.95

1

1.05

1.1

10-4

10-3

10-2

10-1

MSTW08NNLOFFNSDISHT*FFNS+DYHT*FFNSjetHT*FFNSjetZHT*PD

F/M

STW

08 a

t NN

LO

for

u(x

,Q2 )

Restricting higher twist from lowest x value and omitting nuclear targetdata (except dimuon for strangeness). Same trends as for standard fitsbut slightly lower αS


Explains some PDF differences? MSTW FFNS ratios and ABKM ratios.

0.9

0.95

1

1.05

1.1

10-4

10-3

10-2

10-1

MSTW08NLO-0.1202

ABKM09-0.1179

ABM11-0.1180PDF/

MST

W08

at N

LO

for

g(x

,Q2 )

0.9

0.95

1

1.05

1.1

10-4

10-3

10-2

10-1

MSTW08NLO

ABKM09

ABM11PDF/

MST

W08

at N

LO

for

u(x

,Q2 )

0.9

0.95

1

1.05

1.1

10-4

10-3

10-2

10-1

MSTW08NNLO-0.1171

ABKM09-0.1135

ABM11-0.1134PDF/

MST

W08

at N

NL

O f

or g

(x,Q

2 )0.9

0.95

1

1.05

1.1

10-4

10-3

10-2

10-1

MSTW08NNLO

ABKM09

ABM11PDF/

MST

W08

at N

NL

O f

or u

(x,Q

2 )

Better to compare to ABKM09 as mass scheme and data fit are moresimilar.


0

20

40

60

80

100

10-5

10-4

10-3

10-2

10-1

1x

xg(x,Q2=10000GeV2)

-10

-5

0

5

10

15

10-5

10-4

10-3

10-2

10-1x


0

2

4

6

10-5

10-4

10-3

10-2

10-1

1x

xu(x,Q2=10000GeV2)

-10

-5

0

5

10

15

10-5

10-4

10-3

10-2

10-1x


Change in MSTW2008 NNLO PDFs when fitting HERA combined data.


Comparison to LHC data.

Start with ATLAS jets. Use APPLGrid or FastNLO at NLO (Ben Watt)and correlated errors treated as in the formula,

χ2 =Npts.∑i=1

(Di − Ti

σuncorr.i

)2

+Ncorr.∑k=1

r2k,

where Di ≡ Di −∑Ncorr.

k=1 rk σcorr.k,i Di are the data points allowed to shift

by the systematic errors in order to give the best fit, and σcorr.k,i is a

fractional uncertainty. Normalisation is treated as the other correlateduncertainties.

MSTW fit very good (χ2 per point below left ), though numbers lower forinclusive data. Always close to, if not best, particularly for R = 0.6. Nothuge variation in PDFs though.

Scale pT/2 pT 2pTInclusive (R=0.4) 0.752 0.773 0.703Inclusive (R=0.6) 0.845 0.790 0.721Dijet (R=0.4) 2.53 2.24 2.20Dijet (R=0.6) 2.44 2.04 1.74

|rk| < 1 1 < |rk| < 2 2 < |rk| < 3 3 < |rk| < 4Inclusive (R=0.4) 85 2 1 0Inclusive (R=0.6) 87 1 0 0Dijet (R=0.4) 82 6 0 0Dijet (R=0.6) 74 12 2 0


parton distribution functions at the lhc · 2013-02-27 · obtaining pdf sets – general...

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