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Toward NNLO accuracy of parton distribution functions Pavel - - PowerPoint PPT Presentation

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Toward NNLO accuracy of parton distribution functions Pavel Nadolsky Southern Methodist University Dallas, TX, U.S.A. in collaboration with M. Guzzi, F . Olness, J. Huston, H.-L. Lai, Z. Li, J. Pumplin, C.-P . Yuan (CTEQ) September 23, 2011

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SLIDE 1

Toward NNLO accuracy

  • f parton distribution functions

Pavel Nadolsky

Southern Methodist University Dallas, TX, U.S.A.

in collaboration with

  • M. Guzzi, F

. Olness,

  • J. Huston, H.-L. Lai, Z. Li, J. Pumplin, C.-P

. Yuan (CTEQ) September 23, 2011

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 1

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SLIDE 2

NNLO PDF sets as the new norm

PDFs with NNLO (two-loop) QCD corrections to DIS and DY processes are becoming the standard. They are now produced by 6 groups. Our general-purpose set CT10 is obtained at NLO (PRD82,

074024 (2010)). The CT10.1 NNLO set

undergoes pre-publication tests.

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 2

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SLIDE 3

The agreement between NNLO PDF sets is not automatically better than at NLO

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 3

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SLIDE 4

The agreement between NNLO PDF sets is not automatically better than at NLO

  • G. Watt, in PDF4LHC study, arXiv:1101.0536

)

2 Z

(M

S

α

0.11 0.115 0.12 0.125 0.13

(pb)

H

σ

0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3

68% C.L. PDF MSTW08 HERAPDF1.0 ABKM09 GJR08/JR09

= 180 GeV

H

= 1.96 TeV) for M s H at the Tevatron ( → NNLO gg

Open symbols: NLO Closed symbols: NNLO

S

α Outer: PDF+ Inner: PDF only Vertical error bars

)

2 Z

(M

S

α

0.11 0.115 0.12 0.125 0.13

(pb)

H

σ

0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 3

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SLIDE 5

χ2/Ndata points in various experiments (PRELIMINARY)

PDF set Order All Combined BCDMS CDF , D0 D0 Run-2 Ae

ch,

expts. HERA-1 DIS F p,d

2

Run-2 1-jet pe

T > 25 GeV

CT10.1

1.11 1.17

1.10 1.33 3.72

MSTW08 NLO

1.42 1.73

1.16 1.31 11.38

(1.28) (1.4)

(1.17)

NNPDF2.0

1.37 1.32

1.28 1.57 2.79

CT10.1

1.13 1.12

1.14 1.23 2.59

MSTW08

1.34 1.36

1.15 1.38 9.84

NNPDF2.1 NNLO

1.57 1.36

1.30 1.51 5.45

ABM’09 (5f)

1.65 1.4

1.49 2.63 23.78

HERA1.5

1.71 1.15

1.87 ? 5.4

Npoints 2798 579 590 182 12

Cross sections are computed using the CTEQ fitting code and αs, mc, mb values provided by each PDF set. Their agreement does not immediately improve after going to NNLO.

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 4

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SLIDE 6

Origin of differences between PDF sets

NNLO QCD terms (in all 6 PDF fits)

◮ Implementation of heavy-quark mass terms

(N)LO electroweak contributions Selection of data: global analyses (CTEQ, MSTW, NNPDF) vs. restricted (“DIS-based”) analyses (ABM, GJR, HERAPDF) Statistical treatment: Monte-Carlo sampling vs. analytical minimization of χ2; correlated systematic uncertainties; definitions of PDF uncertainties Initial PDF parametrizations: neural networks (NNPDF); 2-5 parameters per flavor (other fits) Values of αs(MZ), mc, and mb and their treatment Differences in NLO codes used by PDF fits

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 5

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SLIDE 7

This talk

Genuine NNLO accuracy requires an earnest effort to calibrate all components of the PDF fits I will provide examples of related activities, focusing on Heavy-quark contributions to DIS at O(α2

s)

M. Guzzi, P .N., H.-L. Lai, C.-P . Yuan, arXiv:1108.5112; additional figures at http://bit.ly/SACOTNNLO11

W charge asymmetry at the Tevatron

  • 1. H.-L. Lai, M. Guzzi, J. Huston, Z. Li, P

. N., J. Pumplin, C.-P . Yuan, Phys.Rev. D82 (2010) 074024.

  • 2. M. Guzzi, P

. N., E. Berger, H.-L. Lai, F . Olness, C.-P . Yuan, arXiv:1101.0561 [hep-ph].

Consistency of DGLAP picture at small x at HERA

  • 1. H.-L. Lai, M. Guzzi, J. Huston, Z. Li, P

. N., J. Pumplin, C.-P . Yuan, Phys.Rev. D82 (2010) 074024. Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 6

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SLIDE 8
  • 1. Heavy quarks in DIS and LHC electroweak cross sections

The latest PDF fits assume mc,b = 0 when evaluating Wilson coeffi- cients in DIS, e±p → e±X and e±p → νX This is needed, in particular, to correctly predict W, Z production rates at the LHC (Tung et al., hep-ph/0611254)

CMS-PAS-EWK-10-005 18.5 19 19.5 20 20.5 21 21.5 22 ΣtotppW{ΝX nb 1.85 1.9 1.95 2 2.05 2.1 2.15 ΣtotppZ0{{

  • X nb

W & Z cross sections at the LHC (14 TeV) CTEQ6.6 (GM) CTEQ6.1 (ZM) NNLLNLO ResBos

  • arXiv:0802.0007

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 7

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SLIDE 9
  • 1. Heavy quarks in DIS and LHC electroweak cross sections

In pp → Z0X at √s = 14 TeV:

σ(general-mass PDFs) σ(zero-mass PDFs) ≈ 1.05 − 1.08;

to be compared with

KNNLO/NLO ≡ σ(α2

s)/σ(αs) ≈ 1.02 CMS-PAS-EWK-10-005 18.5 19 19.5 20 20.5 21 21.5 22 ΣtotppW{ΝX nb 1.85 1.9 1.95 2 2.05 2.1 2.15 ΣtotppZ0{{

  • X nb

W & Z cross sections at the LHC (14 TeV) CTEQ6.6 (GM) CTEQ6.1 (ZM) NNLLNLO ResBos

  • arXiv:0802.0007

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 7

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SLIDE 10

Massive quark contributions to neutral-current DIS

Several heavy-quark factorization schemes

FFN, ACOT, BMSN, CSN, FONLL, TR’... Extensive recent work

Tung et al., hep-ph/0611254; Thorne, hep-ph/0601245; Tung, Thorne, arXiv:0809.0714; P .N., Tung, arXiv:0903.2667; Forte, Laenen, Nason, arXiv:1001.2312; J. Rojo et al., arXiv:1003.1241;Alekhin, Moch, arXiv:1011.5790;...

Is a consistent picture emerging? Will persisting confusions (e.g., about the universality of heavy-quark PDFs) be resolved?

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 8

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SLIDE 11

Heavy-quark NC DIS at NLO: consistent ambiguity

At NLO, the charm mass mc and factorization scale µ of are tuned to best describe the DIS data in each scheme; but the residual differences in the W and Z cross sections remain

  • Scale dependence

long dash: SACOTΧ NLO short dash: FFNS NLO Nf3 MSTW08NLO MSTW08NLOΧ FONLLAΧ FONLLBΧ dotted:SACOT NLO

105 104 103 0.01 0.02 0.05 0.1 0.2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 x 103x0.5 F2cx,Q

LH PDFs Q2 GeV, mc1.41 GeV

2009 Les Houches HQ benchmarks with toy PDFs; default µ = Q

  • G. Watt, PDF4LHC mtg, 26.03.2010

W, Z cross sections; mc = 1.3 GeV in CTEQ6.6

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 9

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SLIDE 12

NNLO: better agreement between the schemes, remaining conceptual differences

arXiv:1108.5112 revisits the QCD factorization theorem for DIS with heavy quarks discusses a scheme (S-ACOT-χ) for a streamlined, algorithmic implementation of NNLO massive contributions

Subtractions c F

(0) (1) h,h h,g

Structure Functions

ch,h

(1)

A(1)

h,g * ch,h (1)

A

(2) h,g ch,h (0) *

A

(2) * c(0) h,h h,l PS PS h,l (2)

F Fh,g

(2)

u,d,s,c c(2)

h,h

A

(1) h,g

c(0)

h,h *

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 10

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SLIDE 13

S-ACOT-χ scheme: merging FFN and ZM

SACOTΧ NNLO FFNS Nf3 NNLO ZM NNLO

0.00 0.05 0.10 0.15 0.20 0.25 F2cx,Q

x0.01

10. 5. 2. 3. 1.5 7. 0.6 0.7 0.8 0.9 1.0 1.1 1.2 Q GeV Ratio to SACOTΧ

S-ACOT-χ reduces to FFN at Q ≈ mc and to ZM at Q mc

Les Houches toy PDFs, evolved at NNLO with threshold matching terms Cancellations between subtractions and other terms at Q ≈ mc and Q mc; details in backup slides

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 11

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SLIDE 14

Are heavy quarks counted as active partons?

Fixed Flavor Number scheme mc = 0 for all µ2 = Q2 ≥ m2

c

PDF "hard" part massive quark

Zero-Mass Variable Flavor Number scheme mc = 0 for all Q2 ≥ m2

c

massless quark

fc/p(ξ, µ2) ∼

∞

X

n=1

αk

S(µ) n

X

k=0

cnk(ξ) lnk „ µ2 m2

c

«

Shown for up to 4 flavors for simplicity

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 12

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SLIDE 15

General-Mass Variable Flavor Number schemes

(Aivasis, Collins, Olness, Tung; Buza et al.; Cacciari, Greco, Nason; Chuvakin et al.; Kniehl et al.; Thorne, Roberts; Forte, Laenen, Nason; ...)

For the ACOT GM scheme, factorization of DIS cross sections is proved to all orders of αs (Collins, 1998) Theorem F2(x, Q, mc) = σ0

  • a=g,q,¯

q

dξ ξ Ca x ξ , Q µF , mc Q

  • fa/p(ξ, µF

mc )+O ΛQCD Q

  • Ca is a Wilson coefficient with an incoming parton a = g, u

(−), ..., c (−)

fa/p is a PDF for Nf flavors Nf = 3 for µ < µ(4)

switch ≈ mc; Nf = 4 for µ ≥ µ(4) switch

limQ→∞ Ca exists; no terms O(mc/Q) in the remainder

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 13

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SLIDE 16

General-Mass Variable Flavor Number schemes

(Aivasis, Collins, Olness, Tung; Buza et al.; Cacciari, Greco, Nason; Chuvakin et al.; Kniehl et al.; Thorne, Roberts; Forte, Laenen, Nason; ...)

For the ACOT GM scheme, factorization of DIS cross sections is proved to all orders of αs (Collins, 1998) Schemes of the ACOT type do not use... PDFs for several Nf values in the same µ range smoothness conditions/damping factors at Q → mc constant terms from higher orders in αs to ensure continuity at the threshold

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 13

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SLIDE 17

Components of inclusive F2,L(x, Q)

Structure of S-ACOT-χ NNLO expressions is reminiscent of the ZM scheme (e.g., in Moch, Vermaseren, Vogt, 2005)

Components of inclusive F2,L(x, Q2) are classified according to the quark couplings to the photon F =

Nl

  • l=1

Fl + Fh (1) Fl = e2

l

  • a
  • Cl,a ⊗ fa/p
  • (x, Q),

Fh = e2

h

  • a
  • Ch,a ⊗ fa/p
  • (x, Q). (2)

e l

h

e

At O(α2

s):

F (2)

h

= e2

h

  • cNS,(2)

h,h

⊗ (fh/p + f¯

h/p) + C(2) h,l ⊗ Σ + C(2) h,g ⊗ fg/p

  • F (2)

l

= e2

l

  • CNS,(2)

l,l

⊗ (fl/p + f¯

l/p) + cP S,(2) ⊗ Σ + c(2) l,g ⊗ fg/p

  • . (3)

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 14

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SLIDE 18

Components of inclusive F2,L(x, Q)

Lower case c(2)

a,b, ˆ

f(k)

a,b → ZM expressions

Zijlstra and Van Neerven PLB272 (1991), NPB383 (1992)

  • S. Moch, J.A.M. Vermaseren and A. Vogt, NPB724 (2005)

Upper case C(2)

a,b, F (k) a,b A(k) a,b → coeff. functions, structure

functions and subtractions with mc = 0,

Buza et al., NPB 472 (1996); EPJC1 (1998); Riemersma, et al. PLB 347 (1995); Laenen et al. NPB392 (1993)

All building blocks are available from literature

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 14

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SLIDE 19

Components of inclusive F2,L(x, Q)

The separation into Fl and Fh (according to the quark’s electric charge e2

i ) is valid at all Q

The “light-quark” Fl contains some subgraphs with heavy-quark lines, denoted by “Gl,l,heavy”. The “heavy-quark” Fh = F c

2:

F c

2 = Fh + (Gl,l,heavy)real,

where Gi,j = C(2)

i,j , F (2) i,j , and A(2) i,j

l

h

(a) (b) l

h

(c) (d) l

h h

l

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 14

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SLIDE 20

Three versions of the ACOT scheme

Scheme Lowest-order graphs Variables in Hc

Full ACOT

Aivasis et. al., hep-ph/9312319 Proof in Collins, hep-ph/9806259

+ −

P 1 m;n=1
  • n
S v nm ln m (Q 2 = M 2 ) m!

mc = 0 ζ =

x 2

  • 1 +
  • 1 + 4m2

c

Q2

  • Simplified

ACOT

Proof in Collins, hep-ph/9806259; Kramer, Olness, Soper, hep-ph/0003035

+ −

mc = 0 ζ = x

S-ACOT-χ

Tung, Kretzer, Schmidt, hep-ph/0110247; proof in Guzzi et al., arXiv:1108.5112

+ −

mc = 0

ζ = x

  • 1 + 4m2

c

Q2

  • ≡ χ

Ha with incoming light partons are unambiguous Hc with incoming c quarks differ by terms of order (m2

c/Q2)p, p > 0

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 15

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SLIDE 21
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SLIDE 22

Three versions of the ACOT scheme

Scheme Lowest-order graphs Variables in Hc

Full ACOT

Aivasis et. al., hep-ph/9312319 Proof in Collins, hep-ph/9806259

+ −

P 1 m;n=1
  • n
S v nm ln m (Q 2 = M 2 ) m!

mc = 0 ζ =

x 2

  • 1 +
  • 1 + 4m2

c

Q2

  • Simplified

ACOT

Proof in Collins, hep-ph/9806259; Kramer, Olness, Soper, hep-ph/0003035

+ −

mc = 0 ζ = x

S-ACOT-χ

Tung, Kretzer, Schmidt, hep-ph/0110247; proof in Guzzi et al., arXiv:1108.5112

+ −

mc = 0

ζ = x

  • 1 + 4m2

c

Q2

  • ≡ χ

Ha with incoming light partons are unambiguous Hc with incoming c quarks differ by terms of order (m2

c/Q2)p, p > 0

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 17

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SLIDE 23

Energy conservation near the threshold

Threshold rate suppression resulting from energy conservation is the most important mass effect at Q ≈ mc

(Tung, Kretzer, Schmidt, hep-ph/0110247; Tung, Thorne, arXiv:0809.0714; P .N., Tung, arXiv:0903.2667)

In DIS, a c¯ c pair is produced only at par- tonic energies W satisfying

  • W 2 = (ξp + q)2 = Q2

ξ x − 1

  • ≥ 4m2

c

⇒χ ≤ ξ ≤ 1, where χ ≡ x

  • 1 + 4m2

c

Q2

  • ≥ x.

Collinear approximation for c¯

c pro- duction may allow to integrate over x ≤ ξ ≤ 1, in violation of energy conservation

e ξp q c ¯ c p

Threshold suppression

x=0.05 µ2=Q2+4 m2 F2

c

Q2 / GeV2 χ=x(1+4 m2 / Q2 ) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 1 10 10

2

10

3

N L O 3

  • f

l v

  • L

O 3

  • f

l v : G F

(1)

  • N

a i v e L O 4

  • f

l v : c ( x ) L O 4

  • f

l v A C O T (c ) : c (c )

  • Pavel Nadolsky (SMU)

Galileo Galilei Institute September 23, 2011 18

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SLIDE 24

Energy conservation near the threshold

Threshold rate suppression resulting from energy conservation is the most important mass effect at Q ≈ mc

(Tung, Kretzer, Schmidt, hep-ph/0110247; Tung, Thorne, arXiv:0809.0714; P .N., Tung, arXiv:0903.2667)

The S-ACOT-χ scheme allows one

to include the energy conservation re- quirement in the QCD factorization the-

  • rem (arXiv:1108.5112)

This is achieved by rescaling, which

restores the correct kinematics of HQ production at the threshold without a posteriori constraints or damping fac- tors imposed by other schemes

e ξp q c ¯ c p

Threshold suppression

x=0.05 µ2=Q2+4 m2 F2

c

Q2 / GeV2 χ=x(1+4 m2 / Q2 ) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 1 10 10

2

10

3

N L O 3

  • f

l v

  • L

O 3

  • f

l v : G F

(1)

  • N

a i v e L O 4

  • f

l v : c ( x ) L O 4

  • f

l v A C O T (c ) : c (c )

  • Pavel Nadolsky (SMU)

Galileo Galilei Institute September 23, 2011 18

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SLIDE 25

Rescaling at the lowest order

+ −

c(ζ) + αs 4π Z 1

χ

dξ ξ g(ξ)C(1)

h,g

„χ ξ « − αs 4π Z 1

ζ

dξ ξ g(ξ)A(1)

h,g

„ζ ξ «

☛ ✡ ✟ ✠ ζ takes place of x in terms 1 and 3

Term 2 (γ∗g fusion) is unambiguous Terms 1 and 3 are essentially

Z 1 dξ ξ δ „ 1 − ζ ξ « c(ξ) − αs 4π Z 1 dξ ξ g(ξ)A(1)

h,g

„ζ ξ « θ „ 1 − ζ ξ « .

Rescaling ζ → κζ, ξ → κ−1ξ changes the ξ range in the Wilson coefficients δ(1 − ζ

ξ) and A(1) h,g

  • ζ

ξ

  • θ
  • 1 − ζ

ξ

  • , but does

not change their magnitude

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 19

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SLIDE 26

Rescaling at the lowest order

+ −

c(ζ) + αs 4π Z 1

χ

dξ ξ g(ξ)C(1)

h,g

„χ ξ « − αs 4π Z 1

ζ

dξ ξ g(ξ)A(1)

h,g

„ζ ξ «

☛ ✡ ✟ ✠ ζ takes place of x in terms 1 and 3

Q2 = 10 GeV2 Red curve: g(ξ)C(1)

h,g(χ/ξ) at χ ≤ ξ ≤ 1

Green: ζ = x; κ = 1 ◮ g(ξ)A(1)

h,g(x/ξ) = 0 at ξ < χ

◮ its integral cancels poorly with c(x) Blue: ζ = χ; κ = 1 + 4m2

c/Q2

◮g(ξ)A(1)

h,g(χ/ξ) = 0 at ξ < χ

◮ its integral cancels better with c(χ)

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 19

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SLIDE 27

Rescaling at the lowest order

+ −

c(ζ) + αs 4π Z 1

χ

dξ ξ g(ξ)C(1)

h,g

„χ ξ « − αs 4π Z 1

ζ

dξ ξ g(ξ)A(1)

h,g

„ζ ξ «

☛ ✡ ✟ ✠ ζ takes place of x in terms 1 and 3

Q2 = 100 GeV2 χ ≈ x g(ξ)A(1)

h,g(ζ/ξ) approximates the logarithmic

growth in g(ξ)C(1)

h,g(χ/ξ)

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 19

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SLIDE 28

Rescaling at the lowest order

+ −

c(ζ) + αs 4π Z 1

χ

dξ ξ g(ξ)C(1)

h,g

„χ ξ « − αs 4π Z 1

ζ

dξ ξ g(ξ)A(1)

h,g

„ζ ξ «

☛ ✡ ✟ ✠ ζ takes place of x in terms 1 and 3

Q2 = 10000 GeV2 χ ≈ x g(ξ)A(1)

h,g(ζ/ξ) approximates the logarithmic

growth in g(ξ)C(1)

h,g(χ/ξ)

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 19

slide-29
SLIDE 29

Rescaling to all orders of αs and the factorization theorem

Z projection

  • k

k

p a = g, q, ¯ q q q p

  • kT

Ta(k, p) Ta(k, p) Ha(q, k) Ha(q, k)

⊗

Rescaling is introduced in the definition of the hard subgraph Hc(q, k) with an incoming c quark in γ∗(q) + c( k) → X. Hard graphs

Hg, Hq with incoming light partons and target graphs Ta(k, p) are not affected

In Hc(q, k), the momentum of the incoming c quark is approxi- mated by b

k = “ ξp+, 0−, 0T ” .

k and Hc(q, k) are invariant under trans- formation p+ → p+/κ, ξ → ξ κ. The physical ξ range is obtained for κ = 1 + 4m2

c/Q2.

This transformation shows that the S-ACOT-χ scheme is valid to all orders on the same grounds as the S-ACOT scheme.

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 20

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SLIDE 30

Rescaling to all orders of αs and the factorization theorem

Z projection

  • k

k

p a = g, q, ¯ q q q p

  • kT

Ta(k, p) Ta(k, p) Ha(q, k) Ha(q, k)

⊗

F(x, Q) = X

a=g,u,d,...,c

Z dξ ξ Ca „x ξ , Q µ , mc Q « fa/p(ξ, µ)

Wilson coefficients with initial heavy quarks are

Cc „x ξ , Q µ , mc Q « ≈ Cc „χ ξ , Q µ , mc = 0 « θ(χ ≤ ξ ≤ 1) where χ ≡ x „ 1 + 4m2

c

Q2 « .

The target (PDF) subgraphs Ta are given by the same universal

  • perator matrix elements in all ACOT schemes

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 20

slide-31
SLIDE 31

NNLO results for F (c)

2 (x, Q2)

At NLO:

  • Scale dependence

long dash: SACOTΧ NLO short dash: FFNS NLO Nf3 MSTW08NLO MSTW08NLOΧ FONLLAΧ FONLLBΧ dotted:SACOT NLO

105 104 103 0.01 0.02 0.05 0.1 0.2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 x 103x0.5 F2cx,Q

LH PDFs Q2 GeV, mc1.41 GeV

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 21

slide-32
SLIDE 32

NNLO results for F (c)

2 (x, Q2)

At NNLO and Q ≈ mc:

S-ACOT-χ (Nf = 4) ≈ FFN (Nf = 3)

without tuning S-ACOT is numerically close to other NNLO schemes NNLO expressions are close to the FONLL-C scheme

(Forte, Laenen, Nason, arXiv:1001.2312).

  • Scale dependence

long dash: SACOTΧ NLO solid: SACOTΧ NNLO short dash: FFNS NNLO Nf3 MSTW08NNLOΧ FONLLCΧ

105 104 103 0.01 0.02 0.05 0.1 0.2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 x 103x0.5 F2cx,Q

LH PDFs Q2 GeV, mc1.41 GeV

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 21

slide-33
SLIDE 33

NNLO results for F (c)

2 (x, Q2)

At NNLO and Q ≈ mc:

S-ACOT-χ (Nf = 4) ≈ FFN (Nf = 3)

without tuning S-ACOT is numerically close to other NNLO schemes NNLO expressions are close to the FONLL-C scheme

(Forte, Laenen, Nason, arXiv:1001.2312).

scalerescaling dependence

blue band: NLO green band: NNLO

105 104 103 0.01 0.02 0.05 0.1 0.2 1 2 3 4 5 x 103x0.5 F2cx,Q

LH PDFs Q2 GeV SACOT

Even without rescaling (a wrong choice!), NNLO cross sections are much closer to FFN at Q ≈ mc than at NLO

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 21

slide-34
SLIDE 34

Main features of the S-ACOT-χ scheme

It is proved to all orders by the QCD factorization theorem for DIS (Collins, 1998) It is relatively simple

◮ One value of Nf (and one PDF set) in each Q range ◮ sets mh = 0 in ME with incoming h = c or b ◮ matching to FFN is implemented as a part of the QCD factorization theorem

Universal PDFs It reduces to the ZM MS scheme at Q2 m2

Q, without

additional renormalization It reduces to the FFN scheme at Q2 ≈ m2

Q

◮ has reduced dependence on tunable parameters at NNLO

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 22

slide-35
SLIDE 35

CT10/CT10W distributions, W charge asymmetry, HERA DIS at small x

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 23

slide-36
SLIDE 36

CT10 parton distribution functions (PRD82, 074024 (2010))

New experimental data, statistical methods, and parametrization forms combined HERA-1 data are included 53 CT10 and 53 CT10W eigenvector sets for αs(MZ) = 0.118 4 CT10AS PDFs for αs(MZ) = 0.116 − 0.120

sufficient for computing the PDF+αs uncertainty, as explained in PRD82,054021 (2010)

CT10/CT10W PDFs with 3 and 4 active flavors extended discussion of the Tevatron Run-2 W asymmetry data search for deviations from NLO DGLAP evolution in the small-x HERA data

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 24

slide-37
SLIDE 37

The puzzle of the W lepton asymmetry

Rapidity asymmetry Aℓ(yℓ) of charged lepton ℓ = e or µ in W boson decay provides important constraints on d(x, Q)/u(x, Q) at x > 0.1

Aℓ(yℓ) = dσ(p¯ p → (W + → ℓ+νℓ)X)/dyℓ − dσ(p¯ p → (W − → ℓ−¯ νℓ)X)/dyℓ dσ(p¯ p → (W + → ℓ+νℓ)X)/dyℓ + dσ(p¯ p → (W − → ℓ−¯ νℓ)X)/dyℓ .

Aℓ(yℓ) is related to the boson-level asymmetry, AW (yW ) = dσ(pp → W +X)/dyW − dσ(pp → W −X)/dyW dσ(pp → W +X)/dyW + dσ(pp → W −X)/dyW , smeared by W ± decay effects

Berger, Halzen, Kim, Willenbrock, PRD 40, 83 (1989); Martin, Roberts, Stirling, 1989 Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 25

slide-38
SLIDE 38

The puzzle of the CDF/D0 W lepton asymmetry

CT10W set reasonably agrees with 3 pTℓ bins of Ae(ye) and

  • ne bin of Aµ(yµ) from D0 Run-2 (2008); shows tension with

NMC, BCDMS F p,d

2

(x, Q) NNPDF 2.0 (arXiv:1012.0836) agrees with Aµ(yµ), disagrees with two pTe bins of Ae(ye). CT10, many other PDFs fail. Agreement of Source or PQCD with D0 Ae(ye) χ2/npt comments

CTEQ6.6, NLO 191/36=5.5

Our study;

CT10W, NLO 78/36=2.2

Resbos, NNLL-NLO

With Aµ(yµ): 88/47=1.9 ABKM’09, NNLO 540/24=22.5

Catani, Ferrera, Grazzini,

MSTW’08, NNLO 205/24=8.6

JHEP 05, 006 (2010)

JR09VF , NNLO 113/24=4.7

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 26

slide-39
SLIDE 39

Why difficulties with fitting Aℓ(yℓ)?

  • 1. Aℓ(yℓ) is very sensitive to the average slope sdu of

d(x, MW )/u(x, MW ) Aℓ(yℓ) ∼ Aℓ(yW )|LO ∝ 1 x1 − x2 d(x1) u(x1) − d(x2) u(x2)

  • ;

x1,2 = Q √se±yW

Berger, Halzen, Kim, Willenbrock, PRD 40, 83 (1989); Martin, Stirling, Roberts, MPLA 4, 1135 (1989); PRD D50, 6734 (1994); Lai et al., PRD 51, 4763 (1995)

  • 2. Constraints on sdu by fixed-target F d

2 (x, Q)/F p 2 (x, Q) are

affected by nuclear and higher-twist effects

Accardi, Christy, Keppel, Monaghan, Melnitchouk, Morfin, Owens, PRD 81, 034016 (2010) Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 27

slide-40
SLIDE 40

Challenges with fitting Aℓ(yℓ)

Small changes in sdu cause significant variations in Aℓ

Lai et al., PRD 51, 4763 (1995)

Alternative constraints on d/u by F d

2 (x, Q)/F p 2 (x, Q) from

fixed-target DIS are affected by nuclear and higher-twist effects

Accardi, Christy, Keppel, Monaghan, Melnitchouk, Morfin, Owens, PRD 81, 034016 (2010) Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 28

slide-41
SLIDE 41

d(x, Q)/u(x, Q) at Q = 85 GeV

η 0.5 1 1.5 2 2.5 3 ) η A(

  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.2

NNPDF20 CT10 MSTW08 >25GeV)

T

D0el(E

  • R. Ball et al., arXiv:1012.0836
  • R. Ball et al., arXiv:1012.0836

H.-L. Lai et al., arXiv:1007.2241

Solid band: CTEQ6.6 uncertainty Hatched band: CT10W uncertainty 105 104 103 0.01 0.02 0.05 0.1 0.2 0.5 0.7 0.7 0.8 0.9 1.0 1.1 1.2 1.3 x Ratio to CTEQ6.6M

d/u at µ = 85 GeV

H.-L. Lai et al., arXiv:1007.2241

D0 Run-2 Aℓ data distinguishes between PDF models, reduces the PDF uncertainty

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 29

slide-42
SLIDE 42

Why difficulties with fitting Aℓ(yℓ)?

  • 3. Existing parametrizations underestimate the PDF uncertainty
  • n d/u

PDFs based on Chebyshov polynomials improve agreement with D0 Run-2 Ae, but are outside of current CTEQ/MSTW bands (Pumplin) This ambiguity is reduced by Aℓ(yℓ) at the LHC, which constrains d/u and ¯ d/¯ u at x ∼ 0.01.

Band: CT10 uncertainty Long dash: Chebyshov, dbar/ubar->1 at x->0 Short dash: Chebyshov, free dbar/ubar at x->0 Solid: MSTW’2008NLO

PRELIMINARY Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 30

slide-43
SLIDE 43

CT10(W) vs. Aℓ at the LHC

|

µ

η | 0.5 1 1.5 2

µ

A 0.15 0.2 0.25 0.3 0.35 |

µ

η | 0.5 1 1.5 2

µ

A 0.15 0.2 0.25 0.3 0.35

=7 TeV) s Data 2010 ( MC@NLO, CTEQ 6.6 MC@NLO, HERA 1.0 MC@NLO, MSTW 2008

  • 1

L dt = 31 pb

∫

ν µ → W

ATLAS

CT10(W) agrees well with the LHC Aℓ; some differences between NLO and NNLL+NLO

| η | 0.5 1 1.5 2 2.5 3 W charge asymmetry 0.1 0.15 0.2 0.25 0.3 0.35

CT10 CT10W CMS electron CMS muon

NNLL−NLO+K, ResBos PRELIMINARY

Zhao Li, 2011

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 31

slide-44
SLIDE 44

CT10(W) vs. Aℓ: LHC-B

| η |

1.5 2 2.5 3 3.5 4 4.5

W charge asymmetry

  • 0.6
  • 0.4
  • 0.2

0.2 0.4 CT10 CT10W LHCB

NNLL−NLO+K, ResBos PRELIMINARY

Zhao Li, 2011

LHC-B marginally prefers CT10W

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 32

slide-45
SLIDE 45

Why difficulties with fitting Aℓ(yℓ)?

  • 4. Experimental Aℓ with lepton pTℓ cuts is sensitive to dσ/dqT of W

boson at transverse momentum qT → 0. Fixed-order (N)NLO calculations (DYNNLO, FEWZ, MCFM,...) predict a wrong shape of dσ/dqT at qT → 0. Small-qT resummation correctly predicts dσ/dqT in this limit. CT10(W) PDFs are fitted using a NNLL-NLO+K resummed prediction for Aℓ (ResBos); must not be used with fixed-order predictions for Aℓ. For example: χ2(CT10W+ResBos) = 1.9 Npt (us); χ2(CT10W+DYNNLO) = 8.4 Npt (NNPDF)

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 33

slide-46
SLIDE 46

Radiative contributions in ResBos + comparison to the Tevatron Z qT distribution

Resummed dσ

` pp → (V → ℓ¯ ℓ′)X ´ /(d3pℓd3p ¯

ℓ′ ) for

V = γ∗, W, Z, with decay and EW interference effects QT ∼ Q: [O(αs) with full decay] × [average O(α2

s) correction (Arnold, Reno)]

QT Q: A(3), B(2), C(1) ◮ no H(2) correction – a ≈ constant error of ±3% in the normalization of Xsecs mass dependence in c, b scattering cross sections (S. Berge, P

. N., F . I. Olness, hep-ph/0509023)

a nonperturbative function from

Konychev, P . N., hep-ph/0506225 (1/GeV)

T

/dp

Z

σ d ×

Z

σ 1/

  • 6

10

  • 5

10

  • 4

10

  • 3

10

  • 2

10

  • 1

10

  • 1

, L=0.97 fb ∅ D NNLO pQCD + corr. PYTHIA Perugia 6 < 115 GeV

µ µ

65 < M > 15 GeV

T

| < 1.7, muon p η muon |

∅ D

(1/GeV)

T

/dp

Z

σ d ×

Z

σ 1/

  • 6

10

  • 5

10

  • 4

10

  • 3

10

  • 2

10

  • 1

10 Ratio to PYTHIA Perugia 6 Ratio to PYTHIA Perugia 6 (GeV)

Z T

p 1 10

2

10 Ratio to PYTHIA Perugia 6 1

  • 1

, L=0.97 fb ∅ D NNLO pQCD Scale & PDF unc. RESBOS NLO pQCD Scale & PDF unc. PYTHIA scale unc. 2 0.7 0.5 (GeV)

Z T

p 1 10

2

10 Ratio to PYTHIA Perugia 6 1 Ratio to PYTHIA Perugia 6 Ratio to PYTHIA Perugia 6 Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 34

slide-47
SLIDE 47

Charge asymmetry in pe

T bins (CDF Run-2, 207 pb−1)

Without the pe

T cut (FEWZ):

Anastasiou et al., 2003 y y of W boson

With pT e cuts imposed, Ach(ye) is sensitive to small-QT resummation

arXiv:1101.0561

0.5 1 1.5 2 2.5 3 y{ 0.6 0.4 0.2 0.2 Charge asymmetry 25 pT{ 35 GeV, CTEQ65

Resummed

NLO LO 0.5 1 1.5 2 2.5 3 y{ 0.4 0.2 0.2 0.4 Charge asymmetry 35 pT{ 45 GeV, CTEQ65 NNLLNLO NLO LO

Balazs, Yuan, 1997; PN, 2007, unpublished; arXiv:1101.0561

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 35

slide-48
SLIDE 48

Acut fits to combined HERA data

x

  • 5

10

  • 4

10

  • 3

10

  • 2

10

  • 1

10 1 ]

2

[ GeV

2 T

/ p

2

/ M

2

Q 1 10

2

10

3

10

4

10

5

10

6

10

NMC-pd NMC SLAC BCDMS HERAI-AV CHORUS FLH108 NTVDMN ZEUS-H2 DYE886 CDFWASY CDFZRAP D0ZRAP CDFR2KT = 0.5

cut

A = 1.0

cut

A = 1.5

cut

A = 3.0

cut

A = 6.0

cut

A

NNPDF2.0 dataset

Ags ≡ Q2x0.3 < Acut Fitting procedure: Include only DIS data above an Acut line Compare the resulting PDFs with DIS data below the Acut line, in a region that is “connected” by DGLAP evolution

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 36

slide-49
SLIDE 49

CT10: Acut fits to DIS data at Q > Q0 = 2 GeV

>

2

<d

0.5 1 1.5 2 2.5 3

= 1.5

cut

Fit with A Fit without cuts

< 1.0

cut

A < 1.5

cut

1.0 < A < 3.0

cut

1.5 < A < 6.0

cut

3.0 < A > 6.0

cut

A Caola, Forte, Rojo, arXiv:1007.5405v2 Q > 1.41 GeV

1.0 1.01.5 1.53.0 3.06.0 6.0

  • CT10 fit

Fit 1 with Acut1.5 Fit 2 with Acut1.5 Q 2 GeV N pts 36 18 30 27 468 Ags ranges 0.5 1.0 1.5 2.0 2.5 3.0 Χdata

2Npts

Motivation Search for deviations from DGLAP evolution at smallest x and Q Follow the procedure proposed by NNPDF (Caola, Forte, Rojo, arXiv:1007.5405)

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 37

slide-50
SLIDE 50

CT10: Acut fits to DIS data at Q > Q0 = 2 GeV

>

2

<d

0.5 1 1.5 2 2.5 3

= 1.5

cut

Fit with A Fit without cuts

< 1.0

cut

A < 1.5

cut

1.0 < A < 3.0

cut

1.5 < A < 6.0

cut

3.0 < A > 6.0

cut

A Caola, Forte, Rojo, arXiv:1007.5405v2 Q > 1.41 GeV

1.0 1.01.5 1.53.0 3.06.0 6.0

  • CT10 fit

Fit 1 with Acut1.5 Fit 2 with Acut1.5 Q 2 GeV N pts 36 18 30 27 468 Ags ranges 0.5 1.0 1.5 2.0 2.5 3.0 Χdata

2Npts

CT10 Two CT10-like fits to data at Ags > 1.5, with different parametrizations of g(x, Q) χ2

i = (Shifted Data − Theory)2

σ2

uncor

Large syst. shifts at Ags < 1.0, in a pattern that could mimic a slower Q2 evolution

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 37

slide-51
SLIDE 51

CT10: Acut fits to DIS data at Q > Q0 = 2 GeV

>

2

<d

0.5 1 1.5 2 2.5 3

= 1.5

cut

Fit with A Fit without cuts

< 1.0

cut

A < 1.5

cut

1.0 < A < 3.0

cut

1.5 < A < 6.0

cut

3.0 < A > 6.0

cut

A Caola, Forte, Rojo, arXiv:1007.5405v2 Q > 1.41 GeV

1.0 1.01.5 1.53.0 3.06.0 6.0

  • CT10 fit

Fit 1 with Acut1.5 Fit 2 with Acut1.5 Q 2 GeV N pts 36 18 30 27 468 Ags ranges 0.5 1.0 1.5 2.0 2.5 3.0 Χdata

2Npts

CT10, cont. δχ2 ∼ 0 at Ags > 1.0 [no difference] δχ2 = 0 − 1.5 at Ags < 1.0, with large uncertainty ⇒ Disagreement with the “DGLAP-connected” data at Ags < Acut is not supported by the CT10 fit

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 37

slide-52
SLIDE 52

Conclusions

Great progress in understanding of heavy-quark contributions to DIS.

◮ S-ACOT-χ recipe-like formulas for implementing NNLO ◮ Energy conservation in DIS is realized as a part of the QCD factorization theorem. Would it be interesting to try to extend to pp processes. ◮ It leads to rescaling of Wilson coefficient functions with incoming heavy quarks. The PDFs are given by universal

  • perator matrix elements.

Progress in understanding of NNLO contributions, new Tevatron and LHC data sets, PDF parametrization issues arXiv:1101.0561: synopsis of recent CTEQ publications

◮ CT10W fit to Run-2 W charge asymmetry; PDFs for leading-order showering programs; constraints on color-octet fermions

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 38

slide-53
SLIDE 53

Conclusion

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 39

slide-54
SLIDE 54

Backup slides

  • 1. Details on S-ACOT-χ scheme at NNLO

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 40

slide-55
SLIDE 55

S-ACOT input parameters

At Q ≈ mc, F c

2 depends significantly on

  • 1. Charm mass: mc = 1.3 GeV in CT10
  • 2. Factorization scale: µ =
  • Q2 + κm2

c ; κ = 1 in CT10

  • 3. Rescaling variable ζ(λ) for matching in γ∗c channels

(Tung et al., hep-ph/0110247; Nadolsky, Tung, PRD79, 113014 (2009)) Fi(x, Q2) =

  • a,b

1

ζ

dξ ξ fa(ξ, µ) Ca

b,λ

ζ ξ , Q µ , mi µ

  • x = ζ
  • 1 + ζλ · (4m2

c)/Q2

, with 0 ≤ λ 1 CT10 uses ζ(0) ≡ χ ≡ x

  • 1 + 4m2

c/Q2

, motivated by momentum conservation

1 Mf

2

Q2 ΖΧACOT Λ0 Ζx Λ Λ0.1 Λ0.2 Λ1 Physical threshold: WMf; Ζ1 104 0.001 0.01 0.1 1 1 x Rescaling factor Ζx

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 41

slide-56
SLIDE 56

Classes of Feynman diagrams I

+ + + + +

NLO Subtraction

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 42

slide-57
SLIDE 57

Classes of Feynman Diagrams II

− − − + + +... −

NNLO Subtractions

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 43

slide-58
SLIDE 58

Cancellations between Feynman diagrams

Validity of the S-ACOT calculation was verified by checking for certain cancellations at Q ≈ mc and Q mc Q ≈ mc: D(2)

C1 D(2) C0 D(1) C0 ≤ F c 2(x, Q)

Q mc: D(2)

g

D(1)

g

< F c

2(x, Q)

These cancellations are indeed observed in our results

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 44

slide-59
SLIDE 59

NNLO: Cancellations at Q2 ≈ m2

c

D

(1) C0

D(1)

C0 = C(0) Q ⊗ c − asC(0) Q ⊗ A(1) Qg ⊗ g;

as = αs (4π) (1)

D

(2) C0

D(2)

C0 = D(1) C0 − a2 sC(0) Q ⊗ A(2),S Qg

⊗ g − a2

sC(0) Q ⊗ A(2),P S QΣ

⊗ Σ (2)

D

(2) C1

D(2)

C1 = C(1) Q ⊗ c − a2 s C(1) Q ⊗ A(1) Qg ⊗ g

(3)

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 45

slide-60
SLIDE 60

NNLO: Cancellations at Q2 ≈ m2

c

FFNS nf3 nnl

LO OΑs DC0

1

DC0

2

DC1

2

105 104 103 0.01 0.02 0.05 0.1 0.2 1 2 3 4 x 103x0.5 F2cx,Q

S−ACOT (no rescaling), Q=2 GeV

FFNS nf3 nnl

LO OΑs DC0

1

DC0

2

DC1

2

105 104 103 0.01 0.02 0.05 0.1 0.2 1 2 3 4 x 103x0.5 F2cx,Q

S−ACOT−chi, Q = 2 GeV

D(2)

C1 D(2) C0 D(1) C0 ≤ FFN at NNLO both for ζ = x and ζ = χ.

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 46

slide-61
SLIDE 61

NNLO: Cancellations at Q mc

D

(1) g

D(1)

g

≡ C(1)

g

= as “ F (1)

g

⊗ g − C(0)

Q ⊗ A(1),S Qg

⊗ g ” (4)

+ +

(2) g

D

D(2)

g

= D(1)

g

+ a2

s

h ˜ F (2)

g

⊗ g + ˜ F (2)

Σ

⊗ Σ − C(1)

Q ⊗ A(1),S Qg

⊗ g −C(0)

Q ⊗ A(2),S Qg

⊗ g − C(0)

Q ⊗ A(2),P S QΣ

⊗ Σ i (5) D(1)

g

is of order of α2

s while D(2) g

is of order of α3

s. Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 47

slide-62
SLIDE 62

F c

2 at NNLO: Cancellations at Q = 10 GeV

FFNS nf3 nnl

Dg

1

Dg

2

105 104 103 0.01 0.02 0.05 0.1 0.2 5 10 15 x 103x0.5 F2cx,Q

S−ACOT (no rescaling), Q = 10 GeV

FFNS nf3 nnl

Dg

1

Dg

2

105 104 103 0.01 0.02 0.05 0.1 0.2 5 10 15 x 103x0.5 F2cx,Q

S−ACOT−chi, Q = 10 GeV

D(2)

g

D(1)

g

< FFN at NNLO < ACOT log Q2

m2

c terms in FFN are cancelled well by subtractions. Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 48

slide-63
SLIDE 63

Comparison of (N)NLO PDF sets with data in the CT10.1 fit

χ2 are computed at NNLO, using the LHAPDF 5.8.6 interface and CTEQ fitting code (very naively!) Whenever possible, adjust settings to reproduce assumptions by other groups

◮ Use αs(Mz), mc,b values suggested by each PDF set ◮ approximate the GM scheme in DIS if possible

Correlated systematic errors are included according to the CTEQ method

χ2 =

  • e={expt.}

 

Npt

  • k=1

1 s2

k

  • Dk − Tk −

Nλ

  • α=1

λαβkα 2 +

Nλ

  • α=1

λ2

α

 

Dk and Tk are data and theory values (k = 1, ..., Npt); sk is the stat.+syst. uncorrelated error; λα are sources of syst. errors

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 49

slide-64
SLIDE 64

χ2 values per experiment (PRELIMINARY)

PDF set Order All Combined BCDMS CDF , D0 D0 Run-2 Ae

ch,

expts. HERA-1 F p,d

2

Run-2 1-jet pe

T > 25 GeV

CT10.1

1.11 1.17

1.10 1.33 3.72

MSTW08 NLO

1.42 1.73

1.16 1.31 11.38

(1.28) (1.4)

(1.17)

NNPDF2.0

1.37 1.32

1.28 1.57 2.79

CT10.2

1.13 1.12

1.14 1.23 2.59

MSTW08

1.34 1.36

1.15 1.38 9.84

NNPDF2.1 NNLO

1.57 1.36

1.30 1.51 5.45

ABM’09 (5f)

1.65 1.4

1.49 2.63 23.78

HERA1.5

1.71 1.15

1.87 ? 5.4

Npoints 2798 579

590 182 12

Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 50