CTEQ-TEA Parton distribution functions and αs

C.-P. Yuan

Michigan State University

in collaboration withH.-L. Lai, M. Guzzi, J. Huston, Z. Li, P. Nadolsky, J. Pumplin, and

D. StumpAlphas Workshop, MPI Munich

Feb 9, 2011

C.-P. Yuan (MSU) Alphas Workshop, MPI Munich Feb 9, 2011 1

CTEQ-Tung Et Al.: recent activities

� Uncertainty induced by αs in the CTEQ-TEA PDF analysis (PRD,

arXiv:1004.4624)

� New PDFs for collider physics

I CTEQ6.6 set (published in 2008)→ CT09→ CT10, CT10W (PRD, arXiv:1007.2241)

I new experimental data, statistical methods, andparametrization forms

� PDFs for Event Generators (JHEP, arXiv:0910.4183)

� Exploration of statistical aspects and PDF parametrizationdependence (PRD, arXiv:0909.0268 and 0909.5176)

C.-P. Yuan (MSU) Alphas Workshop, MPI Munich Feb 9, 2011 2

Uncertainty in αs in the CTEQ-TEA PDF analysisarXiv:1004.4624

� Two leading theoretical uncertainties in LHC processes aredue to αs and the PDFs

� These are not independent uncertainties; how can onequantify their correlation?

� Which central αs(MZ) and which error on αs(MZ) are to beused with the existing PDFs?

� What are the consequences for key LHC processes(gg → H0, etc.)?

C.-P. Yuan (MSU) Alphas Workshop, MPI Munich Feb 9, 2011 3

Uncertainty in αs in the CTEQ-TEA PDF analysisarXiv:1004.4624Recent activity to examine these questions, e.g.:

� MSTW (arXiv:0905.3531)

I αs(MZ) is an output of the global fit (constrained by thehadronic scattering only)

I several sets of error PDFs, each with its own αs(MZ) value⇒lengthier calculations

I The αs uncertainty and PDF uncertainty are inseparable

� NNPDF (in 2009 Les Houches Proceedings, arXiv:1004.0962):I αs(MZ) = 0.119± 0.002 is taken as an inputI αs−PDF correlation is examined with ∼ 1000 PDF replicas and

found to be small

� H1+ZEUS (arXiv:0911.0884): sensitivity of the HERAPDF set toδαs(MZ) = ±0.002 is explored

C.-P. Yuan (MSU) Alphas Workshop, MPI Munich Feb 9, 2011 4

Our findingsTotal PDF+αs errors ∆X are the same when found (a) from a fullfit with floating αs, or (b) by adding ∆XPDF and ∆Xαs inquadrature

10-5 10-4 10-3 0.01 0.02 0.05 0.1 0.2 0.5 0.7

0.6

0.8

1.0

1.2

1.4

x

CT

6.6A

S�C

T6.

6M

g at Q=2 GeV

10-5 10-4 10-3 0.01 0.02 0.05 0.1 0.2 0.5 0.7

0.6

0.8

1.0

1.2

1.4

x

CT

6.6A

S�C

T6.

6M

c at Q=2 GeV

� black – CTEQ6.6 PDFuncertainty

� Blue filled – PDF+αsuncertainty of the fit withfloating αs(MZ)

� Green hatched – PDF+αsuncertainty added inquadrature

Also, agreement in cross sectionpredictions

C.-P. Yuan (MSU) Alphas Workshop, MPI Munich Feb 9, 2011 5

Our findingsTotal PDF+αs errors ∆X are the same if found either from a full fitwith floating αs, or by adding ∆XPDF and ∆Xαs in quadrature

10-5 10-4 10-3 0.01 0.02 0.05 0.1 0.2 0.5 0.7

0.6

0.8

1.0

1.2

1.4

x

CT

6.6A

S�C

T6.

6M

g at Q=2 GeV

10-5 10-4 10-3 0.01 0.02 0.05 0.1 0.2 0.5 0.7

0.6

0.8

1.0

1.2

1.4

x

CT

6.6A

S�C

T6.

6M

c at Q=2 GeV

This agreement isa rigorous consequenceof the quadraticapproximation for χ2

C.-P. Yuan (MSU) Alphas Workshop, MPI Munich Feb 9, 2011 6

Details of the CTEQ6.6FAS analysis� Take the “world-average” αs(MZ) = 0.118± 0.002 as an input:

αs(MZ)|in = 0.118± 0.002 at 90% C.L.

� Find the theory parameter αs(MZ) as an output of a globalfit (CTEQ6.6FAS):

αs(MZ)|out = 0.118± 0.0019 at 90% C.L.

� The combined PDF+αs uncertainty is estimated as

∆X =1

2

√√√√22+1∑i=1

(X

(+)i −X(−)

i

)2� Problem: each PDF set comes with its own αs⇒

cumbersome

� A simple workaround exists!C.-P. Yuan (MSU) Alphas Workshop, MPI Munich Feb 9, 2011 7

A quadrature sum reproduces the αs-PDF correlationH.-L. Lai, J. Pumplin

Theorem

In the quadratic approximation, the total αs+PDF uncertainty ∆σof the CTEQ6.6FAS set, with all correlation, reduces to

∆X =√

∆X2CTEQ6.6 + ∆X2

αs,

where

� ∆XCTEQ6.6 is the CTEQ6.6 PDF uncertainty from 44 PDFs withthe same αs(MZ) = 0.118

� ∆Xαs = (X0.120 −X0.116)/2 is the αs uncertainty computedwith two central CTEQ6.6AS PDFs for αs(MZ) = 0.116 and 0.120

The full proof is given in the paper; the main idea is illustrated forthe αs parameter a0 and 1 PDF parameter a1

C.-P. Yuan (MSU) Alphas Workshop, MPI Munich Feb 9, 2011 8

Illustration of the theorem for 2 parameters

C.-P. Yuan (MSU) Alphas Workshop, MPI Munich Feb 9, 2011 9

Illustration of the theorem for 2 parameters, cont.

C.-P. Yuan (MSU) Alphas Workshop, MPI Munich Feb 9, 2011 10

PDF and αs uncertainties for gg → Hand tt̄ production

)Z

(MSα0.116 0.117 0.118 0.119 0.12

H+X

) [p

b]

→(p

pσ

10.2

10.4

10.6

10.8

11

11.2=120 GeV) at 7 TeV

HHiggs production (M

)Z

(MSα0.116 0.117 0.118 0.119 0.12

+X)

[pb

]t

t→

(pp

σ

830

840

850

860

870

880

890

900

910

920

930=171 GeV) at 14 TeV

t production (mtt

C.-P. Yuan (MSU) Alphas Workshop, MPI Munich Feb 9, 2011 11

Full and reduced fits with variable αs:cross sections

The full and reducedmethods perfectlyagree

C.-P. Yuan (MSU) Alphas Workshop, MPI Munich Feb 9, 2011 12

Correlations between αs and PDFsCorrelation cosine cosφ between the best-fit αs(MZ) and PDFs,plotted as a function of the momentum fraction x.

gHx,Q=2 GeVLSHx,Q=2 GeVLcHx,Q=2 GeVL

10-5 10-4 10-3 0.01 0.02 0.05 0.1 0.2 0.5 0.7-1.0

-0.5

0.0

0.5

1.0

x

cos@jD

CT6.6FAS: correlation of ΑsHMZ L with faHx,QL

gHx,Q=85 GeVLSHx,Q=85 GeVLcHx,Q=85 GeVLbHx,Q=85 GeVL

10-5 10-4 10-3 0.01 0.02 0.05 0.1 0.2 0.5 0.7-1.0

-0.5

0.0

0.5

1.0

x

cos@jD

CT6.6FAS: correlation of ΑsHMZ L with faHx,QL

� solid – gluon PDF;anticorrelated at x ∼ 0.01; fromHERA NC-DIS

� dashed – singlet PDF;correlated at x ∼ 0.4; fromBCDMS and NMC NC-DIS

� dashed-dotted – heavyquark (c, b) PDFs; correlated atx ∼ 0.05 - 0.2; from HERA

At large Q, these correlations arereduced by PDF evolution,except heavy-quark PDFs.

C.-P. Yuan (MSU) Alphas Workshop, MPI Munich Feb 9, 2011 13

Full fit with floating αs in CT10.AS Series

The Minimal χ2 found from a full fit with floating αs in CT10.ASseries.

)Z

(Msα0.11 0.112 0.114 0.116 0.118 0.12 0.122 0.124 0.126

2 χ

3000

3050

3100

3150

3200

3250

3300

3350

3400

CT10

The ∆χ2 = 100 range (at 90% C.L.)is 0.1197± 0.0061which is a rather weak constraintas compare to theworld-averageαs(MZ) = 0.118± 0.002

C.-P. Yuan (MSU) Alphas Workshop, MPI Munich Feb 9, 2011 14

SummaryCTEQ6.6AS and CT10.AS PDF sets (available in LHAPDF):� from 4 alternative CTEQ6.6 fits

(http://hep.pa.msu.edu/cteq/public/cteq6.html) for

αs(MZ) = 0.116, 0.117, 0.119, 0.120

� sufficient to compute uncertainty in αs(MZ) at ≈ 68% and90% C. L., including the world-average αs(MZ) = 0.118± 0.002 asan input data point

� The CTEQ6.6AS αs uncertainty should be combined with theCTEQ6.6 PDF uncertainty as

∆X =√

∆X2CTEQ6.6 + ∆X2

CTEQ6.6AS

� The total uncertainty ∆X reproduces the full correlationbetween αs(MZ) and PDFs

� A larger range (from 0.113 to 0.123) of αs(MZ) for CT10/W.ASseries is given at http://hep.pa.msu.edu/cteq/public/ct10.html

C.-P. Yuan (MSU) Alphas Workshop, MPI Munich Feb 9, 2011 15

Backup slides

C.-P. Yuan (MSU) Alphas Workshop, MPI Munich Feb 9, 2011 16

NMCF d2 /F

p2 and F p

2 data vs. αsMinimal χ2 from a full fit with floating αs in CT10.AS series.

sα0.112 0.114 0.116 0.118 0.12 0.122 0.124 0.126 0.128

NM

C F

2 R

atio

dat

a2 χ

120

121

122

123

124

125

126

127

128

129

130

CT10

sα0.112 0.114 0.116 0.118 0.12 0.122 0.124 0.126 0.128

NM

C F

2 p

roto

n d

ata

2 χ

335

340

345

350

355

360

365

CT10

�We did not find a significanteffect of NMC F d2 /F

p2 data on

αs, though a smaller value ismildly preferred. (χ2is close tothe number of data points,which is 123.)

� NMC F p2 data prefers alarger αs, but χ2is much largerthan the number of datapoints, which is 201.)

�We will examine the NMCreduced cross section (insteadof F2) in the next version ofCTEQ-TEA fits.

C.-P. Yuan (MSU) Alphas Workshop, MPI Munich Feb 9, 2011 17