Parton Distributions for the LHC

RHUL, 12 January 2011

James StirlingCambridge University

(with thanks to Alan Martin, Robert Thorne, Graeme Watt)

parton distribution functions

• introduced by Feynman (1969) in the parton model, to explain Bjorken scaling in deep inelastic scattering data; interpretation as probability distributionsdata; interpretation as probability distributions

• according to the QCD factorisation theorem for inclusive hard scattering processes, universal distributions containing long-distance structure of hadrons; related to parton model distributions at leading order, but with logarithmic scaling violations (DGLAP)

• key ingredients for Tevatron and LHC phenomenology

• deep inelastic scattering and parton distribution functions

• the MSTW08 parton distributions

outline

• the MSTW08 parton distributions

• implications for LHC cross sections

• issues and outlook

deep inelastic scatteringand

1

andparton distributions

deep inelastic scattering

qµ

pµ

X

electron

proton

• variablesQ2 = –q2

x = Q2 /2p·q (Bjorken x)

( y = Q2 /x s )

• resolution •in general, we can write• resolution

at HERA, Q2 < 105 GeV2

⇒ λ > 10-18 m = rp/1000

• inelasticity

⇒ 0 < x ≤ 1

QQ

h GeVm102 16−×==λ

222

2

pX MMQ

Qx

−+=

•in general, we can write

where the Fi(x,Q2) are calledstructure functions

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.0

0.2

0.4

0.6

0.8

1.0

1.2

Q2 (GeV2) 1.5 3.0 5.0 8.0 11.0 8.75 24.5 230 80 800 8000

F2(

x,Q

2 )BjorkenScaling

• photon scatters incoherently off massless, pointlike, spin-1/2 quarks

• probability that a quark carries fraction ξ of parent proton’s momentum is q(ξ), (0< ξ < 1)

the parton model (Feynman 1969)

1

∑∫∑

infinitemomentumframe

6

• the functions u(x), d(x), s(x), … are called parton distribution functions (pdfs) - they encode information about the proton’s deep structure

...)(9

1)(

9

1)(

9

4

)()()()( 2

,

21

0,

2

+++=

=−= ∑∫∑

xsxxdxxux

xqxexqedxF q

qq

q

qq

ξδξξξ

extracting pdfs from experiment• different beams (e,µ,ν,…) &

targets (H,D,Fe,…) measure different combinations of quark pdfs

• thus the individual q(x) can be extracted from a set of [ ]...2

...)(9

1)(

9

4)(

9

1

...)(9

1)(

9

1)(

9

4

2

2

+++=

++++++=

++++++=

usdF

ssdduuF

ssdduuF

p

en

ep

ν

7

be extracted from a set of structure function measurements

• gluon not measured directly, but carries about 1/2 the proton’s momentum

[ ][ ]...2

...2

2

2

+++=

+++=

sduF

usdFn

p

ν

ν

( ) 55.0)()(1

0=+∫∑ xqxqxdx

q

eNN FFss 22 36

5 −== ν

8

40 years of Deep Inelastic Scattering

0.8

1.0

1.2

Q2 (GeV2) 1.5 3.0 5.0 8.0 11.0 8.75

9

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.0

0.2

0.4

0.6 24.5 230 80 800 8000

F2(x

,Q2 )

x

scaling violations and QCD

2

4

6

F2

Q1

Q2 > Q1

The structure function data exhibit systematic violations of Bjorken scaling:

10

quarks emit gluons!0.0 0.2 0.4 0.6 0.8 1.0

0

x

Q1

DGLAPequations

going to higher orders in pQCD is straightforward in principle, since the above structure for F2 and for DGLAP generalises in a straightforward way:

beyond lowest order in pQCD

1972-77 1977-80 2004

The 2004 calculation of the complete set of P(2) splitting functions by Moch, Vermaseren and Vogt (hep-ph/0403192,0404111) completes the calculational tools for a consistent NNLO (massless) pQCD treatment of Tevatron & LHC hard-scattering cross sections

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summary: how pdfs are obtained• choose a factorisation scheme (e.g. MSbar), an order in

perturbation theory (see below, e.g. LO, NLO, NNLO) and a ‘starting scale’ Q0 where pQCD applies (e.g. 1-2 GeV)

• parametrise the quark and gluon distributions at Q0,, e.g.

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• solve DGLAP equations to obtain the pdfs at any x and scale Q > Q0 ; fit data for parameters {Ai,ai, …αS}

• approximate the exact solutions (e.g. interpolation grids, expansions in polynomials etc) for ease of use; thus the output ‘global fits’ are available ‘off the shelf”, e.g.

input | output

SUBROUTINE PDF(X,Q,U,UBAR,D,DBAR,…,BBAR,GLU)

MSTW*

2MSTW*

*Alan Martin, WJS, Robert Thorne, Graeme Watt

arXiv:0901.0002, arXiv:0905.3531, arXiv:1006.2753

the MRS/MRST/MSTW project

• since 1987 (with Alan Martin, Dick Roberts, Robert Thorne, Graeme Watt), to produce ‘state-of-the-art’ pdfs

• combine experimental data with theoretical formalism to perform ‘global fits' to data to extract the pdfs in user-friendly form for the particle physics community

1985

1990

1995

MRS(E,B)MRS(E’,B’)

MRS(D-,D0,S0)

MRS(A)MRS(A’,G)

15

the pdfs in user-friendly form for the particle physics community

• currently widely used at HERA and the Fermilab Tevatron, and in physics simulations for the LHC

• currently, the only available NNLO pdf sets with rigorous treatment of heavy quark flavours

1995

2000

2005

MRS(A’,G)MRS(Rn)MRS(J,J’)MRST1998MRST1999MRSTS2001E

MRST2004MRST2004QED

MRST2006

MSTW2008

MSTW 2008 update

• new data (see next slide)

• new theory/infrastructure

16

− δfi from new dynamic tolerance method: 68%cl (1σ) and 90%cl (cf. MRST) sets available

− new definition of αS (no more ΛQCD)− new GM-VFNS for c, b (see Martin et al., arXiv:0706.0459)− new fitting codes: FEWZ, VRAP, fastNLO− new grids: denser, broader coverage− slightly extended parametrisation at Q0

2 :34-4=30 free parameters including αS

code, text and figures available at: http://projects.hepforge.org/mstwpdf/

data sets used in fit

17

MSTW input parametrisation

18

Note: 20 parameters allowed to go free for eigenvector PDF sets, cf. 15 for MRST sets

which data sets determine which partons?

19

in the MSTW2008 fit

3066/2598 (LO)χ2

global /dof = 2543/2699 (NLO)2480/2615 (NNLO)

LO evolution too slow at small x; NNLO fit marginally better than NLO

LO vs NLO vs NNLO?

20

NNLO fit marginally better than NLO

Note: • an important ingredient missing in the full NNLO global PDF fit is the NNLO correction to the Tevatron high ET jet cross section• LO can be improved (e.g. LO*) for MCs by adding K-factors, relaxing momentum conservation, etc.

21

the asymmetric seaThe ratio of Drell-Yan cross sections for pp,pn → µ+µ- + X provides a measure of the difference between the u and d sea quark distributions

•the sea presumably arises when ‘primordial‘ valence quarks emit gluons which in turn split into quark-antiquark pairs, with suppressed splitting into heavier quark pairs

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...csdu >>>≈

•so we naively expect

• usea, dsea, s obtained from fits to data

• c,b from pQCD, g � Q Q▬

strangeearliest pdf fits had SU(3) symmetry:

later relaxed to include (constant) strange suppression (cf. fragmentation):

with κ = 0.4 – 0.5

nowadays, dimuon production in υN DIS (CCFR, NuTeV) allows ‘direct’ determination:

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in the range 0.01 < x < 0.4

data seem to prefer

theoretical explanation?!

24

MSTW

charm, bottomconsidered sufficiently massive to allow pQCD treatment:

distinguish two regimes:(i) include full mH dependence to get correct threshold behaviour(ii) treat as ~massless partons to resum ααααS

nlogn(Q2/mH2) via DGLAP

FFNS: OK for (i) only ZM-VFNS: OK for (ii) only

consistent GM(=general mass)-VFNS now available (e.g. ACOT(χ), Roberts-

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consistent GM(=general mass)-VFNS now available (e.g. ACOT(χ), Roberts-Thorne) which interpolates smoothly between the two regimes

Note: definition of these is tricky and non-unique (ambiguity in assignment of O(mH

2//Q2) contributions), and the implementation of improved treatment (e.g. in going from MRST2006 to MSTW 2008) can have a big effect on light partons

charm and bottom structure functions

• MSTW 2008 uses fixed values of mc = 1.4 GeV and mb = 4.75 GeV in a GM-VFNS

• the sensitivity of the fit to these values, and impact on LHC cross sections, is discussed in arXiv:1006.2753

the pdf industry• many groups now extracting pdfs from ‘global’

data analyses (MSTW, CTEQ, NNPDF, …)

• broad agreement, but differences due to– choice of data sets (including cuts and corrections)– treatment of data errors–

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– treatment of data errors– treatment of heavy quarks (s,c,b)– order of perturbation theory– parameterisation at Q0

– theoretical assumptions (if any) about: • flavour symmetries• x→0,1 behaviour• …

HERA-DIS

FT-DIS

Drell-Yan

Tevatron jets

Tevatron W,Z

other

MSTW08 CTEQ6.6X NNPDF2.0 HERAPDF1.0 ABKM09 GJR08

HERA DIS � � �* �* � �

F-T DIS � � � ���� � �

F-T DY � � � ���� � �

TEV W,Z � �+ � ���� ���� ����

TEV jets � �+ � ���� ���� �

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TEV jets � � � ���� ���� �

GM-VFNS � � ���� � ���� ����

NNLO � ���� ���� ���� � �

+ Run 1 only* includes new combined H1-ZEUS data � 1 – 2.5% increase in quarks at low x (depending on procedure), similar effect on ααααS(MZ

2) if free and somewhat less on gluon; more stable at NNLO (MSTW prelim.)

X New (July 2010) CT10 includes new combined H1-ZEUS data + Run 2 jet data + extended gluon parametrisation + … � more like MSTW08

impact of Tevatron jet data on fits• a distinguishing feature of pdf sets is whether they use (MRST/MSTW,

CTEQ, NNPDF, GJR,…) or do not use (HERAPDF, ABKM, …) Tevatron jet data in the fit: the impact is on the high-x gluon (Note: Run II data requires slightly softer gluon than Run I data)

• the (still) missing ingredient is the full NNLO pQCD correction to the cross section, but not expected to have much impact in practice [Kidonakis, Owens (2001)]

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dijet mass distribution from D0

30D0 collaboration: arXiv:1002.4594

pdf uncertainties• most groups produce ‘pdfs with errors’

• typically, 20-40 ‘error’ sets based on a ‘best fit’ set to reflect ±1σ variation of all the parameters* {Ai,ai,…,αS}inherent in the fit

•

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• these reflect the uncertainties on the data used in the global fit (e.g. δF2 ≈ ±3% → δu ≈ ±3%)

• however, there are also systematic pdf uncertainties reflecting theoretical assumptions/prejudices in the way the global fit is set up and performed (see earlier slide)

* e.g.

pdfs and αS(MZ2)

• MSTW08, ABKM09 and GJR08: αS(MZ

2) values and uncertainty determined by global fit

• NNLO value about 0.003 − 0.004lower than NLO value, e.g. for MSTW08

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• CTEQ, NNPDF, HERAPDFchoose standard values and uncertainties

• world average (PDG 2009)

• note that the pdfs and αS are correlated!

• e.g. gluon – αS anticorrelation at small x and quark – αS anticorrelation at large x

αS - pdf correlations

• care needed when assessing impact of varying αSon cross sections ~ (αS )n (e.g. top, Higgs)

MSTW: arXiv:0905.3531

implications for LHC cross sections

3implications for LHC cross sections

the QCD factorization theorem for hard-scattering (short-distance) inclusive processes

σ̂

where X=W, Z, H, high-ET jets, SUSY sparticles, black hole, …, and Q is the ‘hard scale’ (e.g. = MX), usually µF = µR = Q, and σ is known …

• to some fixed order in pQCD (and EW), e.g. high-ET jets

• or ‘improved’ by some leading logarithm approximation (LL, NLL, …) to all orders via resummation

^

DGLAP evolution

momentum fractions x1 and x2

determined by mass and rapidity of X

x1P

proton

x2P

proton

M

37

DGLAP evolution

• Benchmarking (‘Standard Candles’)– inclusive SM quantities (V, jets, top,… ), calculated to

the highest precision available (e.g. NNLO, NNLL, etc)– tools needed: robust jet algorithms, kinematics, decays

included, PDFs, …– theory uncertainty in predictions:

precision phenomenology at LHC

• Backgrounds– new physics generally results in some combination of

multijets, multileptons, missing ET

– therefore, we need to know SM cross sections {V,VV,bb,tt,H,…} + jets to high precision � `wish lists’

– ratios can be useful38Note: V = γ*,Z,W±

δσth = δσUHO ⊕ δσPDF ⊕ δσparam ⊕ …

• Learning more about PDFs– high-ET jets, top → gluon? – W+,W–,Z0 → quarks? – very forward Drell-Yan (e.g. LHCb) → small x?– …

39

precision phenomenology at LHC

• LO for generic PS Monte Carlos

• NLO for NLO-MCs and many parton-level signal and background processes

40

and background processes

• NNLO for a limited number of ‘precision observables’ (W, Z, DY, H, …)

+ E/W corrections, resummed HO terms etc…

41

parton luminosity functions• a quick and easy way to assess the mass, collider energy and pdf dependence of production cross sections

√s Xa

b

42

• i.e. all the mass and energy dependence is contained in the X-independent parton luminosity function in [ ]• useful combinations are • and also useful for assessing the uncertainty on cross sections due to uncertainties in the pdfs (see later)

more such luminosity plots available at www.hep.phy.cam.ac.uk/~wjs/plots/plots.html

SSVS

44

e.g.gg�H

45

gg�Hqq�VHqq�qqH

parton luminosity comparisons

Run 1 vs. Run 2 Tevatron jet data

positivity constraint on input gluon

46

Luminosity and cross section plots from Graeme Watt (MSTW, in preparation), available at projects.hepforge.org/mstwpdf/pdf4lhc

momentum sum ruleZM-VFNS

No Tevatron jet data or FT-DIS data in fit

restricted parametrisation

47

no Tevatron jet data in fit

new combined HERA SF data

ZM-VFNS

48

remarkably similar considering the different definitions of

fractional uncertainty comparisons

49

different definitions of pdf uncertainties used by the 3 groups!

LHC benchmark study for Standard Model

50

Standard Model cross sections at 7 TeV LHC

51

W, Z

52

W, Z

differences larger than expected –under investigation!

benchmark W,Z cross sections

53

predictions for σ(W,Z) @ Tevatron, LHC:NLO vs. NNLO

14 TeV

54

55

Higgs

56

Higgs

Harlander,Kilgore

57

Harlander,KilgoreAnastasiou, MelnikovRavindran, Smith, van Neerven…

• only scale variation uncertainty shown

• central values calculated for a fixed set pdfs with a fixed value of ααααS(MZ)

• even at NNLO, a residual ~ ± 10-15% uncertainty in the predictions from QCD scale variations!

benchmark Higgs cross sections

58

… differences from both pdfs and ααααS !... recall additional scale uncertainty of ± 10-15% at NNLO

59

top

60

top

top quark production at hadron colliders

dominates at Tevatron

dominates at LHC

61

NLO known, but awaits full NNLO pQCD; NNLO & NnLL “soft+virtual” fixed order and resummed approximations exist

(Moch, Uwer, Langenfeld; Kidonakis, Vogt; Cacciari, Frixione, Mangano, Nason, Ridolfi; Ahrens, Ferroglia, Neubert, Pecjak, Yang; Beneke, Falgari, Schwinn; …)

LONLONNLOapprox

µf = 2m, m, m/2

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Langenfeld, Moch, Uwer, arXiv:0906.5273

benchmark top cross sections

63

a precision ATLAS/CMS measurement will be very useful in discriminating between pdf sets!

summary and outlook

4summary and outlook

summary and outlook• precision phenomenology at high-energy colliders such

as the LHC requires an accurate knowledge of the distribution functions of partons in hadrons

• determining pdfs from global fits to data is now a major industry… the MSTW collaboration has produced (2008) LO, NLO, NNLO sets specifically for precision phenomenology at LHCphenomenology at LHC

• benchmark studies of standard candle cross sections and ratios

• W,Z cross sections and ratios well predicted (luminosity measurements?); bigger spread for other cross sections (Higgs, top)

• eagerly awaiting precision cross sections at 7,8,...14 TeV!

extra slidesextra slides

general structure of a QCD perturbation series

• choose a renormalisation scheme (e.g. MSbar)• calculate cross section to some order (e.g. NLO)

physical process dependent coefficients renormalisation

• note dσ/dµ=0 “to all orders”, but in practicedσ(N+n)/dµ= O((N+n)αS

N+n+1)

• can try to help convergence by using a “physical scale choice”, µ ~ P , e.g. µ = MZ or µ = ET

jet or µ = mtop

• at hadron colliders, also have factorisation scale µ Falongside µ R

physical variable(s)

process dependent coefficientsdepending on P

renormalisationscale

the impact of NNLO: W,Z

Anastasiou, Dixon, Melnikov, Petriello, 2004

• only scale variation uncertainty shown• central values calculated for a fixed set PDFs with a fixed value of ααααS(MZ

2)

using the W+- charge asymmetry at the LHC

• at the Tevatron σ(W+) = σ(W–), whereas at LHC σ(W+) ~ (1.4 –1.3) σ(W–)

• can use this asymmetry to calibrate backgrounds to new physics, since typically σNP(X → W+ + …) = σNP(X → W– + …)

•

69

• example:

in this case

whereas…

which can in principle help distinguish signal and background

C.H. Kom & WJS, arXiv:1004.3404

70

R± larger at 7 TeV LHC

R± increases with jet pTmin