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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 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
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
q
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
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
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.
13
• 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/
MSTW input parametrisation
18
Note: 20 parameters allowed to go free for eigenvector PDF sets, cf. 15 for MRST sets
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.
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
22
...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:
23
in the range 0.01 < x < 0.4
data seem to prefer
theoretical explanation?!
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-
25
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–
27
– 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 � �+ � ���� ���� �
28
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)]
29
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
•
31
• 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
33
• 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
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…
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)
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
remarkably similar considering the different definitions of
fractional uncertainty comparisons
49
different definitions of pdf uncertainties used by the 3 groups!
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
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; …)
benchmark top cross sections
63
a precision ATLAS/CMS measurement will be very useful in discriminating between pdf sets!
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!
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