Parton Distributions for the LHC

RHUL, 12 January 2011
Parton Distributions for the LHC
James Stirling
Cambridge 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 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
outline
•
deep inelastic scattering and parton
distribution functions
•
the MSTW08 parton distributions
•
implications for LHC cross sections
•
issues and outlook
1
deep inelastic scattering
and
parton distributions
deep inelastic scattering
• variables
electron
qµ
X
h 2 × 10−16 m GeV
λ = =
Q
Q
at HERA, Q2 < 105 GeV2
⇒ λ > 10-18 m = rp/1000
•in general, we can write
where the Fi(x,Q2) are called
structure functions
1.2
2
inelasticity
Q2
x=
Q 2 + M X2 − M p2
⇒0<x≤1
1.0
0.8
Bjorken
Scaling
2
Q (GeV )
1.5
3.0
5.0
8.0
11.0
8.75
24.5
230
80
800
8000
2
•
resolution
proton
F2(x,Q )
•
pµ
Q2 = –q2
x = Q2 /2p·q (Bjorken x)
( y = Q2 /x s )
0.6
0.4
0.2
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
the parton model (Feynman 1969)
• 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)
F2 ( x) =
1
∑ ∫
q ,q
=
•
infinite
momentum
frame
0
dξ eq2 ξ q (ξ ) δ ( x − ξ ) =
∑
eq2 x q( x)
q ,q
4
1
1
x u ( x) + x d ( x) + x s ( x) + ...
9
9
9
the functions u(x), d(x), s(x), … are called parton
distribution functions (pdfs) - they encode
information about the proton’s deep structure
6
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
structure function
measurements
4
1
1
(u + u ) + (d + d ) + ( s + s ) + ...
9
9
9
1
1
4
= (u + u ) + ( d + d ) + ( s + s ) + ...
9
9
9
= 2 d + s + u + ...
F2ep =
F2en
F2νp
F2νn
[
]
= 2 [u + d + s + ...]
• directly,
gluon not measured
but carries about
5 νN
s = s = F2 − 3F2eN
6
1/2 the proton’s momentum
∑ ∫ dx x (q( x) + q( x)) = 0.55
1
q
0
7
8
40 years of Deep Inelastic Scattering
1.2
2
1.0
0.8
2
F2(x,Q )
2
Q (GeV )
1.5
3.0
5.0
8.0
11.0
8.75
24.5
230
80
800
8000
0.6
0.4
0.2
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
x
9
scaling violations and QCD
The structure function data exhibit systematic violations
of Bjorken scaling:
6
4
F2
2
Q2 > Q1
Q1
quarks emit gluons!
0
0.0
0.2
0.4
0.6
0.8
1.0
x
DGLAP
equations
10
beyond lowest order in pQCD
going to higher orders in
pQCD is straightforward in
principle, since the above
structure for F2 and for
DGLAP generalises in a
straightforward way:
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
12
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.
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.
SUBROUTINE PDF(X,Q,U,UBAR,D,DBAR,…,BBAR,GLU)
input |
output
13
2
MSTW*
*Alan Martin, WJS, Robert Thorne, Graeme Watt
arXiv:0901.0002, arXiv:0905.3531, arXiv:1006.2753
the MRS/MRST/MSTW project
1985
•
since 1987 (with Alan Martin, Dick Roberts,
Robert Thorne, Graeme Watt), to produce ‘stateof-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
•
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
MRS(E,B)
MRS(E’,B’)
1990
1995
2000
2005
MRS(D-,D0,S0)
MRS(A)
MRS(A’,G)
MRS(Rn)
MRS(J,J’)
MRST1998
MRST1999
MRSTS2001E
MRST2004
MRST2004QED
MRST2006
MSTW2008
15
MSTW 2008 update
•
new data (see next slide)
•
new theory/infrastructure
− 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 Q02 :34-4=30 free parameters
including αS
code, text and figures available at:
http://projects.hepforge.org/mstwpdf/
16
data sets used in fit
17
MSTW input parametrisation
Note: 20 parameters allowed to go free for
eigenvector PDF sets, cf. 15 for MRST sets
18
which data sets determine which partons?
19
LO vs NLO vs NNLO?
in the MSTW2008 fit
χ2 global /dof =
3066/2598 (LO)
2543/2699 (NLO)
2480/2615 (NNLO)
LO evolution too slow at small x;
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.
20
21
the asymmetric sea
•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
The 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
•so we naively expect
u ≈ d > s > c > ...
• usea, dsea, s
obtained from fits to
data
▬
• c,b from pQCD, g Q Q
22
strange
earliest 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:
in the range 0.01 < x < 0.4
data seem to prefer
theoretical explanation?!
23
MSTW
24
charm, bottom
considered 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 αSnlogn(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), RobertsThorne) which interpolates smoothly between the two regimes
Note: definition of these is tricky and non-unique (ambiguity in
assignment of O(mH2//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
25
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
HERA-DIS
– treatment of heavy quarks (s,c,b)
FT-DIS
– order of perturbation theory
Drell-Yan
– parameterisation at Q0
Tevatron jets
– theoretical assumptions (if any) about: Tevatron
W,Z
• flavour symmetries
• x→0,1 behaviour
•…
other
27
MSTW08
CTEQ6.6X
NNPDF2.0
HERAPDF1.0
ABKM09
GJR08
HERA DIS
*
*
F-T DIS
F-T DY
TEV W,Z
+
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 αS(MZ2) 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
28
+ 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
dijet mass distribution from D0
D0 collaboration: arXiv:1002.4594
30
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
•
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.
31
pdfs and αS(MZ2)
•
•
•
•
MSTW08, ABKM09 and GJR08:
αS(MZ2) values and uncertainty
determined by global fit
NNLO value about 0.003 − 0.004
lower than NLO value, e.g. for
MSTW08
CTEQ, NNPDF, HERAPDF
choose 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
33
αS - pdf correlations
•
MSTW: arXiv:0905.3531
care needed when assessing
impact of varying αS on cross
sections ~ (αS )n (e.g. top, Higgs)
3
implications for LHC cross sections
the QCD factorization theorem for hard-scattering
(short-distance) inclusive processes
^σ
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
momentum fractions x1 and x2
determined by mass and rapidity of X
proton
proton
M
x1 P
x2 P
DGLAP evolution
37
precision phenomenology at LHC
• 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:
δσth = δσUHO ⊕ δσPDF ⊕ δσparam ⊕ …
• 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 useful
Note: V = γ*,Z,W±
38
• 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
•
NNLO for a limited number
of ‘precision observables’
(W, Z, DY, H, …)
+ E/W corrections, resummed
HO terms etc…
40
41
parton luminosity functions
•
a quick and easy way to assess the mass, collider
energy and pdf dependence of production cross sections
√s
a
b
X
•
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)
42
SS
VS
more such luminosity plots available at www.hep.phy.cam.ac.uk/~wjs/plots/plots.html
44
e.g.
ggH
qqVH
qqqqH
45
parton luminosity comparisons
positivity constraint
on input gluon
Run 1 vs. Run 2
Tevatron jet data
No Tevatron jet
data or FT-DIS
data in fit
ZM-VFNS
momentum sum rule
Luminosity and cross section plots from Graeme Watt (MSTW, in
preparation), available at projects.hepforge.org/mstwpdf/pdf4lhc
46
restricted parametrisation
no Tevatron
jet data in fit
47
new combined
HERA SF data
ZM-VFNS
48
fractional uncertainty comparisons
remarkably similar
considering the
different definitions of
pdf uncertainties used
by the 3 groups!
49
LHC benchmark
study for
Standard Model
cross sections at
7 TeV LHC
50
51
W, Z
52
benchmark W,Z cross sections
differences larger
than expected –
under investigation!
53
predictions for σ(W,Z) @ Tevatron, LHC:
NLO vs. NNLO
14 TeV
54
55
Higgs
56
Harlander,Kilgore
Anastasiou, Melnikov
Ravindran, Smith, van Neerven
…
•
only scale variation uncertainty shown
•
central values calculated for a fixed set pdfs with a fixed value of αS(MZ)
•
even at NNLO, a residual ~ ± 10-15% uncertainty in the predictions from
57
QCD scale variations!
benchmark Higgs cross sections
… differences from both pdfs and αS !
... recall additional scale uncertainty of ± 10-15% at NNLO
58
59
top
60
top quark production at hadron colliders
dominates
at Tevatron
dominates
at LHC
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; …)
61
LO
NLO
NNLOapprox
µf = 2m, m, m/2
Langenfeld, Moch, Uwer, arXiv:0906.5273
62
benchmark top cross sections
a precision ATLAS/CMS measurement will be very useful
in discriminating between pdf sets!
63
4
summary 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 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 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
variable(s)
process dependent coefficients
depending on P
renormalisation
scale
•
note dσ/dµ=0 “to all orders”, but in practice
dσ(N+n)/dµ= O((N+n)αSN+n+1)
•
can try to help convergence by using a “physical scale
choice”, µ ~ P , e.g. µ = MZ or µ = ETjet or µ = mtop
•
at hadron colliders, also have factorisation scale µ F
alongside µ R
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 αS(MZ2)
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– + …)
•
example:
in this case
whereas…
which can in principle help distinguish signal and background
69
C.H. Kom & WJS, arXiv:1004.3404
R± increases with jet pTmin
R± larger at 7 TeV LHC
70