1. From the Tevatron to the LHC: Parton Distribution Functions (pdfs)
James Stirling
Cambridge University
(with thanks to Alan Martin, Robert Thorne, Graeme Watt)

2. pdfs: global fits and impact of Tevatron data1
pdfs: global fits and impact of Tevatron data

3. what is a pdf?
the probability of finding a parton of type i ( = u,d,….,g ) carrying a fraction of momentum between x and x+dx in a fast-moving proton when probed at scale Q q p X
electron
proton
Q2 = –q2
x = Q2 /2p·q

4. data sets used in the MSTW2008* fit
*A.D. Martin, W.J. Stirling, R.S. Thorne, G. Watt, arXiv: [hep-ph]

5. which data sets determine which partons?

8. MSTW2008(NLO) vs. CTEQ6.6 Note:
CTEQ error bands comparable with MSTW 90%cl set (different definition of tolerance)
CTEQ light quarks and gluons slightly larger at small x because of imposition of positivity on gluon at Q02
Also:
Alekhin et al
HERAPDF
NNPDF …

9. Note: MSTW are unable to fit the more recent D0 Run II W→e data from arXiv:0807.3367

11. impact of Tevatron jet data on fits
a distinguishing feature of pdf sets is whether they use (MRST/MSTW, CTEQ,…) or do not use (H1, ZEUS, Alekhin, NNPDF,…) 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

12. dijet mass distribution from D0
Rominsky, DIS09

13. 2
pdfs at LHC: — impact on precision phenomenology — can we constrain pdfs further?

14. the QCD factorization theorem for hard-scattering (short-distance) inclusive processes^ 
where X=W, Z, H, high-ET jets, SUSY sparticles, black hole, …, and Q is the ‘hard scale’ (e.g. = MX), usually F = R = Q, and  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 ^

15. momentum fractions x1 and x2 determined by mass and rapidity of X
x1P
proton
x2P M
DGLAP evolution

17. pdfs at LHC – the issues
high precision cross section predictions require accurate knowledge of pdfs: th = pdf + …
→ improved signal and background predictions
→ easier to spot new physics
‘standard candle’ processes (e.g. Z) to
check formalism (factorisation, DGLAP, …)
measure machine luminosity?
learning more about pdfs from LHC measurements. e.g.
high-ET jets → gluon?
W+,W–,Z0 → quarks?
forward Drell-Yan → small x? …

18. how important is pdf precision?
Example 1: σ(MH=120 LHC
σpdf  ±2%, σptNNL0  ± 10%
σptNNLL  ± 8%
→ σtheory  ± 10%
Example 2: LHC
σpdf  ±2%, σptNNL0  ± 2%
→ σtheory  ± 3%
Harlander,Kilgore
Anastasiou, Melnikov
Ravindran, Smith, van Neerven
see talk by Robert Harlander
±2%
MSTW

19. Example 3: σ(tt) @ LHC → σtheory  ± 4%
Moch, Uwer
Example 3: LHC
σpdf  ±2%, σptNNL0approx  ± 3%
→ σtheory  ± 4%

20. pdf uncertainty on (gg→H)
→ typically ± 2-3% pdf uncertainty, except near edges of phase space

21. parton luminosity functions
a quick and easy way to assess the mass and collider energy dependence of production cross sections
s M a b
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

22. LHC / Tevatron
LHC
Tevatron
Huston, Campbell, S (2007)

23. parton luminosity (68%cl) uncertainties at LHC

24. LHC at 10 TeV

25. a note on S world average value (PDG 2008):
MSTW global fit value (minimum 2):
the pdf error sets are generated with S fixed at its ‘best fit’ value, therefore variation of (e.g. jets, top, etc at LHC) cross sections with S is not explicitly included in the ‘pdf error’
Note:

26. new MSTW variable-S sets
allow S to vary in global fit
for fixed S ± S, produce sets with ‘pdf errors’, as before
note gluon – S anticorrelation at small x and quark – S anticorrelation at large x
use resulting sets to quantify combined ‘pdf + S’ error on observables

28. pdf, S uncertainties in jet cross sections

30. pdfs at LHC – the issues
high precision cross section predictions require accurate knowledge of pdfs: th = pdf + …
→ improved signal and background predictions
→ easier to spot new physics
‘standard candle’ processes (e.g. Z) to
check formalism (factorisation, DGLAP, …)
measure machine luminosity?
learning more about pdfs from LHC measurements. e.g.
high-ET jets → gluon?
W+,W–,Z0 → quarks?
forward DY → small x? …

31. standard candles: (W,Z) @ LHC
cross sections (total and rapidity distributions) known to NNLO pQCD and NLO EW; perturbation series seems to be converging quickly
EW parameters well measured at LEP
samples pdfs where they are well measured (in x) in DIS
… although the mix of quark flavours is different: F2 and (W,Z) probe different combinations of u,d,s,c,b → sea quark distributions important
precise measurement of cross section ratios at LHC (e.g. (W+)/(W-), (W)/(Z)) will allow these subtle effects to be explored further

32. LHC
Tevatron
at LHC, ~30% of W and Z total cross sections involves s,c,b quarks

34. Note: at NNLO, factorisation and renormalisation scale variation M/2 → 2M gives an additional ± 2% change in the LHC cross sections

35. predictions for (W,Z) @ LHC (Tevatron)
Bl .W (nb)
Bll .Z (nb)
MSTW 2008 NLO
(2.659)
(0.2426)
MSTW 2008 NNLO
(2.747)
(0.2507)
MRST 2006 NLO
(2.645)
(0.2426)
MRST 2006 NNLO
(2.759)
(0.2535)
MRST 2004 NLO
(2.632)
(0.2424)
MRST 2004 NNLO
(2.724)
(0.2519)
CTEQ6.6 NLO
(2.599)
(0.2393)
Alekhin 2002 NLO
(2.733)
(0.2543)
Alekhin 2002 NNLO
(2.805)
(0.2611)
MSTW

36. predictions for (W,Z) @ Tevatron, LHC
MRST/MSTW NNLO: 2008 ~ 2006 > 2004 mainly due to changes in treatment of charm
CTEQ: 6.6 ~ 6.5 > 6.1 due to changes in treatment of s,c,b
NLO: CTEQ6.6 2% higher than MSTW 2008 at LHC, because of slight differences in quark (u,d,s,c) pdfs, difference within quoted uncertainty

37. Note: still dominated by scale variation uncertainty - see earlier

38. R± = (W+→l+) / (W→l)
σth  σpdf  ±1%,
σexpt  ???

42. using the W+- charge asymmetry at the LHC
at the Tevatron (W+) = (W–), whereas at LHC (W+) ~ 1.3(W–)
can use this asymmetry to calibrate backgrounds to new physics, since typically NP(X → W+ + …) = NP(X → W– + …)
example:
in this case
whereas…
which can in principle help distinguish signal and background

43. pdfs at LHC – the issues
high precision cross section predictions require accurate knowledge of pdfs: th = pdf + …
→ improved signal and background predictions
→ easier to spot new physics
‘standard candle’ processes (e.g. Z) to
check formalism (factorisation, DGLAP, …)
measure machine luminosity?
learning more about pdfs from LHC measurements. e.g.
high-ET jets → gluon?
W+,W–,Z0 → quarks?
forward DY → small x? …

44. impact of LHC measurements on pdfs
the standard candles: central (W,Z,tt,jets) as a probe and test of pdfs in the x ~ 10 -2±1, Q2 ~ GeV2 range where most New Physics is expected (H, SUSY, ….)
forward production of (relatively) low-mass states (e.g. *,dijets,…) to access partons at x<<1 (and x~1)
W,Z *

45. LHCb Unique features pseudo-rapidity range 1.9 - 4.9
complementary to ATLAS/CMS
> 2.5 unique to LHCb
beam defocused at LHCb: 1 year of running = 2 fb-1
trigger on low momentum muons: p > 8 GeV, pT > 1 GeV
access to unique range of (x,Q2)

46. LHCb
→ detect forward, low pT muons from

47. Ronan McNulty et al, at DIS08

48. Impact of 1 fb-1 LHCb data for forward Z and 
Impact of 1 fb-1 LHCb data for forward Z and * (M = 14 GeV) production on the gluon distribution uncertainty

49. summary
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 – there are a number of sets available on the market: LHC phenomenology requires full NNLO, with proper treatment of heavy quarks in the proton
pdf uncertainty for ‘new physics’ cross sections not expected to be too important (few % level), apart from at very high mass; this results from the accuracy of the data used in the global fits (inc. Tevatron data)
ongoing high-precision studies of standard candle cross sections and ratios
potential of LHCb to access very small x via low-mass Drell-Yan lepton pair production – ongoing study
still to do: an update of the 2004 MRST ‘pdfs with QED effects’ to allow for consistent treatment of O() E/W effects on cross sections

50. “Proton Structure at the LHC”
IoP Half─day meeting
“Proton Structure at the LHC”
Cavendish Laboratory, Cambridge
Wednesday, 3 June 2009
2.00pm – 5.30pm
The purpose of the meeting is to engage the wider UK (LHC) experimental community with the UK's "pdf experts" to identify topics of mutual interest, in particular: (i) the importance of a detailed knowledge of pdfs for precision SM and BSM LHC predictions and measurements, (ii) what can we learn about pdfs from LHC measurements, and (iii) how can the UK HEP community best contribute to these topics.
IOP Institute of Physics

51. extra slides

54. using the W+- charge asymmetry at the LHC
at the Tevatron (W+) = (W–), whereas at LHC (W+) ~ 1.3(W–)
can use this asymmetry to calibrate backgrounds to new physics, since typically NP(X → W+ + …) = NP(X → W– + …)
example:
in this case
whereas…
which can in principle help distinguish signal and background

55. for njet > 1 dominant subprocess is:
W+- + n LHC
for njet > 1 dominant subprocess is:
W+nj
%qq
%qg
%gg
W+0j
100
W+1j 75 25
W+2j 18 7
W+3j 72 10

56. W+/W- ratio:
very sensitive to u/d pdf ratio
varies with yW
depends slightly on njet and ETj(min)
fairly independent of scale choice etc

60. 70 GeV  7 TeV ep

62. MSTW2008 vs MRST2006

63. MSTW2008 vs Alekhin2002

64. MSTW2008 vs NNPDF1.0

65. R(W/Z)=(W)/(Z) @ Tevatron & LHC
CDF 2007: R = ± 0.15 (stat) ± 0.14 (sys)

66. scaling violations measured at HERA
NLO DGLAP fit

67. MSTW 2008 update new data (see next slide) new theory/infrastructure
fi from new dynamic tolerance method: 68%cl (1) and 90%cl (cf. MRST) sets available
new definition of S (no more QCD)
new GM-VFNS for c, b (see Martin et al., arXiv: )
new fitting codes: FEWZ, VRAP, fastNLO
new grids: denser, broader coverage
slightly extended parameterisation at Q02 :34-4=30 free parameters including S
code, text and figures available at:
and in latest version LHAPDF:

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

69. pdf uncertainties
this defines a set of n ‘error’ pdfs, spanning the allowed variation in the parameters, as determined by T:
rather than using a fixed value of T (cf. MRST, CTEQ), we determine the ‘dynamic’ tolerance for each eigenvector from the condition that all data sets should be described within their 68% or 90% or … confidence limit

71. 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
so we naively expect
but why such a big d-u asymmetry? Meson cloud, Pauli exclusion, …?
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

72. strange earliest pdf fits had SU(3) symmetry:
later relaxed to include (constant) strange suppression (cf. fragmentation):
with  = 0.4 – 0.5
nowadays, dimuon production in N DIS (CCFR, NuTeV) allows ‘direct’ determination:
in the range 0.01 < x < 0.4
data seem to prefer
theoretical explanation?!

73. MSTW

74. MSTW

75. 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 Snlogn(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(), Roberts-Thorne) 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

76. charm and bottom structure functions
F2bb
F2cc
MSTW 2008

77. Pumplin et al., arXiv:0904.2424 [hep-ph]

78. Pumplin et al., arXiv:0904.2424 [hep-ph]
similar, but not identical, to published CDF Run-2 midpoint jet data

79. A. D. Martin, W. J. Stirling, R. S. Thorne, G. Watt, arXiv:0901
A.D. Martin, W.J. Stirling, R.S. Thorne, G. Watt, arXiv: [hep-ph]

80. A. D. Martin, W. J. Stirling, R. S. Thorne, G. Watt, arXiv:0901
A.D. Martin, W.J. Stirling, R.S. Thorne, G. Watt, arXiv: [hep-ph] (revised)