Toward NNLO accuracy of parton distribution functions Pavel Nadolsky Southern Methodist University Dallas, TX, U.S.A. in collaboration with M. Guzzi, F . Olness, J. Huston, H.-L. Lai, Z. Li, J. Pumplin, C.-P . Yuan (CTEQ) September 23, 2011 Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 1
NNLO PDF sets as the new norm � PDFs with NNLO (two-loop) QCD corrections to DIS and DY processes are becoming the standard. They are now produced by 6 groups. � Our general-purpose set CT10 is obtained at NLO (PRD82, 074024 (2010)) . The CT10.1 NNLO set undergoes pre-publication tests. Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 2
The agreement between NNLO PDF sets is not automatically better than at NLO Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 3
The agreement between NNLO PDF sets is not automatically better than at NLO NNLO gg → H at the Tevatron ( s = 1.96 TeV) for M = 180 GeV H (pb) (pb) 0.3 0.3 0.28 0.28 H H σ σ 0.26 0.26 0.24 0.24 0.22 0.22 68% C.L. PDF 0.2 0.2 MSTW08 0.18 0.18 HERAPDF1.0 0.16 0.16 ABKM09 GJR08/JR09 Vertical error bars 0.14 0.14 Inner: PDF only Closed symbols: NNLO Outer: PDF+ α 0.12 0.12 S Open symbols: NLO 0.11 0.11 0.115 0.115 0.12 0.12 0.125 0.125 0.13 0.13 2 2 α α (M (M ) ) G. Watt, in PDF4LHC study, arXiv:1101.0536 S S Z Z Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 3
χ 2 /N data points in various experiments (PRELIMINARY) D0 Run-2 A e PDF set Order All Combined BCDMS CDF , D0 ch , F p,d p e expts. HERA-1 DIS Run-2 1-jet T > 25 GeV 2 1.11 1.17 1.10 1.33 3.72 CT10.1 1.16 1.31 11.38 1.42 1.73 MSTW08 NLO (1.28) (1.4) (1.17) 1.28 1.57 2.79 1.37 1.32 NNPDF2.0 1.13 1.12 1.14 1.23 2.59 CT10.1 1.15 1.38 9.84 1.34 1.36 MSTW08 1.57 1.36 1.30 1.51 5.45 NNPDF2.1 NNLO 1.49 2.63 23.78 1.65 1.4 ABM’09 (5f) 1.87 ? 5.4 1.71 1.15 HERA1.5 N points 2798 579 590 182 12 Cross sections are computed using the CTEQ fitting code and α s, m c , m b values provided by each PDF set. Their agreement does not immediately improve after going to NNLO. Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 4
Origin of differences between PDF sets � NNLO QCD terms (in all 6 PDF fits) ◮ Implementation of heavy-quark mass terms � (N)LO electroweak contributions � Selection of data : global analyses (CTEQ, MSTW, NNPDF) vs. restricted (“DIS-based”) analyses (ABM, GJR, HERAPDF) � Statistical treatment : Monte-Carlo sampling vs. analytical minimization of χ 2 ; correlated systematic uncertainties; definitions of PDF uncertainties � Initial PDF parametrizations : neural networks (NNPDF) ; 2-5 parameters per flavor (other fits) � Values of α s ( M Z ) , m c , and m b and their treatment � Differences in NLO codes used by PDF fits Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 5
This talk Genuine NNLO accuracy requires an earnest effort to calibrate all components of the PDF fits I will provide examples of related activities, focusing on � Heavy-quark contributions to DIS at O ( α 2 s ) M. Guzzi, P .N., H.-L. Lai, C.-P . Yuan, arXiv:1108.5112 ; additional figures at http://bit.ly/SACOTNNLO1 1 � W charge asymmetry at the Tevatron 1. H.-L. Lai, M. Guzzi, J. Huston, Z. Li, P . N., J. Pumplin, C.-P . Yuan, Phys.Rev. D82 (2010) 074024. 2. M. Guzzi, P . N., E. Berger, H.-L. Lai, F . Olness, C.-P . Yuan, arXiv:1101.0561 [hep-ph]. � Consistency of DGLAP picture at small x at HERA 1. H.-L. Lai, M. Guzzi, J. Huston, Z. Li, P . N., J. Pumplin, C.-P . Yuan, Phys.Rev. D82 (2010) 074024. Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 6
1. Heavy quarks in DIS and LHC electroweak cross sections The latest PDF fits assume m c,b � = 0 when evaluating Wilson coeffi- cients in DIS, e ± p → e ± X and e ± p → νX This is needed, in particular, to correctly predict W, Z production rates at the LHC (Tung et al., hep-ph/0611254) W � & Z cross sections at the LHC (14 TeV) 2.15 NNLL � NLO ResBos 2.1 CTEQ6.6 (GM) � X � � nb � 2.05 � Σ tot � pp � � Z 0 � {{ 2 1.95 CTEQ6.1 (ZM) 1.9 1.85 18.5 19 19.5 20 20.5 21 21.5 22 CMS-PAS-EWK-10-005 Σ tot � pp � � W � � { Ν � X � � nb � - arXiv:0802.0007 Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 7
1. Heavy quarks in DIS and LHC electroweak cross sections In pp → Z 0 X at √ s = 14 TeV: σ ( general-mass PDFs ) ≈ 1 . 05 − 1 . 08; σ ( zero-mass PDFs ) to be compared with W � & Z cross sections at the LHC (14 TeV) K NNLO/NLO ≡ σ ( α 2 2.15 NNLL � NLO ResBos s ) /σ ( α s ) ≈ 1 . 02 2.1 CTEQ6.6 (GM) � X � � nb � 2.05 � Σ tot � pp � � Z 0 � {{ 2 1.95 CTEQ6.1 (ZM) 1.9 1.85 18.5 19 19.5 20 20.5 21 21.5 22 CMS-PAS-EWK-10-005 Σ tot � pp � � W � � { Ν � X � � nb � - arXiv:0802.0007 Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 7
Massive quark contributions to neutral-current DIS Extensive recent work Several heavy-quark factorization schemes Tung et al., hep-ph/0611254; Thorne, hep-ph/0601245; Tung, Thorne, arXiv:0809.0714; P .N., Tung, arXiv:0903.2667; Forte, Laenen, Nason, FFN, ACOT, BMSN, CSN, arXiv:1001.2312; J. Rojo et al., arXiv:1003.1241;Alekhin, FONLL, TR’... Moch, arXiv:1011.5790;... Is a consistent picture emerging? Will persisting confusions (e.g., about the universality of heavy-quark PDFs) be resolved? Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 8
Heavy-quark NC DIS at NLO: consistent ambiguity At NLO, the charm mass m c and factorization scale µ of are tuned to best describe the DIS data in each scheme; but the residual differences in the W and Z cross sections remain LH PDFs Q � 2 GeV, m c � 1.41 GeV 3.5 Scale dependence long dash: S � ACOT �Χ NLO � short dash: FFNS NLO Nf � 3 � 3.0 � MSTW08 � NLO � MSTW08 � NLO �Χ 2.5 � � FONLL � A �Χ � 10 3 x 0.5 F 2 � c � x,Q � � FONLL � B �Χ dotted:S � ACOT NLO � 2.0 � � � � 1.5 � � � 1.0 � � � � 0.5 � � � G. Watt, PDF4LHC mtg, 26.03.2010 � 0.0 10 � 5 10 � 4 0.01 0.02 0.05 0.1 0.2 10 � 3 x W, Z cross sections; 2009 Les Houches HQ benchmarks m c = 1 . 3 GeV in CTEQ6.6 with toy PDFs; default µ = Q Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 9
NNLO: better agreement between the schemes, remaining conceptual differences Structure Functions Subtractions arXiv:1108.5112 � revisits the QCD factorization c (0) F (1) A (1) c (0) h,h h,g * h,g h,h theorem for DIS with heavy (1) c h,h quarks A (1) h,g * c h,h (1) � discusses a scheme PS (2) F (2) (2) h,g c h,h (0) PS (2) (S-ACOT- χ ) for a streamlined, F h,g A A * c (0) h,l * h,l h,h algorithmic implementation of u,d,s,c NNLO massive contributions c (2) h,h Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 10
S-ACOT- χ scheme: merging FFN and ZM x � 0.01 0.25 S-ACOT- χ reduces S � ACOT �Χ NNLO FFNS Nf � 3 NNLO to FFN at Q ≈ m c 0.20 ZM NNLO and to ZM at Q � m c 0.15 F 2 � c � x,Q � 0.10 Les Houches toy PDFs, evolved at NNLO with threshold 0.05 matching terms 0.00 Ratio to SACOT Χ 1.2 1.1 Cancellations between 1.0 0.9 subtractions and other terms at 0.8 0.7 Q ≈ m c and Q � m c ; details in 0.6 5. 7. 1.5 2. 3. 10. backup slides Q � GeV � Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 11
Are heavy quarks counted as active partons? Fixed Flavor Number scheme Zero-Mass Variable Flavor m c � = 0 for all µ 2 = Q 2 ≥ m 2 Number scheme c m c = 0 for all Q 2 ≥ m 2 c massless quark „ µ 2 n ∞ « f c/p ( ξ, µ 2 ) ∼ X α k X c nk ( ξ ) ln k S ( µ ) massive quark "hard" part PDF m 2 c n =1 k =0 Shown for up to 4 flavors for simplicity Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 12
General-Mass Variable Flavor Number schemes (Aivasis, Collins, Olness, Tung; Buza et al.; Cacciari, Greco, Nason; Chuvakin et al.; Kniehl et al.; Thorne, Roberts; Forte, Laenen, Nason; ...) For the ACOT GM scheme, factorization of DIS cross sections is proved to all orders of α s (Collins, 1998) Theorem � dξ � x � � Λ QCD � � ξ , Q , m c f a/p ( ξ, µ F F 2 ( x, Q, m c ) = σ 0 ξ C a )+ O µ F Q m c Q a = g,q, ¯ q ( − ) , ..., c ( − ) � C a is a Wilson coefficient with an incoming parton a = g, u � f a/p is a PDF for N f flavors � N f = 3 for µ < µ (4) switch ≈ m c ; N f = 4 for µ ≥ µ (4) switch � lim Q →∞ C a exists; no terms O ( m c /Q ) in the remainder Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 13
General-Mass Variable Flavor Number schemes (Aivasis, Collins, Olness, Tung; Buza et al.; Cacciari, Greco, Nason; Chuvakin et al.; Kniehl et al.; Thorne, Roberts; Forte, Laenen, Nason; ...) For the ACOT GM scheme, factorization of DIS cross sections is proved to all orders of α s (Collins, 1998) Schemes of the ACOT type do not use... � PDFs for several N f values in the same µ range � smoothness conditions/damping factors at Q → m c � constant terms from higher orders in α s to ensure continuity at the threshold Pavel Nadolsky (SMU) Galileo Galilei Institute September 23, 2011 13
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