fastNLO: Fast pQCD Calculations for PDF Fits Contributed by Kluge, Rabbertz, Wobisch

3.5 fastNLO: Fast pQCD Calculations for PDF Fits

0.6 0.8 1 1.2 1.4

250 500

k-factor

200 400 100 200

p_T (GeV/c) 0.0 < |y| < 0.1 0.7 < |y| < 1.1 1.6 < |y| < 2.1

CTEQ6.1M PDFs

k-factor:

µ_r,f = p_T/2

total gg gq + qg all qq proc.

Fig. 3.5.33: Thek-factor for the inclusive p¯p jet cross section at√s = 1.96 TeV as a function of pTat different rapiditiesy for the total cross section (solid line) and for different partonic subprocesses: gluon-gluon (dashed), gluon-quark (dotted) and the sum of all quark and/or anti-quark induced subprocesses (dashed-dotted).

• Even the LO Monte-Carlo integration of (3.5.6) is a trade-off between speed and precision. With finite statistical errors, however, theory predictions are not ideally smooth functions of the fit pa-rameters. This contributes to numerical noise in theχ²calculations [91] distorting theχ²contour during the PDF error analysis, especially for fit parameters with small errors.

• The procedure can only be used for observables for which LO calculations are fast. Currently, this prevents the global PDF analyses from using Tevatron dijet data and DIS jet data.

In a time when phenomenology is aiming towards NNLO precision [89, 90], thek-factor approximation is clearly not satisfying concerning both its limitation in precision and its restrictions concerning data sets.

The fastNLO Solution

A better solution is implemented in the fastNLO project. The basic idea is to transform the convolution in (3.5.6) into the factorized expression (3.5.9). Many proposals for this have been made in the past, originally related to solving the DGLAP parton evolution equations [92] and later to computing of jet cross sections [93, 94, 95, 96, 97]. The fastNLO method is an extension of the concepts developed for DIS jet production [93, 96] which have been applied at HERA to determine the gluon density in the proton from DIS jet data [85]. Starting from (3.5.6) for the following discussion the renormalization scale is set equal to the factorization scale (µ_r,f = µ). The extension to µ_r 6= µf is, however, trivial. The x dependence of the PDFs and the scale dependence of αⁿ_s and the PDFs can be approximated using an interpolation between sets of fixed valuesx^(k)andµ^(m)(k = 1, · · · , kmax; m = 1, · · · , mmax)

αⁿ_s(µ) · Fⁱ(xa, x_b, µ) ≃ [“=” is true for kmax, lmax, mmax→ ∞]

X

k,l,m

αⁿ_s(µ^(m)) · Fi(x^(k)_a , x^(l)_b , µ^(m)) · e^(k)(x_a) · e^(l)(x_b) · b^(m)(µ) (3.5.7)

wheree^(k,l)(x) and b^(m)(µ) are interpolation functions for the x and the µ dependence, respectively. All information of the perturbatively calculable piece (including phase space restrictions, jet definition, etc.

but excludingα_sand the PDFs) is fully contained in the quantity

˜

σ_n,i,k,l,m(µ) = c_n,i(x_a, x_b, µ) ⊗h

e^(k)(x_a) · e^(l)(x_b) · b^(m)(µ)i

. (3.5.8)

0 0.25 0.5 0.75 1

fractional contribution 10 ₁₀² ₁₀² ₁₀³

partonic subprocesses for hadron-hadron → jets

gg → jets gq → jets (x_g < x_q) gq → jets (x_g > x_q)

q_iq_j→ jets q_iq_i→ jets

q_iq_i→ jets q_iq_j→ jets

RHIC Tevatron LHC

0.0 < |y| < 1.0 0.0 < |y| < 0.4 0.0 < |y| < 0.5

pp at √s = 0.2 TeV pp-bar at √s = 1.96 TeV pp at √s = 14 TeV

p_T (GeV/c) x_T

20 40 80 50 5×10²

0.1 0.2 0.5 0.03 0.1 0.2 0.5 0.02 0.1 0.2 0.5

CTEQ6.1M NLOJET++

fastNLO

Fig. 3.5.34: Contributions of different partonic subprocesses to the inclusive jet cross section at RHIC (left), the Tevatron (middle) and the LHC (right) as a function ofpTandxT = 2pT/√

s. The subprocess gq → jets has been separated into the contributions (2) and (3) where either the quark- or the gluon momentum fraction is larger.

In the final prediction for the cross section the convolution in (3.5.6) is then reduced to a simple product

σ(µ) ≃ X

n,i,k,l,m

˜

σ_n,i,k,l,m(µ) · αⁿs(µ^(m)) · Fi(x^(k)_a , x^(l)_b , µ^(m)) . (3.5.9)

The time-consuming step involving the calculation of the universal (PDF andα_sindependent)σ is there-˜ fore factorized and needs to be done only once. Any further calculation of the pQCD prediction for arbitrary PDFs and α_s values can later be done very fast by computing the simple sum of products in (3.5.9). While the extension of the method from one initial-state hadron [96] to two hadrons was conceptually trivial, the case of two hadrons requires additional efforts to improve the efficiency and precision of the interpolation. Both, the efficiency and the precision, are directly related to the choices of the points x^(k,l), µ^(m) and the interpolation functions e(x), b(µ). The implementation in fastNLO achieves a precision of better than 0.1% for k_max, l_max = 10 and m_max ≤ 4. Computation times for cross sections in fastNLO are roughly 40-200µs per order α_s(depending onm_max). Further details are given in Ref [98].

The σ in (3.5.8) are computed using NLOJET++ [99, 100]. A unique feature in fastNLO is the˜ inclusion of the O(α⁴_s) threshold correction terms to the inclusive jet cross section [101], a first step towards a full NNLO calculation.

Results

Calculations by fastNLO are available at http://hepforge.cedar.ac.uk/fastnlo for a large set of (published, ongoing, or planned) jet cross section measurements at HERA, RHIC, the Tevatron, and the LHC (either online or as computer code for inclusion in PDF fits). Some fastNLO results for the inclusive jet cross

-1 -0.5 0 0.5 1

10

relative uncertainty

10² 10² 10³

PDF uncertainties for hadron-hadron → jets

RHIC Tevatron LHC

0.0 < |y| < 1.0 0.0 < |y| < 0.4 0.0 < |y| < 0.5

pp at √s = 0.2 TeV pp-bar at √s = 1.96 TeV pp at √s = 14 TeV

p_T (GeV/c) x_T

20 40 80 50 5×10²

0.1 0.2 0.5 0.03 0.1 0.2 0.5 0.02 0.1 0.2 0.5

CTEQ6.1M NLOJET++

fastNLO

Fig. 3.5.35: Comparison of PDF uncertainties for the inclusive jet cross section at RHIC (left), the Tevatron (middle) and the LHC (right). The uncertainty band is obtained for the CTEQ6.1M parton density functions and the results are shown as a function ofpTandxT= 2pT/√s.

section in different reactions are shown in this section. The contributions from different partonic sub-processes to the central inclusive jet cross section are compared in Fig. 3.5.34 for different colliders: For pp collisions at RHIC and the LHC, and for p¯p scattering at Tevatron Run II energies. It is seen that the quark-induced subprocesses are dominated by the valence quarks: In proton-proton collisions (RHIC, LHC) the quark-quark subprocesses (4,5) give much larger contributions than the quark-antiquark sub-processes (6,7) while exactly the opposite is true for proton-antiproton collisions at the Tevatron. The contribution from gluon-induced subprocesses is significant at all colliders over the whole p_T ranges.

It is interesting to note that at fixedx_T = 2p_T/√s the gluon contributions are largest at RHIC. Here, the jet cross section atxT = 0.5 still receives 55% contributions from gluon-induced subprocesses, as compared to only35% at the Tevatron or 38% at the LHC. As shown in Fig. 3.5.35, this results in much larger PDF uncertainties for the highx_T inclusive jet cross section at RHIC, as compared to the Tevatron and the LHC for which PDF uncertainties are roughly of the same size (at the samex_T). This indicates that the PDF sensitivity at the samex_T is about the same at the Tevatron and at the LHC, while it is much higher at RHIC.

An overview over published measurements of the inclusive jet cross section in different reactions and at different center-of-mass energies is given in Fig. 3.5.36. The results are shown as ratios of data over theory. The theory calculations include the best available perturbative predictions (NLO for DIS data and NLO +O(α⁴s) threshold corrections for p¯p data) which have been corrected for non-perturbative effects.

Over the whole phase space of8 < pT < 700 GeV jet data in DIS and p¯p collisions are well-described by the theory predictions using CTEQ6.1M PDFs [53]. The phase space inx and p_T covered by these measurements is shown in Fig. 3.5.37, demonstrating what can be gained by using fastNLO to include these data sets in future PDF fits. A first study using fastNLO on the future potential of LHC jet data has been published in Ref. [102].

1 10 10 ²

10 10 ² 10 ³

inclusive jet production

in hadron-induced processes

fastNLO

hepforge.cedar.ac.uk/fastnlo

DIS

pp-bar

√ s = 300 GeV

√ s = 318 GeV

√ s = 546 GeV

√ s = 630 GeV

√ s = 1800 GeV

√ s = 1960 GeV

H1 150 < Q² < 200 GeV² H1 200 < Q² < 300 GeV² H1 300 < Q² < 600 GeV² H1 600 < Q² < 3000 GeV² ZEUS 125 < Q² < 250 GeV² ZEUS 250 < Q² < 500 GeV² ZEUS 500 < Q² < 1000 GeV² ZEUS 1000 < Q² < 2000 GeV² ZEUS 2000 < Q² < 5000 GeV²

CDF 0.1 < |y| < 0.7

DØ |y| < 0.5

CDF 0.1 < |y| < 0.7 DØ 0.0 < |y| < 0.5 DØ 0.5 < |y| < 1.0

CDF cone algorithm CDF k_T algorithm

(× 100)

(× 35) (× 16)

(× 6)

(× 3)

(× 1)

all pQCD calculations using NLOJET++ with fastNLO:

α_s(M_Z)=0.118 | CTEQ6.1M PDFs | µ_r = µ_f = p_{T jet}

NLO plus non-perturbative corrections | pp: incl. threshold corrections (2-loop)

p

_T

(GeV/c)

data / theory

Fig. 3.5.36: An overview of data over theory ratios for inclusive jet cross sections, measured in different processes at different center-of-mass energies. The data are compared to calculations obtained by fastNLO in NLO precision (for DIS data) and includingO(α⁴s) threshold corrections (for p¯p data). The inner error bars represent the statistical errors and the outer error bars correspond to the quadratic sum of all experimental uncertainties. In all cases the perturbative predictions have been corrected for non-perturbative effects.

10 10

²

10

^-2

10

^-1

1

x p

T

(GeV/c)

CDF (546 GeV) DØ (630 GeV) CDF (1800 GeV) DØ (1800 GeV) CDF (1960 GeV)

ZEUS (300 GeV) H1 (300 GeV)

inclusive jet production

hepforge.cedar.ac.uk/fastnlo

fastNLO

Fig. 3.5.37: The phase space inx and pTcovered by the data sets shown in the previous figure.

4 Event Generator Tuning

4.1 Dijet Azimuthal Decorrelations and Monte Carlo Tuning

In document Tevatron-for-LHC Report of the QCD Working Group (Page 58-64)