CTEQ6.6 pdf s etc J. Huston Michigan State University 1
Parton distribution functions and global fits Calculation of production cross sections at the LHC relies upon knowledge of pdf s in the relevant kinematic region Pdf s are determined by global analyses of data from DIS, DY and jet production Two major groups that provide semi-regular updates to parton distributions when new data/theory becomes available MRS->MRST98->MRST99 ->MRST2001->MRST2002 ->MRST2003->MRST2004 ->MSTW2008 CTEQ->CTEQ5->CTEQ6 ->CTEQ6.1->CTEQ6.5 ->CTEQ6.6 CTEQ6: circa 2002; full error treatment CTEQ6.1: circa 2003; technical improvements CTEQ6.5: circa 2006; full heavy quark mass CTEQ6.6: circa 2008; above + additional freedom for strange quark 2
Global fitting: best fit Using our 2794 data points, we do our global fit by performing a χ 2 minimization where D i are the data points and T i are the theoretical predictions; we allow for a normalization shift f N for each experimental data set but we provide a quadratic penalty for any normalization shift where there are k systematic errors β for each data point in a particular data set and where we allow the data points to be shifted by the systematic errors with the shifts given by the s j parameters but we give a quadratic penalty for non-zero values of the shifts s j where σ i is the statistical error for data point i For each data set, we calculate χ 2 = i f N D i k β ij s j j =1 T i k 2 + s σ j i For a set of theory parameters it is possible to analytically solve for the shifts s j,and therefore, continually update them as the fit proceeds To make matters more complicated, we may give additional weights to some experiments due to the utility of the data in those experiments (i.e. NA-51), so we adjust the χ 2 to be 1 f χ 2 = w k χ 2 k + w N N,k k k σ N norm where w k is a weight given to the experimental data and w N,k is a weight given to the normalization 2 2 j =1 2 3
Minimization and errors Free parameters in the fit are parameters for quark and gluon distributions f (x) = x (a 1 1) (1 x) a 2 ea 3x [1 + e a 4 x]a 5 Too many parameters to allow all to remain free some are fixed at reasonable values or determined by sum rules 20 free parameters for CTEQ6.1, 22 for CTEQ6.6 2 additional parameters for strange quark distributions Result is a global χ 2 /dof on the order of 1 for a NLO fit worse for a LO fit, since the LO pdf s can not make up for the deficiencies in the LO matrix elements 4
PDF Errors: old way Make plots of lots of pdf s (no matter how old) and take spread as a measure of the error Can either underestimate or overestimate the error Review sources of uncertainty on pdf s data set choice kinematic cuts parametrization choices treatment of heavy quarks order of perturbation theory errors on the data There are now more sophisticated techniques to deal with at least the errors due to the experimental data uncertainties 5
PDF Errors: new way So we have optimal values (minimum χ 2 ) for the d=20 (22) free pdf parameters in the global fit {a µ },µ=1, d Varying any of the free parameters from its optimal value will increase the χ 2 It s much easier to work in an orthonormal eigenvector space determined by diagonalizing the Hessian matrix, determined in the fitting process H uv = 1 2 χ 2 a µ a ν To estimate the error on an observable X(a), due to the experimental uncertainties of the data used in the fit, we use the Master Formula ( ΔX ) 2 X = Δχ 2 µ,ν a µ ( H 1 ) µν X a ν 6
PDF Errors: new way Recap: 20 (22) eigenvectors with the eigenvalues having a range of >1E6 Largest eigenvalues (low number eigenvectors) correspond to best determined directions; smallest eigenvalues (high number eigenvectors) correspond to worst determined directions Easiest to use Master Formula in eigenvector basis To estimate the error on an observable X(a), from the experimental errors, we use the Master Formula ( ΔX ) 2 X = Δχ 2 µ,ν a µ ( H 1 ) µν X a ν where X i + and X i - are the values for the observable X when traversing a distance corresponding to the tolerance T(=sqrt(Δχ 2 )) along the i th direction 7
PDF Errors: new way What is the tolerance T? This is one of the most controversial questions in global pdf fitting? We have 2794 data points in the CTEQ6.6 data set (on order of 2000 for CTEQ6.1) Technically speaking, a 1-sigma error corresponds to a tolerance T(=sqrt(Δχ 2 ))=1 This results in far too small an uncertainty from the global fit with data from a variety of processes from a variety of experiments from a variety of accelerators For CTQE6.1, we chose a Δχ 2 of 100 to correspond to a 90% CL limit with an appropriate scaling for the larger data set for CTEQ6.6 MSTW has chosen a Δχ 2 of 50 for the same limit so CTEQ errors will be larger than MSTW errors 8
What do the eigenvectors mean? Each eigenvector corresponds to a linear combination of all 20 (22) pdf parameters, so in general each eigenvector doesn t mean anything? However, with 20 (22) dimensions, often eigenvectors will have a large component from a particular direction Take eigenvector 1 (for CTEQ6.1); error pdf s 1 and 2 It has a large component sensitive to the small x behavior of the u quark valence distribution Not surprising since this is the best determined direction 9
What do the eigenvectors mean? Take eigenvector 15 (for CTEQ6.1); error pdf s 29 and 30 Probes high x gluon distribution creates largest uncertainty for high p T jet cross sections at both the Tevatron and LHC I haven t done this exercise yet for CTEQ6.6 10
Aside: PDF re-weighting Any physical cross section at a hadron-hadron collider depends on the product of the two pdf s for the partons participating in the collision convoluted with the hard partonic cross section Nominally, if one wants to evaluate the pdf uncertainty for a cross section, this convolution should be carried out 41 times (for CTEQ6.1); once for the central pdf and 40 times for the error pdf s However, the partonic cross section is not changing, only the product of the pdf s So one can evaluate the full cross section for one pdf (the central pdf) and then evaluate the pdf uncertainty for a particular cross section by taking the ratio of the product of the pdf s (the pdf luminosity) for each of the error pdf s compared to the central pdf s f i is the error pdf and f 0 the central pdf f i a / A (x a,q 2 ) f i b / B (x b,q 2 ) f 0 a / A (x a,q 2 ) f 0 b / B (x b,q 2 ) This works exactly for fixed order calculations and works well enough for parton shower Monte Carlo calculations (if you stay at NLO; I don t know the size of the error for mixing LO and NLO pdf s). Most experiments now have code to easily do this and many programs will do it for you (MCFM) 11
Cross sections at the LHC Note that the data from HERA and fixed target cover only part of kinematic range accessible at the LHC We will access pdf s down to 1E -6 (crucial for the underlying event) and Q 2 up to 100 TeV 2 We can use the DGLAP equations to evolve to the relevant x and Q 2 range, but we re somewhat blind in extrapolating to lower x values than present in the HERA data, so uncertainty may be larger than currently estimated we re assuming that DGLAP is all there is; at low x BFKL type of logarithms may become important DGLAP BFKL? 12
Parton kinematics at the LHC To serve as a handy look-up table, it s useful to define a parton-parton luminosity (a la EHLQ) Equation 3 can be used to estimate the production rate for a hard scattering at the LHC as the product of a differential parton luminosity and a scaled hard scatter matrix element this is from the CHS review paper 13
Cross section estimates gg qq gq for p T =0.1* sqrt(s-hat) 14
PDF uncertainties at the LHC gg tt Note that for much of the SM/discovery range, the pdf luminosity uncertainty is small Need similar level of precision in theory calculations It will be a while, i.e. not in the first fb -1, before the LHC data starts to constrain pdf s qq W/Z NBIII: tt uncertainty is of the same order as W/Z production Rule-of-thumb (CHS): uncertainties in acceptances factor of 5-10 less than uncertainties in cross sections gq NB I: the errors are determined using the Hessian method for a Δχ 2 of 100 using only experimental uncertainties,i.e. no theory uncertainties NB II: the pdf uncertainties for W/Z cross sections are not the smallest 15
Precision benchmarks: W/Z cross sections at the LHC CTEQ6.1 and MRST NLO predictions in good agreement with each other NNLO corrections are small and negative NNLO mostly a K-factor; NLO predictions adequate for most predictions at the LHC 16
Heavy quark mass effects in global fits CTEQ6.1 (and previous generations of global fits) used zero-mass VFNS scheme With new sets of pdf s (CTEQ6.5/6.6), heavy quark mass effects consistently taken into account in global fitting cross sections and in pdf evolution In most cases, resulting pdf s are within CTEQ6.1 pdf error bands But not at low x (in range of W and Z production at LHC) Heavy quark mass effects only appreciable near threshold ex: prediction for F 2 at low x,q at HERA smaller if mass of c,b quarks taken into account thus, quark pdf s have to be bigger in this region to have an equivalent fit to the HERA data 17 implications for LHC phenomenology
CTEQ6.5(6) Inclusion of heavy quark mass effects affects DIS data in x range appropriate for W/Z production at the LHC Cross sections for W/Z increase by 7-8% now CTEQ and MRST2004 in disagreement and relative uncertainties of W/Z increase although individual uncertainties of W and Z decrease somewhat Two new free parameters in fit dealing with strangeness degrees of freedom so now have 44 error pdf s rather than 40 CTEQ6.5(6) Note importance of strange quark uncertainty for ratio 18
but Inclusion of heavy quark mass effects affects DIS data in x range appropriate for W/Z production at the LHC but MSTW2008 has also lead to increased W/Z cross sections at the LHC now CTEQ6.6 and MSTW2008 in better agreement CTEQ6.5(6) MSTW08 Beware of (precision) predictions that use earlier generations of pdf s. 19
Correlations Consider a cross section X(a), a function of the Hessian eigenvectors i th component of gradient of X is Now take 2 cross sections X and Y or one or both can be pdf s Consider the projection of gradients of X and Y onto a circle of radius 1 in the plane of the gradients in the parton parameter space The circle maps onto an ellipse in the XY plane The angle φ between the gradients of X and Y is given by If two cross sections are very correlated, then cosφ~1 uncorrelated, then cosφ~0 anti-correlated, then cosφ~-1 The ellipse itself is given by 20
Correlations with Z, tt Define a correlation cosine between two quantities Z tt If two cross sections are very correlated, then cosφ~1 uncorrelated, then cosφ~0 anti-correlated, then cosφ~-1 Note: correlations with acceptances will probably be smaller than correlations for cross sections (see Manuela s talk) 21
Correlations with Z, tt Define a correlation cosine between two quantities tt Z If two cross sections are very correlated, then cosφ~1 uncorrelated, then cosφ~0 anti-correlated, then cosφ~-1 Note that correlation curves to Z and to tt are mirror images of each other By knowing the pdf correlations, can reduce the uncertainty for a given cross section in ratio to a benchmark cross section iff cos φ > 0;e.g. Δ(σ W +/σ Z )~1% If cos φ < 0, pdf uncertainty for one cross section normalized to a benchmark cross section is larger So, for gg->h(500 GeV); pdf uncertainty is 4%; Δ(σ H /σ Z )~8% 22
CTEQ plans New data from Tevatron have been added to global fit at some point in near future, there will be a CTEQ6.7 NLO pdf or equivalent more HERA data? combined data sets? NNLO pdf s (for first time from CTEQ) Re-summed pdf fits so far we have been fitting only the longitudinal degrees of freedom of the pdf s we have set up the machinery to fit the transverse degrees of freedom as well may be important for precision measurements such as W mass Modified LO fits 23
CTEQ modified LO pdf s (LO*) Include in LO* fit (weighted) pseudo-data for characteristic LHC processes produced using CTEQ6.6 NLO pdf s with NLO matrix elements (using MCFM), along with full CTEQ6.6 dataset (2885 points) low mass bb fix low x gluon for UE tt over full mass range higher x gluon W +,W -,Z 0 rapidity distributions quark distributions gg->h (120 GeV) rapidity distribution Choices Use of 2-loop or 1-loop α s Herwig preference for 2-loop Pythia preference for 1-loop Fixed momentum sum rule, or not re-arrange momentum within proton and/or add extra momentum extra momentum appreciated by some of pseudo-data sets but not others and may lose some useful correlations Fix pseudo-data normalizations to K-factors expected from higher order corrections, or let float Scale variation within reasonable range for fine-tuning of agreement with pseudo-data for example, let vector boson scale vary from 0.5 m B to 2.0 m B Will provide pdf s with several of these options for user 24
Some observations Pseudo-data has conflicts with global data set that s the motivation of the modified pdf s Requiring better fit to pseudo-data increases chisquare of LO fit to global data set (although this is not the primary concern; the fit to the pseudo-data is) χ 2 improves with α s free in fit no real preference for 1-loop or 2-loop α s that I can see χ 2 improves with momentum sum rule free prefers more momentum (~1.05) normalization of pseudo-data (needed K-factor) gets closer to 1 (since the chisquare gets better if that happens) still some conflicts with DIS data that don t prefer more momentum 25
Some results (2-loop α s ) 26
Collate/create cross section predictions for LHC processes such as W/Z/ Higgs(both SM and BSM)/ diboson/tt/single top/photons/ jets at LO, NLO, NNLO (where available) CTEQ4LHC/FROOT new: W/Z production to NNLO QCD and NLO EW pdf uncertainty, scale uncertainty, correlations, correlation tools impacts of resummation (q T and threshold) As prelude towards comparison with actual data Using programs such as: MCFM ResBos Pythia/Herwig/Sherpa private codes with CTEQ First on webpage and later as a report Primary goal: have all theorists write out parton level output into ROOT ntuples Secondary goal: make libraries of prediction ntuples available FROOT: a simple interface for writing Monte-Carlo events into a ROOT ntuple file Written by Pavel Nadolsky (nadolsky@physics.smu.edu) CONTENTS ======== froot.c -- the C file with FROOT functions taste_froot.f -- a sample Fortran program writing 3 events into a ROOT ntuple taste_froot0.c -- an alternative toplevel C wrapper (see the compilation notes below) Makefile 27
PDF Uncertainties and FROOT Z production in ResBos new way, all pdf weights stored in ntuple, events generated once old way independent ntuple for each pdf 28
Ratio of Z p T distributions to that from CTEQ6.6 This type of sensitivity not possible with independent generation pdf s 1,2 pdf s 11,12 pdf s 3,4 29
MCFM 5.3 has FROOT built in mcfm.fnal.gov in principle can store scale uncertainties at same time 30
CHS Some references
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