aMCfast: Automation of Fast NLO Computations for PDF fits

aMCfast:
Automation of fast NLO
calculations for PDF fits
Juan Rojo
STFC Rutherford Fellow
Rudolf Peierls Center for Theoretical Physics, University of Oxford
arXiv:1406.7693
V. Bertone, R. Frederix, S. Frixione, JR and M. Sutton
ATLAS PDF fit Forum
CERN, 01/07/2014
Juan Rojo ATLAS PDF fit Forum, CERN, 01/07/2013

2 Juan Rojo ATLAS PDF ﬁt Forum, CERN, 01/07/2013

(N)NLO QCD calculations are too CPU-time intensive to be used directly into PDF analysis
The traditional solution, LO supplemented by iterative bin-by-bin K-factors, is not suitable in general
to match the precision of LHC data
In the recent years, various approaches have been proposed to provide fast interfaces to NLO
calculations, that can be used directly in PDF analysis, the main ones being:
APPLgrid: interfaced to MCFM and NLOJet++
FastNLO: interfaced to NLOJet++
Basic strategy: interpolate PDFs in a suitable basis, and precompute the partonic cross-section into a set
of grids, reconstructing the ﬁnal distributions via a fast convolution. The same ideas underlie most x-space
PDF evolution codes: HOPPET, QCDNUM, APFEL, ....
Main limitations of present tools:
Restricted to a limited number of processes, implementation and debugging of each new process
is time consuming
Only QCD corrections, no QED and electroweak corrections available, important for many LHC
processes
Only Fixed Order processes, cannot account for Monte Carlo parton shower effects. These are
required both to estimate non-perturbative corrections and to account for QCD shower
resummation effects

4
MadGraph5_aMCatNLO provides calculations with NLO accuracy for arbitrary processes, their
matching with Parton Showers, and the merging of exclusive NLO+PS samples of different multiplicity
Built upon the MadGraph framework, it uses MadFKS for subtraction of soft/collinear divergences,
MadLoop (with CutTools) for the computation of virtual corrections, MC@NLO method to match matrix
elements with parton showers, and the FxFx merging of NLO+PS samples with different multiplicies
PDF and scale uncertainties provided at no extra cost for each run via reweighting
Ongoing developments include extending loop corrections to electroweak theory as well as to generic
renormalizable BSM Lagrangians (via FeynRules@NLO)
arxiv:1405.0301
http://amcatnlo.web.cern.ch/amcatnlo/

MadGraph5_aMCatNLO provides theory predictions with NLO accuracy for arbitrary processes
Available fast interfaces are restricted to a limited number of processes
The implementation of each new process is time consuming and error-prone
A fast interface to MadGraph5_aMCatNLO would allow to include arbitrarily complicated LHC
processes into a NLO global PDF fit
MadGraph5_ aMCatNLO provides an automatic matching of NLO events with various parton showers
Available fast interfaces allow only fixed-order computations
NLO+PS computations are not only more accurate, they also provide an exclusive events
description, and allow a more direct data/theory comparisons with reduced extrapolations
MadGraph5_ aMCatNLO will soon include not only NLO QCD but also NLO electroweak corrections
QED and electroweak corrections are important to fit TeV scale data, and are not available in form
of a fast interface to PDF fits. Also photon-induced effects are sizable for many EW processes
Therefore, a fast interface to MadGraph5_ aMCatNLO is of outmost important for global PDF analysis:
Increase the number of LHC processes for which fast NLO interfaces are available, and that thus can
be used to constrain PDFs.
Allow to perform PDF fits with NLO+PS accuracy, study the stability of PDF fits wrt higher order
corrections, increase the number of observables that can be used in PDF fits, and eventually provide
specific PDF sets for NLO event generators
Include consistently electroweak corrections in PDF fits at the matrix element level, and include QED
effects that allow to perform a precision determination of the photon PDF from LHC data

The fast interface to MadGraph5_aMC@NLO, which we denote by aMCfast, is constructed using the
routines provided by the APPLgrid library
The key idea is to use a generic higher-order Lagrange interpolation
We want to use this expansion to compute a generic integral, deﬁned by the convolution of two terms
“Slow” function, CPU-time intensive, to be
precomputed only one (ie NLO cmatrix element)
“Fast” function, quick evaluation
To be expanded in Lagrange polynomials
(ie PDFs)

Now we can show that the integral J can be written as
That is, in terms of the function F(z) evaluated only at a ﬁnite subset of interpolation grid nodes,
weighted by a factor that depends on the node position, and is by
The idea of aMCfast is to extract the information on the hard-scattering matrix elements from
MadGraph5_aMC@NLO, use this to ﬁll the APPLgrid interpolating grids, and then reconstruct the
original distributions a posteriori with arbitrary PDFs and scales.
Given the automated nature of MadGraph5_aMC@NLO, this is a task that needs to be performed only
once, and then it will be valid for a generic process

In the FKS subtraction formalism, a generic 2 -> n short distance cross-section is given by
with the contribution of the fully resolved event (E) and of the soft (S), collinear (C) and soft-collinear
(SC) counterevents
The information on partonic matrix-elements, including scale dependence, is encoded in the functions:
where the various pieces are the Born, the scale independent NLO term, and the terms which encode the
renormalization and factorization scale dependences of the NLO calculation
In aMCfast, the information on the event-by-event W weights is extracted and used to ﬁll the APPLgrid
interpolation grids

Skipping technicalities (see paper for full details), we can ﬁnally write the four components of the FKS
cross-section in terms of PDFs and strong coupling evaluated only at the grid nodes, and grid weight
factors which encode all dependence on matrix elements, and that need to be computed only once
As opposed to other NLO calculations, in the FKS formalism we require four independent
interpolation grids rather than two, to account for the most general possible scale variations

Once the APPLgrid interpolating grids have been produced with aMCfast, we can recompute the
original kinematical distributions for arbitrary PDFs, scales and value of S
We assume that scales vary wrt the central value by a constant factor in all phase space
We end up with the following ﬁnal expressions in terms of PDFs in the grid and node weight factors:
This is a different approach for scale variations as compared to the default APPLgrid method, which requires
using an external code, HOPPET, and is valid only in the limit of very high statistics
The aMCfast method is fully generic, valid on a event-by-event basis and completely stand-alone. Also, if
needed, it can be generalized to more complex choices of scale variations

13
For all the processes we have explicitly tested, the aMCfast+APPLgrid interface to
MadGraph5_aMC@NLO has an accuracy well below the permille level, more than enough for any
phenomenological application (and could be further increased by using a ﬁner grid)
This is the case both for the central scale, dynamically chosen on an event-by-event basis, and for
arbitrary scale variations. In the examples I show we have used HT as central scale
The aMCfast interpolation is correct on a event-by-event basis: the original MadGraph5_aMC@NLO
distributions are successfully reproduced even in very low statistics runs (of course useful only for
validation purposes)
We use NNPDF2.3/2.1 as input PDFs, with the number of active ﬂavors consistent with the process
under consideration
APPLgrid settings

Useful to constrain the gluon PDF
For this process, we have the contribution from 7 independent
PDF luminosities. Lumis determined automatically at run time
Excellent agreement with original distributions both at low
and at high statistics

Could have PDF discrimination power if experimental measurements are precise enough
Same generation commands as before, adding the correct jet deﬁnition for a 5-ﬂavor scheme
Again, excellent agreement for all distributions and choices of scales
Uses the Frixione isolation criterion to remove the fragmentation component
Generation-level cuts and analysis parameters (including isolation) accessible from the run card

Due to the complexity of this process, here we have
33 independent PDF luminosities
Determined at run time, this information is
completely transparent to the user

Useful to constrain the gluon and the antiquarks
Feature around pT = 30 GeV understood from perturbative instability in fNLO calculation, requires
resummation
Ratios of this process with W+jets could provide important information on quark ﬂavor separation

Example of a quite complicated ﬁnal state that beneﬁts
from the complete automation of fast NLO calculations
in aMCfast
Calculation needs to be performed in a scheme with
massive b quark
In this case we have 13 independent PDF lumis

Crucial experimental data to constrain the strange PDF
Illustrates that resonance decays can the trivially included in aMCfast
Any theoretical reﬁnement of the original MadGraph5_aMC@NLO calculation, such as the use of
complex mass scheme to account for ﬁnite widths of resonances, translates automatically into aMCfast

Just some representative examples
Huge range of interesting pheno applications to be explored

21
Estimating non-perturbative corrections, for instance due to hadronization or underlying event, is
required to unfold measurements from hadron level to parton level
With MadGraph5_aMC@NLO, one should generate the same process in NLO and NLO+PS, with
identical settings, and extract consistently the bin-by-bin correction due to NP effects
Of course, this is only possible in kinematic regions where the effects of perturbative resummation from
the parton shower are small
This could be applied to the ATLAS W+charm data
Run MadGraph5_aMC@NLO at fNLO with aMCfast activated, generate the APPLgrid tables
corresponding to your measurement
Re-run MadGraph5_aMC@NLO in the NLO+PS mode in identical settings
In the PDF ﬁt, use the fast NLO grid calculation supplemented with the parton-to-hadron correction
Can be included now in PDF ﬁt thanks
to MadGraph5_aMC@NLO and the
aMCfast interface

22
Top quark data provides useful constraints on the large-x gluon PDF
Studies so far use total cross-sections, now moving towards including top quark differential
distributions in PDF fits
Thanks to MadGraph5_aMC@NLO + aMCfast, one should be able to include in the PDF fit fully
differential distributions including finite width effects of the top quark, for example, in the fully leptonic
final state
Once the full NNLO calculation is available, correct exclusive NLO calculation with a suitable K-factor
(which has very reduced PDF sensitivity)
With the same method as for W+charm, estimating uncertainties due to shower and non-perturbative
effects becomes an easy talk
PS Absolute distributions have greatly
improved PDF constraining power as
compared to normalized distributions

23
The transverse momentum distribution of the Z boson is very useful to constrain PDFs, both gluons and
antiquarks, and can be measured very precisely in terms of leptonic variables only
It is crucial to identify the region of validity of ﬁxed order QCD, and this can be studied for example
running MadGraph5_aMC@NLO at ﬁxed order and then matched to parton showers
NLO electroweak and mixed QCD/EW corrections are important at the highest pt, will become
available in a future release of MadGraph5_aMC@NLO and therefore also in aMCfast

25
The automation of fast NLO QCD calculations needed in PDF fits for arbitrary processes can be
considered now as fully completed
Next step is to extend the aMCfast interface to NLO+PS calculations.
While there are no conceptual problems, extra technical work is required to extend the NLO+PS
reweighting format propagating all the information that allow to fill APPLgrids starting from showered
events, and modify LHEF/HepMC formats
In parallel, once the mixed QCD/EW expansion becomes available in MadGraph5_aMC@NLO, we will
generalize aMCfast to processes with both QCD and QED/EWK corrections
Again, no conceptual obstructions here, the generalization of the aMCfast formulae to a mixed
expansions are trivial
From the phenomenology point of view, once aMCfast with QCD/EW corrections is available it should
be possible to use it to provide stringent constraints to the photon PDF using LHC data sensitive to
photon-initiated contributions, and use QCD+EW corrections to fit high-ET data like jets, high mass tt,
high-mass DY etc where NLO electroweak corrections are substantial

aMCfast will be make available in the next days in its HepForge website
It also requires the use of the latest APPLgrid release
The version of MadGraph5_aMC@NLO with the complete MCfast functionalities will be make available
in one of the future releases
In the meantime, a development version of MadGraph5_aMC@NLO with these functionalities is
available from the authors upon request

aMCfast: Automation of Fast NLO Computations for PDF fits

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