DESY Report 14-188

HERAFitterOpen Source QCD Fit Project

S. Alekhin1,2 · O. Behnke3 · P. Belov3,4 · S. Borroni3 · M. Botje5 · D. Britzger3 ·S. Camarda3 · A.M. Cooper-Sarkar6 · K. Daum7,8 · C. Diaconu9 · J. Feltesse10 ·A. Gizhko3 · A. Glazov3 · A. Guffanti11 · M. Guzzi3 · F. Hautmann12,13,14 ·A. Jung15 · H. Jung3,16 · V. Kolesnikov17 · H. Kowalski3 · O. Kuprash3 ·A. Kusina18 · S. Levonian3 · K. Lipka3 · B. Lobodzinski19 · K. Lohwasser1,3 ·A. Luszczak20 · B. Malaescu21 · R. McNulty22 · V. Myronenko3 · S. Naumann-Emme3 · K. Nowak3,6 · F. Olness18 · E. Perez23 · H. Pirumov3 · R. Placakyte3 ·K. Rabbertz24 · V. Radescu3 · R. Sadykov17 · G.P. Salam25,26 · A. Sapronov17 ·A. Schoning27 · T. Schorner-Sadenius3 · S. Shushkevich3 · W. Slominski28 ·H. Spiesberger29 · P. Starovoitov3 · M. Sutton30 · J. Tomaszewska31 · O. Turkot3 ·A. Vargas3 · G. Watt32 · K. Wichmann3

1 Deutsches Elektronen-Synchrotron (DESY), Platanenallee 6,D–15738 Zeuthen, Germany2 Institute for High Energy Physics,142281 Protvino, Moscow region,Russia3 Deutsches Elektronen-Synchrotron (DESY), Hamburg, Germany4 Current address: Department of Physics, St. Petersburg State Univer-sity, Ulyanovskaya 1, 198504 St. Petersburg, Russia5 Nikhef, Science Park, Amsterdam, the Netherlands6 Department of Physics, University of Oxford, Oxford, United King-dom7 Fachbereich C, Universitat Wuppertal, Wuppertal, Germany8 Rechenzentrum, Universitat Wuppertal, Wuppertal, Germany9 Aix Marseille Universite, CNRS/IN2P3, CPPM UMR 7346, 13288Marseille, France10 CEA, DSM/Irfu, CE-Saclay, Gif-sur-Yvette, France11 Niels Bohr International Academy and Discovery Center, NielsBohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100Copenhagen, Denmark12 School of Physics and Astronomy, University of Southampton, UK13 Rutherford Appleton Laboratory, Chilton OX11 0QX, United King-dom14 Dept. of Theoretical Physics, University of Oxford, Oxford OX13NP, United Kingdom15 FERMILAB, Batavia, IL, 60510, USA16 Elementaire Deeltjes Fysica, Universiteit Antwerpen, B 2020Antwerpen, Belgium17 Joint Institute for Nuclear Research (JINR), Joliot-Curie 6, 141980,Dubna, Moscow Region, Russia18 Southern Methodist University, Dallas, Texas19 Max Planck Institut Fur Physik, Werner Heisenberg Institut,Fohringer Ring 6, Munchen20 T. Kosciuszko Cracow University of Technology21 Laboratoire de Physique Nucleaire et de Hautes Energies, UPMCand Universite, Paris-Diderot and CNRS/IN2P3, Paris, France22 University College Dublin, Dublin 4, Ireland23 CERN, European Organization for Nuclear Research, Geneva,Switzerland24 Institut fur Experimentelle Kernphysik, Karlsruhe, Germany

Abstract HERAFitter is an open-source package that pro-vides a framework for the determination of the parton distri-bution functions (PDFs) of the proton and for many differ-ent kinds of analyses in Quantum Chromodynamics (QCD).It encodes results from a wide range of experimental mea-surements in lepton-proton deep inelastic scattering andproton-proton (proton-antiproton) collisions at hadron col-liders. These are complemented with a variety of theoreticaloptions for calculating PDF-dependent cross section predic-tions corresponding to the measurements. The frameworkcovers a large number of the existing methods and schemesused for PDF determination. The data and theoretical predic-tions are brought together through numerous methodologi-cal options for carrying out PDF fits and plotting tools tohelp visualise the results. While primarily based on the ap-proach of collinear factorisation, HERAFitter also providesfacilities for fits of dipole models and transverse-momentum

25 CERN, PH-TH, CH-1211 Geneva 23, Switzerland26 leave from LPTHE; CNRS UMR 7589; UPMC Univ. Paris 6; Paris75252, France27 Physikalisches Institut, Universitat Heidelberg, Heidelberg, Ger-many28 Jagiellonian University, Institute of Physics, Reymonta 4, PL-30-059 Cracow, Poland29 PRISMA Cluster of Excellence, Institut fur Physik (WA THEP),Johannes-Gutenberg-Universitat, D-55099 Mainz, Germany30 University of Sussex, Department of Physics and Astronomy, Sus-sex House, Brighton BN1 9RH, United Kingdom31 Warsaw University of Technology, Faculty of Physics, Koszykowa75, 00-662 Warsaw, Poland32 Institute for Particle Physics Phenomenology, Durham University,Durham, DH1 3LE, United Kingdom

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dependent PDFs. The package can be used to study the im-pact of new precise measurements from hadron colliders.This paper describes the general structure of HERAFitterand its wide choice of options.

1 Introduction

The recent discovery of the Higgs boson [1, 2] and the ex-tensive searches for signals of new physics in LHC proton-proton collisions demand high-precision calculations to testthe validity of the Standard Model (SM) and factorisation inQuantum Chromodynamics (QCD). Using collinear factori-sation, inclusive cross sections in hadron collisions may bewritten as

σ(αs(µ2R),µ

2R,µ

2F) = ∑

a,b

1∫0

dx1 dx2 fa(x1,µ2F) fb(x2,µ

2F)

× σab(x1,x2;αs(µ

2R),µ

2R,µ

2F)

+ O

(Λ 2

QCD

Q2

)(1)

where the cross section σ is expressed as a convolution ofParton Distribution Functions (PDFs) fa and fb with theparton cross section σab, involving a momentum transfer qsuch that Q2 = |q2| �Λ 2

QCD, where ΛQCD is the QCD scale.At Leading-Order (LO) in the perturbative expansion of thestrong-coupling constant, the PDFs represent the probabil-ity of finding a specific parton a (b) in the first (second)hadron carrying a fraction x1 (x2) of its momentum. Theindices a and b in Eq. 1 indicate the various kinds of par-tons, i.e. gluons, quarks and antiquarks of different flavoursthat are considered as the constituents of the proton. ThePDFs depend on the factorisation scale, µF, while the par-ton cross sections depend on the strong coupling constant,αs, and the factorisation and renormalisation scales, µF andµR. The parton cross sections σab are calculable in pertur-bative QCD (pQCD) whereas PDFs are usually constrainedby global fits to a variety of experimental data. The assump-tion that PDFs are universal, within a particular factorisationscheme [3–7], is crucial to this procedure. Recent review ar-ticles on PDFs can be found in Refs. [8, 9].

A precise determination of PDFs as a function of x re-quires large amounts of experimental data that cover a widekinematic region and that are sensitive to different kinds ofpartons. Measurements of inclusive Neutral Current (NC)and Charge Current (CC) Deep Inelastic Scattering (DIS)at the lepton-proton (ep) collider HERA provide crucial in-formation for determining the PDFs. Different processes inproton-proton (pp) and proton-antiproton (pp) collisions atthe LHC and the Tevatron, respectively, provide comple-mentary information to the DIS measurements. The PDFsare determined from χ2 fits of the theoretical predictions

to the data. The rapid flow of new data from the LHC ex-periments and the corresponding theoretical developments,which are providing predictions for more complex processesat increasingly higher orders, has motivated the developmentof a tool to combine them together in a fast, efficient, open-source framework.

This paper describes the open-source QCD fit frame-work HERAFitter [10], which includes a set of tools to fa-cilitate global QCD analyses of pp, pp and ep scatteringdata. It has been developed for the determination of PDFsand the extraction of fundamental parameters of QCD suchas the heavy quark masses and the strong coupling constant.It also provides a common framework for the comparison ofdifferent theoretical approaches. Furthermore, it can be usedto test the impact of new experimental data on the PDFs andon the SM parameters.

This paper is organised as follows: The general structureof HERAFitter is presented in Sec. 2. In Sec. 3 the variousprocesses available in HERAFitter and the correspondingtheoretical calculations, performed within the framework ofcollinear factorisation and the DGLAP [11–15] formalism,are discussed. In Sec. 4 tools for fast calculations of thetheoretical predictions are presented. In Sec. 5 the method-ology to determine PDFs through fits based on various χ2

definitions is described. In particular, different treatments ofcorrelated experimental uncertainties are presented. Alter-native approaches to the DGLAP formalism are presentedin Sec. 6. The organisation of the HERAFitter code is dis-cussed in Sec. 7, specific applications of the package are pre-sented in Sec. 8, which is followed by a summary in Sec. 9.

2 The HERAFitter Structure

The diagram in Fig. 1 gives a schematic overview of theHERAFitter structure and functionality, which can be di-vided into four main blocks:

Data: Measurements from various processes are providedin the HERAFitter package including the information ontheir uncorrelated and correlated uncertainties. HERA inclu-sive scattering data are directly sensitive to quark PDFs andindirectly sensitive to the gluon PDF through scaling viola-tions and the longitudinal structure function FL. These dataare the basis of any proton PDF extraction, and are used inall current PDF sets from MSTW [16], CT [17], NNPDF[18], ABM [19], JR [20] and HERAPDF [21] groups. Mea-surements of charm and beauty quark production at HERAare sensitive to heavy quark PDFs and jet measurementshave direct sensitivity to the gluon PDF. However, the kine-matic range of HERA data mostly covers low and mediumranges in x. Measurements from the fixed target experi-ments, the Tevatron and the LHC provide additional con-straints on the gluon and quark distributions at high-x, better

HERAFitter 3

Initialisation

Data– Collider, Fixed Target:

ep, µ p– Collider: pp, pp

Theory– PDF Parametrisation– QCD Evolution:

DGLAP (QCDNUM),non-DGLAP (CCFM, dipole)

– Cross Section Calculation

QCD Analysis– Treatment of the Uncertainties– Fast χ2 Computation– Minimisation (MINUIT)

Results– PDFs, LHAPDF, TMDlib Grids– αs, mC , . . .– Data vs. Predictions– χ2, Pulls, Shifts

Fig. 1 Schematic overview of the HERAFitter program.

Experimental Process Reaction Theory schemesData calculationsHERA, DIS NC ep→ eX TR′, ACOT,Fixed Target µ p→ µX ZM (QCDNUM),

FFN (OPENQCDRAD,QCDNUM),TMD (uPDFevolv)

HERA DIS CC ep→ νeX ACOT, ZM (QCDNUM),FFN (OPENQCDRAD)

DIS jets ep→ e jetsX NLOJet++ (fastNLO)

DIS heavy ep→ eccX , TR′, ACOT,quarks ep→ ebbX ZM (QCDNUM),

FFN (OPENQCDRAD,QCDNUM)

Tevatron, Drell-Yan pp(p)→ llX , MCFM (APPLGRID)LHC pp(p)→ lνX

top pair pp(p)→ ttX MCFM (APPLGRID),HATHOR, DiffTop

single top pp(p)→ tlνX , MCFM (APPLGRID)pp(p)→ tX ,pp(p)→ tWX

jets pp(p)→ jetsX NLOJet++ (APPLGRID),NLOJet++ (fastNLO)

LHC DY heavy pp→V hX MCFM (APPLGRID)quarks

Table 1 The list of experimental data and theory calculations imple-mented in the HERAFitter package. The references for the individualcalculations and schemes are given in the text.

understanding of heavy quark distributions and decompo-sition of the light-quark sea. For these purposes, measure-ments from fixed-target experiments, the Tevatron and theLHC are included.

The processes that are currently available within theHERAFitter framework are listed in Tab. 1.

Theory: The PDFs are parametrised at a starting scale, Q20,

using a functional form and a set of free parameters p.These PDFs are evolved to the scale of the measurementsQ2, Q2 > Q2

0. By default, the evolution uses the DGLAPformalism [11–15] as implemented in QCDNUM [22]. Alter-

natively, the CCFM evolution [23–26] as implemented inuPDFevolv [27] can be chosen. The prediction of the crosssection for a particular process is obtained, assuming factori-sation, by the convolution of the evolved PDFs with the cor-responding parton scattering cross section. Available theorycalculations for each process are listed in Tab. 1. Predictionsusing dipole models [28–30] can also be obtained.

QCD Analysis: The PDFs are determined in a least squaresfit: a χ2 function, which compares the input data and theorypredictions, is minimised with the MINUIT [31] program. InHERAFitter various choices are available for the treatmentof experimental uncertainties in the χ2 definition. Correlatedexperimental uncertainties can be accounted for using a nui-sance parameter method or a covariance matrix method asdescribed in Sec. 5.2. Different statistical assumptions forthe distributions of the systematic uncertainties, e.g. Gaus-sian or LogNormal [32], can also be studied (see Sec. 5.3).

Results: The resulting PDFs are provided in a format readyto be used by the LHAPDF library [33, 34] or by TMDlib [35].HERAFitter drawing tools can be used to display the PDFswith their uncertainties at a chosen scale. As an example, thefirst set of PDFs extracted using HERAFitter from HERAI data, HERAPDF1.0 [21], is shown in Fig. 2 (taken fromRef. [21]). Note that following conventions, the PDFs aredisplayed as parton momentum distributions x f (x,µ2

F).

Fig. 2 Distributions of valence (xuv, xdv), sea (xS) and the gluon (xg)PDFs in HERAPDF1.0 [21]. The gluon and the sea distributions arescaled down by a factor of 20. The experimental, model and parametri-sation uncertainties are shown as coloured bands.

4

3 Theoretical formalism using DGLAP evolution

In this section the theoretical formalism based on DGLAP[11–15] equations is described.

A direct consequence of factorisation (Eq. 1) is that thescale dependence or “evolution” of the PDFs can be pre-dicted by the renormalisation group equations. By requiringphysical observables to be independent of µF, a representa-tion of the parton evolution in terms of the DGLAP equa-tions is obtained:

d fa(x,µ2F)

d log µ2F

= ∑b=q,q,g

∫ 1

x

dzz

Pab

(xz

; µ2F

)fb(z,µ2

F) , (2)

where the functions Pab are the evolution kernels or splittingfunctions, which represent the probability of finding par-ton a in parton b. They can be calculated as a perturbativeexpansion in αs. Once PDFs are determined at the initialscale µ2

F = Q20, their evolution to any other scale Q2 > Q2

0is entirely determined by the DGLAP equations. The PDFsare then used to calculate cross sections for various differ-ent processes. Alternative approaches to DGLAP evolutionequations, valid in different kinematic regimes, are also im-plemented in HERAFitter and will be discussed in Sec. 6.

3.1 Deep Inelastic Scattering and Proton Structure

The formalism that relates the DIS measurements to pQCDand the PDFs has been described in detail in many exten-sive reviews (see e.g. Ref. [36]) and it is only briefly sum-marised here. DIS is the process where a lepton scatters offthe partons in the proton by the virtual exchange of a neu-tral (γ/Z) or charged (W±) vector boson and, as a result, ascattered lepton and a hadronic final state are produced. Thecommon DIS kinematic variables are the scale of the pro-cess Q2, which is the absolute squared four-momentum ofthe exchanged boson, Bjorken x, which can be related in theparton model to the momentum fraction that is carried bythe struck quark, and the inelasticity y. These are related byy = Q2/sx, where s is the squared centre-of-mass energy.The NC cross section can be expressed in terms of gener-alised structure functions:

d2σe±pNC

dxdQ2 =2πα2Y+

xQ4 σe±pr,NC, (3)

σe±pr,NC = F±2 ∓

Y−Y+

xF±3 −y2

Y+F±L , (4)

where Y± = 1± (1− y)2 and α is the electromagnetic cou-pling constant. The generalised structure functions F2,3 canbe written as linear combinations of the proton structurefunctions Fγ

2 ,FγZ2,3 and FZ

2,3, which are associated with purephoton exchange terms, photon-Z interference terms and

pure Z exchange terms, respectively. The structure functionF2 is the dominant contribution to the cross section, xF3 be-comes important at high Q2 and FL is sizable only at highy. In the framework of pQCD, the structure functions are di-rectly related to the PDFs: at LO F2 is the weighted momen-tum sum of quark and anti-quark distributions, xF3 is relatedto their difference, and FL vanishes. At higher orders, termsrelated to the gluon distribution appear, in particular FL isstrongly related to the low-x gluon.The inclusive CC ep cross section, analogous to the NC epcase, can be expressed in terms of another set of structurefunctions, W :

d2σe±pCC

dxdQ2 =1±P

2G2

F2πx

[ m2W

m2W +Q2

]σ

e±pr,CC (5)

σe±pr,CC = Y+W±2 ∓Y−xW±3 − y2W±L , (6)

where P represents the lepton beam polarisation. At LO inαs, the CC e+p and e−p cross sections are sensitive to dif-ferent combinations of the quark flavour densities.

Beyond LO, the QCD predictions for the DIS structurefunctions are obtained by convoluting the PDFs with appro-priate hard-process scattering matrix elements, which are re-ferred to as coefficient functions.

The DIS measurements span a large range of Q2 from afew GeV2 to about 105 GeV2, crossing heavy quark massthresholds, thus the treatment of heavy quark (charm andbeauty) production and the chosen values of their massesbecome important. There are different schemes for the treat-ment of heavy quark production. Several variants of theseschemes are implemented in HERAFitter and they arebriefly discussed below.

Zero-Mass Variable Flavour Number (ZM-VFN):In this scheme [37], the heavy quarks appear as partons inthe proton at Q2 values above ∼m2

h (heavy quark mass) andthey are then treated as massless in both the initial and fi-nal states of the hard scattering process. The lowest orderprocess is the scattering of the lepton off the heavy quarkvia electroweak boson exchange. This scheme is expectedto be reliable in the region where Q2�m2

h. In HERAFitter

this scheme is available for the DIS structure function cal-culation via the interface to the QCDNUM [22] package, thusit benefits from the fast QCDNUM convolution engine.

Fixed Flavour Number (FFN):In this rigorous quantum field theory scheme [38–40], onlythe gluon and the light quarks are considered as partonswithin the proton and massive quarks are produced pertur-batively in the final state. The lowest order process is theheavy quark-antiquark pair production via boson-gluon fu-sion. In HERAFitter this scheme can be accessed via theQCDNUM implementation or through the interface to the open-source code OPENQCDRAD [41] as implemented by the ABM

HERAFitter 5

group. This scheme is reliable for Q2 ∼ m2h. In QCDNUM,

the calculation of the heavy quark contributions to DISstructure functions are available at Next-to-Leading Order(NLO) and only electromagnetic exchange contributionsare taken into account. In the OPENQCDRAD implementa-tion the heavy quark contributions to CC structure functionsare also available and, for the NC case, the QCD correc-tions to the coefficient functions in Next-to-Next-to Lead-ing Order (NNLO) are provided in the best currently knownapproximation [42, 43]. The OPENQCDRAD implementationuses in addition the running heavy quark mass in the MSscheme [44].

It is sometimes argued that this scheme reduces the sen-sitivity of the DIS cross sections to higher order corrections[42, 43]. It is also known to have smaller non-perturbativecorrections than the pole mass scheme [45].

General-Mass Variable Flavour Number (GM-VFN):In this scheme [46], heavy quark production is treated forQ2 ∼ m2

h in the FFN scheme and for Q2� m2h in the mass-

less scheme with a suitable interpolation in between. Thedetails of this interpolation differ between implementations.The groups that use GM-VFN schemes in PDFs are MSTW,CT (CTEQ), NNPDF, and HERAPDF. HERAFitter imple-ments different variants of the GM-VFN scheme.

– GM-VFN Thorne-Roberts scheme: The Thorne-Roberts(TR) scheme [47] was designed to provide a smoothtransition from the massive FFN scheme at low scalesQ2 ∼ m2

h to the massless ZM-VFNS scheme at highscales Q2� m2

h. Because the original version was tech-nically difficult to implement beyond NLO, it was up-dated to the TR′ scheme [48]. There are two variantsof the TR′ schemes: TR′ standard (as used in MSTWPDF sets [16, 48]) and TR′ optimal [49], with a smoothertransition across the heavy quark threshold region. BothTR′ variants are accessible within the HERAFitter

package at LO, NLO and NNLO.

– GM-VFN ACOT scheme: The Aivazis-Collins-Olness-Tung (ACOT) scheme belongs to the group of VFN fac-torisation schemes that use the renormalisation methodof Collins-Wilczek-Zee (CWZ) [50]. This scheme uni-fies the low scale Q2 ∼ m2

h and high scale Q2 > m2h re-

gions in a coherent framework across the full energyrange. Within the ACOT package, the following vari-ants of the ACOT MS scheme are available at LO andNLO: ACOT-Full [51], S-ACOT-χ [52, 53] and ACOT-ZM [51]. For the longitudinal structure function higherorder calculations are also available. A comparison ofPDFs extracted from QCD fits to the HERA data withthe TR′ and ACOT-Full schemes is illustrated in Fig. 3(taken from [21]).

Fig. 3 Distributions of valence (xuv, xdv), sea (xS) and the gluon (xg)PDFs in HERAPDF1.0 [21] with their total uncertainties at the scaleof Q2 = 10 GeV2 obtained using the TR′ scheme and compared to thePDFs obtained with the ACOT-Full scheme using the k-factor tech-nique (red). The gluon and the sea distributions are scaled down by afactor of 20.

3.2 Electroweak Corrections to DIS

Calculations of higher-order electroweak corrections to DISat HERA are available in HERAFitter in the on-shellscheme. In this scheme, the masses of the gauge bosons mWand mZ are treated as basic parameters together with the top,Higgs and fermion masses. These electroweak correctionsare based on the EPRC package [54]. The code calculates therunning of the electromagnetic coupling α using the mostrecent parametrisation of the hadronic contribution [55] aswell as an older version from Burkhard [56].

3.3 Diffractive PDFs

About 10% of deep inelastic interactions at HERA arediffractive, such that the interacting proton stays intact(ep→ eX p). The outgoing proton is separated from the restof the final hadronic system, X , by a large rapidity gap. Suchevents are a subset of DIS where the hadronic state X comesfrom the interaction of the virtual photon with a colour-neutral cluster stripped off the proton [57]. The process canbe described analogously to the inclusive DIS, by meansof the diffractive parton distributions (DPDFs) [58]. Theparametrization of the colour-neutral exchange in terms offactorisable ‘hard’ Pomeron and a secondary Reggeon [59],both having a hadron-like partonic structure, has proved re-markably successful in the description of most of the diffrac-tive data. It has also provided a practical method to deter-mine DPDFs from fits to the diffractive cross sections.

6

In addition to the usual DIS variables x, Q2, extra kine-matic variables are needed to describe the diffractive pro-cess. These are the squared four-momentum transfer of theexchanged Pomeron or Reggeon, t, and the mass mX ofthe diffractively produced final state. In practice, the vari-able mX is often replaced by the dimensionless quantityβ = Q2

m2X+Q2−t

. In models based on a factorisable Pomeron, β

may be viewed at LO as the fraction of the Pomeron longitu-dinal momentum, xIP, which is carried by the struck parton,x = βxIP, where P denotes the momentum of the proton.For the inclusive case, the diffractive cross section reads as:

d4σ

dβ dQ2dxIP dt =2πα2

βQ4

(1+(1− y)2

)σ

D(4)(β ,Q2,xIP, t) (7)

with the “reduced cross section”:

σD(4) = FD(4)

2 − y2

1+(1−y)2 FD(4)L . (8)

The diffractive structure functions can be expressed asconvolutions of calculable coefficient functions with thediffractive quark and gluon distribution functions, which ingeneral depend on xIP, Q2, β and t.

The DPDFs [60, 61] in HERAFitter are implemented asa sum of two factorised contributions:

ΦIP(xIP, t) f IPa (β ,Q2)+ΦIR(xIP, t) f IR

a (β ,Q2) , (9)

where Φ(xIP, t) are the Reggeon and Pomeron fluxes. TheReggeon PDFs, f IR

a are fixed as those of the pion, while thePomeron PDFs, f IP

a , can be obtained from a fit to the data.

3.4 Drell-Yan Processes in pp or pp Collisions

The Drell-Yan (DY) process provides valuable informationabout PDFs. In pp and pp scattering, the Z/γ∗ and W pro-duction probe bi-linear combinations of quarks. Comple-mentary information on the different quark densities can beobtained from the W± asymmetry (d, u and their ratio), theratio of the W and Z cross sections (sensitive to the flavourcomposition of the quark sea, in particular to the s-quark dis-tribution), and associated W and Z production with heavyquarks (sensitive to s, c- and b-quark densities). Measure-ments at large boson transverse momentum pT & mW,Z arepotentially sensitive to the gluon distribution [62].

At LO the DY NC cross section triple differential in in-variant mass m, boson rapidity y and lepton scattering an-gle cosθ in the parton centre-of-mass frame can be writtenas [63, 64]:

d3σ

dmdyd cosθ=

πα2

3ms ∑q

σq(cosθ ,m)

×[

fq(x1,m2) fq(x2,m2)+(q↔ q)], (10)

where s is the squared centre-of-mass beam energy, theparton momentum fractions are given by x1,2 =

m√s exp(±y),

fq(x1,m2) are the PDFs at the scale of the invariant mass,and σq is the parton-parton hard scattering cross section.

The corresponding triple differential CC cross sectionhas the form:

d3σ

dmdyd cosθ=

πα2

48ssin4θW

m3(1− cosθ)2

(m2−m2W )+Γ 2

W m2W

× ∑q1,q2

V 2q1q2

fq1(x1,m2) fq2(x2,m2), (11)

where Vq1q2 is the Cabibbo-Kobayashi-Maskawa (CKM)quark mixing matrix and mW and ΓW are the W boson massand decay width, respectively.

The simple LO form of these expressions allows for theanalytic calculations of integrated cross sections. In both NCand CC expressions the PDFs depend only on the boson ra-pidity y and invariant mass m, while the integral in cosθ canbe evaluated analytically even for the case of realistic kine-matic cuts.

Beyond LO, the calculations are often time-consumingand Monte Carlo generators are employed. Currently, thepredictions for W and Z/γ∗ production are available up toNNLO and the predictions for W and Z production in asso-ciation with heavy flavour quarks are available to NLO.

There are several possibilities to obtain the theoreticalpredictions for DY production in HERAFitter. The NLOand NNLO calculations can be implemented using k-factoror fast grid techniques (see Sec. 4 for details), which areinterfaced to programs such as MCFM [65–67], available forNLO calculations, or FEWZ [68] and DYNNLO[69] for NLOand NNLO, with electroweak corrections estimated usingMCSANC [70, 71].

3.5 Jet Production in ep and pp or pp Collisions

The cross section for production of high pT hadronic jetsis sensitive to the high-x gluon PDF (see e.g. Ref. [16]).Therefore this process can be used to improve the determi-nation of the gluon PDF, which is particularly important forHiggs production and searches for new physics. Jet produc-tion cross sections are currently known only to NLO. Calcu-lations for higher-order contributions to jet production in ppcollisions are in progress [72–74]. Within HERAFitter, theNLOJet++ program [75, 76] may be used for calculationsof jet production. Similarly to the DY case, the calculationis very demanding in terms of computing power. Thereforefast grid techniques are used to facilitate the QCD analysesincluding jet cross section measurements in ep, pp and ppcollisions. For details see Sec. 4.

HERAFitter 7

3.6 Top-quark Production in pp or pp Collisions

At the LHC, top-quark pairs (tt) are produced dominantlyvia gg fusion. Thus, LHC measurements of the tt cross sec-tion provide additional constraints on the gluon distributionat medium to high values of x, on αs and on the top-quarkmass, mt [77]. Precise predictions for the total inclusive ttcross section are available up to NNLO [78, 79]. Currently,they can be computed within HERAFitter via an interfaceto the program HATHOR [80].

Fixed-order QCD predictions for the differential tt crosssection at NLO can be obtained by using the programMCFM [67, 81–84] interfaced to HERAFitter with fast gridtechniques.

Single top quarks are produced by exchanging elec-troweak bosons and the measurement of their productioncross section can be used, for example, to probe the ratio ofthe u and d distributions in the proton as well as the b-quarkPDF. Predictions for single-top production are available atthe NLO accuracy by using MCFM.

Approximate predictions up to NNLO in QCD for thedifferential tt cross section in one-particle inclusive kine-matics are available in HERAFitter through an interfaceto the program DiffTop [85, 86]. It uses methods of QCDthreshold resummation beyond the leading logarithmic ap-proximation. This allows the users to estimate the impact ofthe recent tt differential cross section measurements on theuncertainty of the gluon density within a QCD PDF fit atNNLO. A fast evaluation of the DiffTop differential crosssections is possible via an interface to fast grid computations[87].

4 Computational Techniques

Precise measurements require accurate theoretical predic-tions in order to maximise their impact in PDF fits. Per-turbative calculations become more complex and time-consuming at higher orders due to the increasing number ofrelevant Feynman diagrams. The direct inclusion of compu-tationally demanding higher-order calculations into iterativefits is thus not possible currently. However, a full repetitionof the perturbative calculation for small changes in input pa-rameters is not necessary at each step of the iteration. Twomethods have been developed which take advantage of thisto solve the problem: the k-factor technique and the fast gridtechnique. Both are available in HERAFitter.

4.1 k-factor Technique

The k-factors are defined as the ratio of the prediction of ahigher-order (slow) pQCD calculation to a lower-order (fast)

calculation using the same PDF. Because the k-factors de-pend on the phase space probed by the measurement, theyhave to be stored including their dependence on the rele-vant kinematic variables. Before the start of a fitting proce-dure, a table of k-factors is computed once for a fixed PDFwith the time consuming higher-order code. In subsequentiteration steps the theory prediction is derived from the fastlower-order calculation by multiplying by the pre-tabulatedk-factors.

This procedure, however, neglects the fact that the k-factors are PDF dependent, and as a consequence, they haveto be re-evaluated for the newly determined PDF at the endof the fit for a consistency check. The fit must be repeateduntil input and output k-factors have converged. In sum-mary, this technique avoids iteration of the higher-order cal-culation at each step, but still requires typically a few re-evaluations.

In HERAFitter, the k-factor technique can also be usedfor the fast computation of the time-consuming GM-VFNschemes for heavy quarks in DIS. “FAST” heavy-flavourschemes are implemented with k-factors defined as the ratioof calculations at the same perturbative order but for massivevs. massless quarks, e.g. NLO (massive)/NLO (massless).These k-factors are calculated only for the starting PDF andhence, the “FAST” heavy flavour schemes should only beused for quick checks. Full heavy flavour schemes shouldbe used by default. However, for the ACOT scheme, due toexceptionally long computation times, the k-factors are usedin the default setup of HERAFitter.

4.2 Fast Grid Techniques

Fast grid techniques exploit the fact that iterative PDF fit-ting procedures do not impose completely arbitrary changesto the types and shapes of the parameterised functions thatrepresent each PDF. Instead, it can be assumed that a genericPDF can be approximated by a set of interpolating func-tions with a sufficient number of judiciously chosen sup-port points. The accuracy of this approximation is checkedand optimised such that the approximation bias is negligiblysmall compared to the experimental and theoretical accu-racy. This method can be used to perform the time consum-ing higher-order calculations (Eq. 1) only once for the set ofinterpolating functions. Further iterations of the calculationfor a particular PDF set are fast, involving only sums overthe set of interpolators multiplied by factors depending onthe PDF. This approach can be used to calculate the crosssections of processes involving one or two hadrons in theinitial state and to assess their renormalisation and factori-sation scale variation.

This technique serves to facilitate the inclusion of timeconsuming NLO jet cross section predictions into PDF fitsand has been implemented in the two projects, fastNLO

8

[88, 89] and APPLGRID [90, 91]. The packages differ in theirinterpolation and optimisation strategies, but both of themconstruct tables with grids for each bin of an observable intwo steps: in the first step, the accessible phase space in theparton momentum fractions x and the renormalisation andfactorisation scales µR and µF is explored in order to op-timise the table size. In the second step the grid is filledfor the requested observables. Higher-order cross sectionscan then be obtained very efficiently from the pre-producedgrids while varying externally provided PDF sets, µR andµF, or αs(µR). This approach can in principle be extendedto arbitrary processes. This requires an interface betweenthe higher-order theory programs and the fast interpolationframeworks. For the HERAFitter implementations of thetwo packages, the evaluation of αs is done consistently withthe PDF evolution code. A brief description of each packageis given below:

– The fastNLO project [89] has been interfaced to theNLOJet++ program [75] for the calculation of jet pro-duction in DIS [92] as well as 2- and 3-jet productionin hadron-hadron collisions at NLO [76, 93]. Thresholdcorrections at 2-loop order, which approximate NNLOfor the inclusive jet cross section for pp and pp, havealso been included into the framework [94] followingRef. [95].The latest version of the fastNLO convolution program[96] allows for the creation of tables in which renormal-isation and factorisation scales can be varied as a func-tion of two pre-defined observables, e.g. jet transversemomentum p⊥ and Q for DIS. Recently, the differen-tial calculation of top-pair production in hadron colli-sions at approximate NNLO [85] has been interfacedto fastNLO [87]. The fastNLO code is available online[97]. Jet cross section grids computed for the kinemat-ics of various experiments can be downloaded from thissite.The fastNLO libraries and tables with theory pre-dictions for comparison to particular cross sectionmeasurements are included in the HERAFitter pack-age. The interface to the fastNLO tables from withinHERAFitter was used in a recent CMS analysis, wherethe impact on extraction of the PDFs from the inclusivejet cross section is investigated [98].

– In the APPLGRID package [91, 99], in addition to jetcross sections for pp(pp) and DIS processes, calcula-tions of DY production and other processes are also im-plemented using an interface to the standard MCFM par-ton level generator [65–67]. Variation of the renormal-isation and factorisation scales is possible a posteriori,when calculating theory predictions with the APPLGRIDtables, and independent variation of αS is also allowed.

For predictions beyond NLO, the k-factors technique canalso be applied within the APPLGRID framework.As an example, the HERAFitter interface to APPLGRID

was used by the ATLAS [100] and CMS [101] collabora-tions to extract the strange quark distribution of the pro-ton. The ATLAS strange PDF extracted employing thesetechniques is displayed in Fig. 4 together with a compar-ison to the global PDF sets CT10 [17] and NNPDF2.1[18] (taken from [100]).

Fig. 4 The strange antiquark distribution versus x for the ATLASepWZ free s NNLO fit [100] (magenta band) compared to predic-tions from NNPDF2.1 (blue hatched) and CT10 (green hatched) atQ2 = 1.9 GeV2. The ATLAS fit was performed using a k-factor ap-proach for NNLO corrections.

5 Fit Methodology

When performing a QCD analysis to determine PDFs thereare various assumptions and choices to be made concerning,for example, the functional form of the input parametrisa-tion, the treatment of heavy quarks and their mass values, al-ternative theoretical calculations, alternative representationsof the fit χ2 and for different ways of treating correlated sys-tematic uncertainties. It is useful to discriminate or quantifythe effect of a chosen ansatz within a common frameworkand HERAFitter is optimally designed for such tests. Themethodology employed by HERAFitter relies on a flexibleand modular framework that allows independent integrationof state-of-the-art techniques, either related to the inclusionof a new theoretical calculation, or of new approaches totreat data and their uncertainties.

In this section we describe the available options for thefit methodology in HERAFitter. In addition, as an alterna-

HERAFitter 9

tive approach to a complete QCD fit, the Bayesian reweight-ing method, which is also available in HERAFitter, is de-scribed.

5.1 Functional Forms for PDF Parametrisation

The PDFs can be parametrised using several predefinedfunctional forms and flavour decompositions:

Standard Polynomials: The standard polynomial form is themost commonly used. A polynomial functional form is usedto parametrise the x-dependence of the PDFs, where index jdenotes each parametrised PDF flavour:

x f j(x) = A jxB j(1− x)C j Pj(x). (12)

The parametrised PDFs are the valence distributions xuv andxdv, the gluon distribution xg, and the u-type and d-type sea,xU , xD, where xU = xu, xD = xd + xs at the starting scale,which is chosen below the charm mass threshold. The formof polynomials Pj(x) can be varied. The form (1+ ε j

√x+

D jx+E jx2) is used for the HERAPDF [21] with additionalconstraints relating to the flavour decomposition of the lightsea. This parametrisation is termed HERAPDF-style. Thepolynomial can also be parametrised in the CTEQ-style,where Pj(x) takes the form ea3x(1 + ea4x + ea5x2) and, incontrast to the HERAPDF-style, this is positive by construc-tion. QCD number and momentum sum rules are used todetermine the normalisations A for the valence and gluondistributions, and the sum-rule integrals are solved analyti-cally.

Bi-Log-Normal Distributions: This parametrisation is moti-vated by multi-particle statistics and has the following func-tional form:

x f j(x) = a jxp j−b j log(x)(1− x)q j−d j log(1−x). (13)

This function can be regarded as a generalisation of the stan-dard polynomial form described above, however, numericalintegration of Eq. 13 is required in order to impose the QCDsum rules.

Chebyshev Polynomials: A flexible parametrisation basedon the Chebyshev polynomials can be employed for thegluon and sea distributions. Polynomials with argumentlog(x) are considered for better modelling the low-x asymp-totic behaviour of those PDFs. The polynomials are mul-tiplied by a factor of (1− x) to ensure that they vanish asx→ 1. The resulting parametric form reads

xg(x) = Ag (1− x)Ng−1

∑i=0

AgiTi

(−2logx− logxmin

logxmin

), (14)

xS(x) = (1− x)NS−1

∑i=0

ASiTi

(−2logx− logxmin

logxmin

), (15)

where Ti are first-type Chebyshev polynomials of order i.The normalisation factor Ag is derived from the momentumsum rule analytically. Values of Ng,S to 15 are allowed, how-ever the fit quality is already similar to that of the standard-polynomial parametrisation from Ng,S ≥ 5 and has a sim-ilar number of free parameters. Fig. 5 (taken from [102])shows a comparison of the gluon distribution obtained withthe parametrisation Eqs. 14, 15 to the standard-polynomialone, for Ng,S = 9.

Cheb=9, Q2 = 1.9. GeV2

-5

-4

-3

-2

-1

0

1

2

3

4

5

10-4

10-3

10-2

10-1

1x

xG(x

)

Fig. 5 The gluon density is shown at the starting scale Q2 = 1.9 GeV2.The black lines correspond to the uncertainty band of the gluon distri-bution using a standard parametrisation and it is compared to the caseof the Chebyshev parametrisation [102]. The uncertainty band for thelatter case is estimated using the Monte Carlo technique (see Sec. 5.3)with the green lines denoting fits to data replica. Red lines indicate thestandard deviation about the mean value of these replicas.

External PDFs: HERAFitter also provides the possibilityto access external PDF sets, which can be used to computetheoretical predictions for the cross sections for all the pro-cesses available in HERAFitter. This is possible via an in-terface to LHAPDF [33, 34] providing access to the globalPDF sets. HERAFitter also allows one to evolve PDFs fromLHAPDF using QCDNUM. Fig. 6 illustrates a comparison ofvarious gluon PDFs accessed from LHAPDF as produced withthe drawing tools available in HERAFitter.

5.2 Representation of χ2

The PDF parameters are determined in HERAFitter byminimisation of a χ2 function taking into account correlatedand uncorrelated measurement uncertainties. There are vari-ous forms of χ2, e.g. using a covariance matrix or providingnuisance parameters to encode the dependence of each cor-related systematic uncertainty for each measured data point.The options available in HERAFitter are the following:

10

x -310 -210 -110

)2 x

g(x,

Q

0

1

2

3

4

5

6

7

2 = 4.0 GeV2Q

CT10_NNLOMSTW2008_NNLOABM12_4N_NNLOHERAPDF1.5_NNLONNPDF2.3_NNLO

Fig. 6 The gluon PDF as extracted by various groups at the scale ofQ2 = 4 GeV2, plotted using the drawing tools from HERAFitter.

Covariance Matrix Representation: For a data point µiwith a corresponding theory prediction mi, the χ2 func-tion can be expressed in the following form:

χ2(m) = ∑

i,k(mi−µi)C−1

ik (mk−µk), (16)

where the experimental uncertainties are given as a co-variance matrix Cik for measurements in bins i and k. Thecovariance matrix Cik is given by a sum of statistical, un-correlated and correlated systematic contributions:

Cik = Cstatik +Cuncor

ik +Csysik . (17)

Using this representation one cannot distinguish the ef-fect of each source of systematic uncertainty.

Nuisance Parameter Representation: In this case, the χ2 isexpressed as

χ2 (m,b) =∑

i

[µi−mi

(1−∑ j γ i

jb j

)]2

δ 2i,uncm2

i +δ 2i,stat µimi

(1−∑ j γ i

jb j

)+∑j

b2j ,

(18)

where, δi,stat and δi,unc are relative statistical and un-correlated systematic uncertainties of the measurementi. Further, γ i

j quantifies the sensitivity of the measure-ment to the correlated systematic source j. The functionχ2 depends on the set of systematic nuisance parame-ters b j. This definition of the χ2 function assumes thatsystematic uncertainties are proportional to the centralprediction values (multiplicative uncertainties, mi(1−∑ j γ i

jb j)), whereas the statistical uncertainties scale with

the square root of the expected number of events. How-ever, additive treatment of uncertainties is also possiblein HERAFitter.During the χ2 minimisation, the nuisance parameters b jand the PDFs are determined, such that the effect of dif-ferent sources of systematic uncertainties can be distin-guished.

Mixed Form Representation: In some cases, the statisti-cal and systematic uncertainties of experimental dataare provided in different forms. For example, the corre-lated experimental systematic uncertainties are availableas nuisance parameters, but the bin-to-bin statistical cor-relations are given in the form of a covariance matrix.HERAFitter offers the possibility to include such mixedforms of information.

Any source of measured systematic uncertainty can betreated as additive or multiplicative, as described above. Thestatistical uncertainties can be included as additive or fol-lowing the Poisson statistics. Minimisation with respect tonuisance parameters is performed analytically, however, formore detailed studies of correlations individual nuisance pa-rameters can be included into the MINUIT minimisation.

5.3 Treatment of the Experimental Uncertainties

Three distinct methods for propagating experimental uncer-tainties to PDFs are implemented in HERAFitter and re-viewed here: the Hessian, Offset, and Monte Carlo method.

Hessian (Eigenvector) method: The PDF uncertaintiesreflecting the data experimental uncertainties are esti-mated by examining the shape of the χ2 function in theneighbourhood of the minimum [103]. Following theapproach of Ref. [103], the Hessian matrix is definedby the second derivatives of χ2 on the fitted PDFparameters. The matrix is diagonalised and the Hessianeigenvectors are computed. Due to orthogonality thesevectors correspond to independent sources of uncer-tainty in the obtained PDFs.

Offset method: The Offset method [104] uses the χ2 func-tion for the central fit, but only uncorrelated uncertain-ties are taken into account. The goodness of the fit can nolonger be judged from the χ2 since correlated uncertain-ties are ignored. The correlated uncertainties are propa-gated into the PDF uncertainties by performing variantsof the fit with the experimental data varied by±1σ fromthe central value for each systematic source. The result-ing deviations of the PDF parameters from the ones ob-tained in the central fit are statistically independent, andthey can be combined in quadrature to derive a total PDFsystematic uncertainty.

HERAFitter 11

The uncertainties estimated by the offset method aregenerally larger than those from the Hessian method.

Monte Carlo method: The Monte Carlo (MC) technique[105, 106] can also be used to determine PDF uncer-tainties. The uncertainties are estimated using pseudo-data replicas (typically > 100) randomly generated fromthe measurement central values and their systematic andstatistical uncertainties taking into account all point-to-point correlations. The QCD fit is performed for eachreplica and the PDF central values and their experimen-tal uncertainties are estimated from the distribution ofthe PDF parameters obtained in these fits, by taking themean values and standard deviations over the replicas.The MC method has been checked against the standarderror estimation of the PDF uncertainties obtained bythe Hessian method. A good agreement was found be-tween the methods provided that Gaussian distributionsof statistical and systematic uncertainties are assumedin the MC approach [32]. A comparison is illustratedin Fig. 7. Similar findings were reported by the MSTWglobal analysis [107].

Fit vs H1PDF2000, Q2 = 4. GeV2

0

1

2

3

4

5

6

7

8

9

10

10-4

10-3

10-2

10-1

1x

xG(x

)

Fig. 7 Comparison between the standard error calculations as em-ployed by the Hessian approach (black lines) and the MC approach(with more than 100 replicas) assuming Gaussian distribution for un-certainty distributions, shown here for each replica (green lines) to-gether with the evaluated standard deviation (red lines) [32]. The blackand red lines in the figure are superimposed because agreement of themethods is so good that it is hard to distinguish them.

Since the MC method requires large number of replicas,the eigenvector representation is a more convenient wayto store the PDF uncertainties. It is possible to transformMC to eigenvector representation as shown by [108].Tools to perform this transformation are provided withHERAFitter and were recently employed for the repre-

sentation of correlated sets of PDFs at different pertur-bative orders [109].

The nuisance parameter representation of χ2 in Eq. 18 isderived assuming symmetric experimental errors, however,the published systematic uncertainties are often asymmet-ric. HERAFitter provides the possibility to use asymmetricsystematic uncertainties. The implementation relies on theassumption that asymmetric uncertainties can be describedby a parabolic function. The nuisance parameter in Eq. 18 ismodified as follows

γij→ ω

ijb j + γ

ij, (19)

where the coefficients ω ij, γ i

j are defined from the maximumand minimum shifts of the cross sections due to a variationof the systematic uncertainty j, S±i j ,

ωij =

12

(S+i j +S−i j

), γ

ij =

12

(S+i j −S−i j

). (20)

5.4 Treatment of the Theoretical Input Parameters

The results of a QCD fit depend not only on the input databut also on the input parameters used in the theoretical cal-culations. Nowadays, PDF groups address the impact of thechoices of theoretical parameters by providing alternativePDFs with different choices of the mass of the charm quarks,mc, mass of the bottom quarks, mb, and the value of αs(mZ).Other important aspects are the choice of the functional formfor the PDFs at the starting scale and the value of the startingscale itself. HERAFitter provides the possibility of differ-ent user choices of all these inputs.

5.5 Bayesian Reweighting Techniques

As an alternative to performing a full QCD fit, HERAFitterallows the user to assess the impact of including new datain an existing fit using the Bayesian Reweighting technique.The method provides a fast estimate of the impact of newdata on PDFs. Bayesian Reweighting was first proposed forPDF sets delivered in the form of MC replicas by [105] andfurther developed by the NNPDF Collaboration [110, 111].More recently, a method to perform Bayesian Reweightingstudies starting from PDF fits for which uncertainties areprovided in the eigenvector representation has been also de-veloped [107]. The latter is based on generating replica setsby introducing Gaussian fluctuations on the central PDF setwith a variance determined by the PDF uncertainty givenby the eigenvectors. Both reweighting methods are imple-mented in HERAFitter. Note that the precise form of theweights used by both methods has recently been questioned[112, 113].

12

The Bayesian Reweighting technique relies on the factthat MC replicas of a PDF set give a representation of theprobability distribution in the space of PDFs. In particular,the PDFs are represented as ensembles of Nrep equiprobable(i.e. having weights equal to unity) replicas, { f}. The centralvalue for a given observable, O({ f}), is computed as theaverage of the predictions obtained from the ensemble as

〈O({ f})〉= 1Nrep

Nrep

∑k=1

O( f k), (21)

and the uncertainty as the standard deviation of the sample.Upon inclusion of new data the prior probability distri-

bution, given by the original PDF set, is modified accordingto Bayes Theorem such that the weight of each replica, wk,is updated according to

wk =(χ2

k )12 (Ndata−1)e−

12 χ2

k

1Nrep

∑Nrepk=1(χ

2k )

12 (Ndata−1)e−

12 χ2

k, (22)

where Ndata is the number of new data points, k denotes thespecific replica for which the weight is calculated and χ2

k isthe χ2 of the new data obtained using the k-th PDF replica.Given a PDF set and a corresponding set of weights, whichdescribes the impact of the inclusion of new data, the pre-diction for a given observable after inclusion of the new datacan be computed as the weighted average,

〈O({ f})〉= 1Nrep

Nrep

∑k=1

wkO( f k). (23)

To simplify the use of a reweighted set, an unweightedset (i.e. a set of equiprobable replicas which incorporates theinformation contained in the weights) is generated accordingto the unweighting procedure described in [110]. The num-ber of effective replicas of a reweighted set is measured byits Shannon Entropy [111]

Neff ≡ exp

{1

Nrep

Nrep

∑k=1

wk ln(Nrep/wk)

}, (24)

which corresponds to the size of a refitted equiprobablereplica set containing the same amount of information. Thisnumber of effective replicas, Neff, gives an indicative mea-sure of the optimal size of an unweighted replica set pro-duced with the reweighting/unweighting procedure. No ex-tra information is gained by producing a final unweighted setthat has a number of replicas (significantly) larger than Neff.If Neff is much smaller than the original number of replicasthe new data have great impact, however, it is unreliable touse the new reweighted set. In this case, instead, a full refitshould be performed.

6 Alternatives to DGLAP Formalism

QCD calculations based on the DGLAP [11–15] evolutionequations are very successful in describing all relevant hardscattering data in the perturbative region Q2 & few GeV2. Atsmall-x (x < 0.01) and small-Q2 DGLAP dynamics may bemodified by saturation and other (non-perturbative) higher-twist effects. Different approaches alternative to the DGLAPformalism can be used to analyse DIS data in HERAFitter.These include several dipole models and the use of trans-verse momentum dependent, or unintegrated PDFs (uPDFs).

6.1 Dipole Models

The dipole picture provides an alternative approach toproton-virtual photon scattering at low x which can be ap-plied to both inclusive and diffractive processes. In this ap-proach, the virtual photon fluctuates into a qq (or qqg) dipolewhich interacts with the proton [114, 115]. The dipoles canbe considered as quasi-stable quantum mechanical states,which have very long life time ∝ 1/mpx and a size whichis not changed by scattering with the proton. The dynamicsof the interaction are embedded in a dipole scattering ampli-tude.

Several dipole models, which assume different be-haviours of the dipole-proton cross section, are implementedin HERAFitter: the Golec-Biernat-Wusthoff (GBW) dipolesaturation model [28], a modified GBW model which takesinto account the effects of DGLAP evolution, termed theBartels-Golec-Kowalski (BGK) dipole model [30] and thecolour glass condensate approach to the high parton den-sity regime, named the Iancu-Itakura-Munier (IIM) dipolemodel [29].

GBW model: In the GBW model the dipole-proton crosssection σdip is given by

σdip(x,r2) = σ0

(1− exp

[− r2

4R20(x)

]), (25)

where r corresponds to the transverse separation betweenthe quark and the antiquark, and R2

0 is an x-dependent scaleparameter which represents the spacing of the gluons in theproton. R2

0 takes the form, R20(x) = (x/x0)

λ 1/GeV2, and iscalled the saturation radius. The cross-section normalisa-tion σ0, x0, and λ are parameters of the model fitted to theDIS data. This model gives exact Bjorken scaling when thedipole size r is small.

BGK model: The BGK model is a modification of the GBWmodel assuming that the spacing R0 is inverse to the gluondistribution and taking into account the DGLAP evolutionof the latter. The gluon distribution, parametrised at some

HERAFitter 13

starting scale by Eq. 12, is evolved to larger scales usingDGLAP evolution.

BGK model with valence quarks: The dipole models arevalid in the low-x region only, where the valence quark con-tribution to the total proton momentum is 5% to 15% for xfrom 0.0001 to 0.01 [116]. The inclusive HERA measure-ments have a precision which is better than 2%. Therefore,HERAFitter provides the option of taking into account thecontribution of the valence quarks

IIM model: The IIM model assumes an expression forthe dipole cross section which is based on the Balitsky-Kovchegov equation [117]. The explicit formula for σdip canbe found in [29]. The alternative scale parameter R, x0 andλ are fitted parameters of the model.

6.2 Transverse Momentum Dependent PDFs

QCD calculations of multiple-scale processes and complexfinal-states can necessitate the use of transverse-momentumdependent (TMD) [7], or unintegrated parton distributionand parton decay functions [118–126]. TMD factorisationhas been proven recently [7] for inclusive DIS. TMD fac-torisation has also been proven in the high-energy (small-x)limit [127–129] for particular hadron-hadron scattering pro-cesses, like heavy flavour, vector boson and Higgs produc-tion.

In the framework of high-energy factorisation [127, 130,131] the DIS cross section can be written as a convolutionin both longitudinal and transverse momenta of the TMDparton distribution function A

(x,kt ,µ

2F)

with the off-shellparton scattering matrix elements as follows

σ j(x,Q2) =∫ 1

xdz∫

d2kt σ j(x,Q2,z,kt) A(z,kt ,µ

2F), (26)

where the DIS cross sections σ j( j = 2,L) are related to thestructure functions F2 and FL by σ j = 4π2Fj/Q2, and thehard-scattering kernels σ j of Eq. 26 are kt -dependent.

The factorisation formula in Eq. 26 allows resummationof logarithmically enhanced small-x contributions to all or-ders in perturbation theory, both in the hard scattering coef-ficients and in the parton evolution, fully taking into accountthe dependence on the factorisation scale µF and on the fac-torisation scheme [132, 133].

Phenomenological applications of this approach requirematching of small-x contributions with finite-x contribu-tions. To this end, the evolution of the transverse momentumdependent gluon density A is obtained by combining theresummation of small-x logarithmic corrections [134–136]with medium-x and large-x contributions to parton split-ting [11, 14, 15] according to the CCFM evolution equa-tion [23–26].

The cross section σ j, ( j = 2,L) is calculated in a FFNscheme, using the boson-gluon fusion process (γ∗g∗→ qq).The masses of the quarks are explicitly included as param-eters of the model. In addition to γ∗g∗ → qq, the contribu-tion from valence quarks is included via γ∗q→ q by using aCCFM evolution of valence quarks [137–139].

CCFM Grid Techniques: The CCFM evolution cannot bewritten easily in an analytic closed form. For this rea-son, a MC method is employed, which is, however, time-consuming, and thus cannot be used directly in a fit program.

Following the convolution method introduced in [139,140], the kernel ˜A (x′′,kt , p) is determined from the MC so-lution of the CCFM evolution equation, and then folded witha non-perturbative starting distribution A0(x)

xA (x,kt , p) = x∫

dx′∫

dx′′A0(x′) ˜A(x′′,kt , p

)δ (x′x′′− x)

=∫

dx′A0(x′)xx′

˜A( x

x′,kt , p

), (27)

where kt denotes the transverse momentum of the propaga-tor gluon and p is the evolution variable.

The kernel ˜A incorporates all of the dynamics of theevolution. It is defined on a grid of 50⊗ 50⊗ 50 bins inx,kt , p. The binning in the grid is logarithmic, except forthe longitudinal variable x for which 40 bins in logarithmicspacing below 0.1, and 10 bins in linear spacing above 0.1are used.

Calculation of the cross section according to Eq. 26 in-volves a time-consuming multidimensional MC integration,which suffers from numerical fluctuations. This cannot beemployed directly in a fit procedure. Instead the followingequation is applied:

σ(x,Q2) =∫ 1

xdxgA (xg,kt , p)σ(x,xg,Q2)

=∫ 1

xdx′A0(x′)σ(x/x′,Q2), (28)

where first σ(x′,Q2) is calculated numerically with a MCintegration on a grid in x for the values of Q2 used in thefit. Then the last step in Eq. 28 is performed with a fast nu-merical Gauss integration, which can be used directly in thefit.

Functional Forms for TMD parametrisation: For the start-ing distribution A0, at the starting scale Q2

0, the followingform is used:

xA0(x,kt) = Nx−B(1− x)C(1−Dx+E

√x)

× exp[−k2t /σ

2], (29)

where σ2 = Q20/2 and N,B,C,D,E are free parameters. Va-

lence quarks are treated using the method of Ref. [137] asdescribed in Ref. [139] with a starting distribution takenfrom any collinear PDF and imposition of the flavour sum

14

rule at every scale p.The TMD parton densities can be plotted either with HERA-

Fitter tools or with TMDplotter [35].

7 HERAFitter Code Organisation

HERAFitter is an open source code under the GNU generalpublic licence. It can be downloaded from a dedicated web-page [10] together with its supporting documentation andfast grid theory files (described in Sec. 4) associated withdata files. The source code contains all the relevant infor-mation to perform QCD fits with HERA DIS data as a de-fault set. 1 The execution time depends on the fitting optionsand varies from 10 minutes (using “FAST” techniques asdescribed in Sec. 4) to several hours when full uncertaintiesare estimated. The HERAFitter code is a combination ofC++ and Fortran 77 libraries with minimal dependencies,i.e. for the default fitting options no external dependenciesare required except the QCDNUM evolution program [22]. TheROOT libraries are only required for the drawing tools andwhen invoking APPLGRID. Drawing tools built into HERA-

Fitter provide a qualitative and quantitative assessment ofthe results. Fig. 8 shows an illustration of a comparison be-tween the inclusive NC data from HERA I with the predic-tions based on HERAPDF1.0 PDFs. The consistency of themeasurements and the theory can be expressed by pulls, de-fined as the difference between data and theory divided bythe uncorrelated error of the data. In each kinematic bin ofthe measurement, pulls are provided in units of standard de-viations. The pulls are also illustrated in Fig. 8.

x 0.01 0.1

/dx

σ d

­0.4

­0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

NC →p +e

2 = 150 GeV2H1 ZEUS Data Q

uncorrelatedδ

totalδ

Theory

Theory + shifts

Theory_RT

Theory_ACOT

Th

eo

ry/D

ata

0.95

1

1.05

Da

ta

Th

eo

ry+

sh

ifts

0.96

0.98

1

1.02

1.04

x 0.01 0.1

pu

lls

­2

0

2

Fig. 8 An illustration of the consistency of HERA measurements [21]and the theory predictions, obtained in HERAFitter with the defaultdrawing tool.

In HERAFitter there are also available cache options forfast retrieval, fast evolution kernels, and the OpenMP (OpenMulti-Processing) interface which allows parallel applica-tions of the GM-VFNS theory predictions in DIS.

1 Default settings in HERAFitter are tuned to reproduce the centralHERAPDF1.0 set.

8 Applications of HERAFitter

The HERAFitter program has been used in a number ofexperimental and theoretical analyses. This list includes sev-eral LHC analyses of SM processes, namely inclusive Drell-Yan and Wand Z production [100, 101, 141–143], inclu-sive jet production [98, 144], and inclusive photon pro-duction [145]. The results of QCD analyses using HERA-

Fitter were also published by HERA experiments for in-clusive [21, 146] and heavy flavour production measure-ments [147, 148]. The following phenomenological studieshave been performed with HERAFitter: a determination ofthe transverse momentum dependent gluon distribution us-ing precision HERA data [139], an analysis of HERA datawithin a dipole model [149], the study of the low-x uncer-tainties in PDFs determined from the HERA data using dif-ferent parametrisations [102] and the impact of QED radia-tive corrections on PDFs [150]. A recent study based on a setof PDFs determined with HERAFitter and addressing thecorrelated uncertainties between different orders has beenpublished in [109]. An application of the TMDs obtainedwith HERAFitterW production at the LHC can be found in[151].

The HERAFitter framework has been used to pro-duce PDF grids from QCD analyses performed at HERA[21, 152] and at the LHC [153], using measurements fromATLAS [100, 144]. These PDFs can be used to study predic-tions for SM or beyond SM processes. Furthermore, HERA-Fitter provides the possibility to perform various bench-marking exercises [154] and impact studies for possiblefuture colliders as demonstrated by QCD studies at theLHeC [155].

9 Summary

HERAFitter is the first open-source code designed for stud-ies of the structure of the proton. It provides a unique andflexible framework with a wide variety of QCD tools to fa-cilitate analyses of the experimental data and theoretical cal-culations.

The HERAFitter code, in version 1.1.0, has sufficientoptions to reproduce the majority of the different theoreti-cal choices made in MSTW, CTEQ and ABM fits. This willpotentially make it a valuable tool for benchmarking andunderstanding differences between PDF fits. Such a studywould however need to consider a range of further questions,such as the choices of data sets, treatments of uncertainties,input parameter values, χ2 definitions, nuclear corrections,etc.The further progress of HERAFitter will be driven by thelatest QCD advances in theoretical calculations and in theprecision of experimental data.

HERAFitter 15

Acknowledgements HERAFitter developers team acknowledges thekind hospitality of DESY and funding by the Helmholtz Alliance”Physics at the Terascale” of the Helmholtz Association. We are grate-ful to the DESY IT department for their support of the HERAFitter

developers. We thank the H1 and ZEUS collaborations for the supportin the initial stage of the project. Additional support was received fromthe BMBF-JINR cooperation program, the Heisenberg-Landau pro-gram, the RFBR grant 12-02-91526-CERN a, the Polish NSC projectDEC-2011/03/B/ST2/00220 and a dedicated funding of the Initiativeand Networking Fond of Helmholtz Association SO-072. We also ac-knowledge Nathan Hartland with Luigi Del Debbio for contributing tothe implementation of the Bayesian Reweighting technique and wouldlike to thank R. Thorne for fruitful discussions.

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