TMDlib and TMDplotter: library and plotting tools for transversemomentumdependent parton distributions
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Abstract
Transversemomentumdependent distributions (TMDs) are extensions of collinear parton distributions and are important in highenergy physics from both theoretical and phenomenological points of view. In this manual we introduce the library \({TMDlib }\), a tool to collect transversemomentumdependent parton distribution functions (TMD PDFs) and fragmentation functions (TMD FFs) together with an online plotting tool, TMDplotter. We provide a description of the program components and of the different physical frameworks the user can access via the available parameterisations.
Keywords
Transverse Momentum Fragmentation Function Factorisation Theorem Hard Scale Hadronic Tensor1 PROGRAM SUMMARY
Computer for which the program is designed and others on which it is operable: any with standard C++, tested on Linux and Mac OSX systems
Programming Language used: C++
Highspeed storage required: No
Separate documentation available: No
Keywords: QCD, TMD factorisation, highenergy factorisation, TMD PDFs, TMD FFs, unintegrated PDFs, small\(x\) physics.
Other programs used: LHAPDF (version 6) for access to collinear parton distributions, Boost (required by LHAPDF version 6), Root (version higher than 5.30) for plotting the results
Download of the program: http://tmdlib.hepforge.org
Unusual features of the program: None
Contacts: H. Jung (hannes.jung@desy.de), E. Nocera (emanuele.nocera@edu.unige.it), A. Signori (asignori@nikhef.nl)
Citation policy: please cite the current version of the manual and the paper(s) related to the parameterisation(s).
2 Introduction
The Quantum Chromodynamics (QCD) interpretation of highenergy particle reactions requires a simultaneous treatment of processes at different energy scales. Factorisation theorems provide the mathematical framework to properly separate the physical regimes. For instance, when two protons collide in a Drell–Yan (DY) event the highenergy partonic cross section is described with a perturbative QCD expansion and the soft physics underlying the structure of the hadrons is treated with parton distribution functions (PDFs), supplemented by QCD evolution. “Evolution”, in this context, refers to the scale dependence of parton distributions (and similar nonperturbative objects) that arises in a detailed treatment of factorisation in QCD perturbation theory. A classic example of a consequence of QCD evolution is the violation of Bjorkenscaling in inclusive deepinelastic leptonhadron scattering (DIS), predicted by the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) evolution equations [1, 2, 3].
The same basic picture applies to other (semi)inclusive processes, like semiinclusive DIS (SIDIS), and \(e^+e^\) annihilation into hadrons. A PDF describes the likelihood for finding a parton of a particular momentum inside an incoming hadron. In processes with observed hadrons in the final state, fragmentation functions (FFs) enter to describe the transition from a partonic state to an observed finalstate hadron.
For sufficiently inclusive processes, only the component of parton momentum collinear to the momentum of its parent hadron is relevant at leading power (leading twist) in the hard scale. Factorisation theorems for such processes are traditionally called collinear factorisation theorems. In less inclusive processes, however, sensitivity to the partonic motion transverse to the direction of the parent hadron can become important. In such cases, the PDFs and FFs must carry information about transverse parton momentum in addition to the collinear momentum. One must introduce transversemomentumdependent (TMD) PDFs and FFs and use them in the context of new factorisation theorems, called TMD factorisation theorems. TMD factorisation has been formulated for a number of semiinclusive processes including SIDIS, DY and \(e^+e^\) annihilation [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. For particular processes in hadronic collisions, like heavy flavour or heavy boson (including Higgs) production, TMD factorisation has also been formulated in the highenergy (small\(x\)) limit [17, 18, 19, 20]. In this context, the functions encoding the hadronic structure are more often referred to as unintegrated parton distribution functions (uPDFs), see e.g. Refs. [21, 22, 23, 24, 25, 26, 27, 28, 29].
The presence of a large variety of TMD factorisation and evolution frameworks complicates efforts to compare different TMD PDFs/FFs and uPDFs parameterisations. In some cases, the differences arise because different formalisms employ similar TMD concepts, but are tailored to specific physical applications. An example is the difference between the Collins–Soper–Sterman (CSS) style of TMD factorisation discussed in Sect. 2.1 compared with the highenergy TMD factorisation style discussed in Sect. 2.2. The former is designed for semiinclusive processes differential in a particular physical transverse momentum and with a finite and nonzero ratio between the hard scale and the overall energy. The latter (highenergy TMD factorisation) is designed for the limit of a fixed hard scale and very high energies. Moreover, within each category there are also competing subcategories of approaches. For instances, the detailed phenomenological methods that employ a CSSstyle of approach in Refs. [30, 31, 32, 33, 34, 35, 36, 37] are rather different.
In this paper, we describe a new tool for collecting different fits and parameterisations into a single library, TMDlib , and the online plotter tool, TMDplotter. Provided that the user takes into account all the possible differences between formalisms, collecting parameterisations for both the objects in TMDlib and TMDplotter will also make phenomenological comparisons easier.
The paper is organised as follows: In Sect. 2, we briefly introduce the theoretical framework for both TMD and highenergy factorisation and evolution. In Sect. 3, we present a concise documentation of the TMDlib library and TMDplotter tool, discussing the basic procedure to readily use them.
3 Theoretical framework
In this section, we briefly describe two different commonlyused frameworks for factorisation and evolution of parton distributions. Specifically, we discuss TMD and highenergy factorisation theorems and evolution equations.
3.1 TMD factorisation and evolution
When one hard scale enters a highenergy process (like the invariant mass of the exchanged virtual photon in DIS) and the relevant transverse momenta are integrated over, one applies collinear factorisation to separate the hard partonic physics from the soft hadronic physics. When sensitivity to intrinsic transverse momentum is important, one must go beyond the collinear framework to factorise perturbative and nonperturbative dynamics. For example, this is the case in processes with observed transverse momenta in the final states, like SIDIS and DY lepton pair production at low transverse momentum. In these cases the low transverse momentum provides greater access to novel QCD dynamics as compared to the collinear case. If the observable transverse momenta are much larger than \(\Lambda _\mathrm{QCD}\), then often the cross section may be expressed entirely in collinear factorisation, though supplemented by transverse momentum resummation.
The term \(Y_\mathrm{SIDIS}(\mathbf{q}_\perp ,Q)\) is a correction for the region of \(q_\perp \sim Q\) where a separation into TMDs is not valid, and all transverse momentum is generated inside the hard scattering. This socalled \(Y\)term is calculable in collinear factorisation. With it included, the corrections are suppressed by powers of \(\Lambda _\mathrm{QCD}/Q\), pointbypoint in \(\mathbf{q}_\perp \), as indicated by the last term, where \(p > 0\). Taking into account all the possible combinations of polarisation (parton, target and detected hadron), there are eight TMD PDFs and eight TMD FFs at leadingtwist, although the number of operator combinations could be larger [38, 39]. The expression of the hadronic tensor is modified accordingly [40, 41, 42].
TMD parton distributions or fragmentation functions depend on two types of auxiliary scales, \(\zeta _{f,D}\) and \(\mu \), and they satisfy evolution equations with respect to both of them. The evolution with respect to \(\zeta _{f}\) and \(\zeta _D\) corresponds to Collins–Soper (CS) evolution and is determined by a processindependent soft factor [9, 15, 31, 43, 44, 45, 46, 47, 48, 49]. The scales \(\zeta _f,\ \zeta _D\) must satisfy the constraint \(\zeta _f \zeta _D = Q^4\). The evolution in \(\mu \), instead, is determined by standard renormalisation group methods.
When the energy range covered by the experimental data is not large (see, e.g., Ref. [50, 51]) fits of TMD PDFs and FFs can be performed without taking into account effects induced by evolution. These fits rely essentialy on a simple parton model approach and are oriented towards investigations of hadron structure at a relatively lowenergy scale. Recent examples are Refs. [35, 52]. In order to explore the evolution of hadron structure with the energy scale, these fixed scale fits can be incorporated into a Collins–Soper–Sterman (CSS) style of factorisation theorem like Eq. (1), as described in Refs. [53, 54]. There, fixed scale fits from [55, 56, 57, 58, 59] are combined with traditional CSS style fits from Refs. [30, 31].
3.2 Highenergy factorisation and evolution
A form of TMD factorisation holds at high energy [17, 60, 61] and has been applied to several processes in photonhadron, leptonhadron and hadronhadron collisions. For instance, the highenergy factorisation expresses the heavyquark leptoproduction cross section in terms of the TMD gluon density via wellprescribed, calculable perturbative coefficients [60]. This framework is extended to deepinelastic structure functions in Refs. [62, 63]. Perturbative applications of the method include the resummation of small\(x\) logarithmic corrections to DIS to all orders in \(\alpha _s\) at leading and nexttoleading \(\ln x\) level [62, 63, 64, 65]. In hadronhadron scattering, highenergy factorisation has been applied to processes such as heavy flavour and Higgs boson production [20, 60].
The factorisation formula, Eq. (2), allows for resummation of logarithmically enhanced \(x\rightarrow 0 \) contributions to all orders in perturbation theory, both in the hardscattering coefficients and in the parton evolution, taking into account the full dependence on the factorisation scale \(\mu \) and on the factorisation scheme [62, 63].
Realistic applications of this approach at collider energies require matching of \(x \rightarrow 0\) contributions with finite\(x\) contributions. To this end, the evolution of the gluon uPDF \(\mathcal{A} \) is obtained by combining the resummation of small\(x\) logarithmic contributions [66, 67, 68] with medium and large\(x\) contributions to parton splitting [1, 2, 3], according to the CCFM evolution equations [73].
The cross section \(\sigma _j\) (\(j= 2, L\)) is usually computed in a Fixed Flavour Number (FFN) scheme, where the photongluon fusion process (\(\gamma ^* g^* \rightarrow q \bar{q}\)) is included. The masses of the quarks are explicitly included with the light and heavy quark masses being free parameters. In addition to \(\gamma ^* g^* \rightarrow q\bar{q}\), the contribution from valence quarks is included via \(\gamma ^* q \rightarrow q\) by using CCFM evolution of valence quarks [73, 74, 75]. A fit of CCFM uPDFs to the combined DIS precision data [76, 77] has been recently presented in Ref. [75] using the evolution given in Ref. [78]. Earlier CCFM fits to DIS were presented in Ref. [79]. In Ref. [80] the unintegrated gluon distribution has been obtained by means of a saturation ansatz.
4 TMDlib documentation
TMDlib is a C++ library which provides a framework and an interface to a collection of different uPDF/TMD parameterisations. The parameterisations of TMDs in TMDlib are explicitly authorised for distribution in TMDlib by the authors. No explicit QCD evolution code is included: the parameterizations are as given in the corresponding references. In the present version of TMDlib no attempt is made to unify grid files and the interpolation procedure; both are those provided by the authors.
The source code of TMDlib is available from http://tmdlib.hepforge.org/ and can be installed using the standard autotools sequence configure, make, make install, with options to specify the installation path and the location of the LHAPDF PDF library [81, 82] and the ROOT data analysis framework library [83, 84] (which is used optionally for plotting). If ROOT is not found via rootconfig, the plotting option is disabled. After installation, TMDlibconfig gives access to necessary environment variables.

Initialisation. The first step consists in initialising the desired uPDF/TMD set. Initialisation assigns the chosen uPDF/TMD set, specified by its name, an identifying number proper to that set.^{1} This number is stored into memory and called each time the identification of the uPDF/TMD set is needed by any TMDlib internal function. The complete list of uPDF/TMD sets available in is given in Table 1 with the corresponding name, identifier, kinematic coverage, and reference. This list will be constantly updated at http://tmdlib.hepforge.org/pdfsets.html as soon as new uPDF/TMD sets will become available. The TMD fit of Ref. [52] is provided as a Monte Carlo ensemble of \(N_{\mathrm {rep}}=200\) equally probable replicas, as both a grid with polynomial interpolation and the analytic form with the bestfit parameters for each replica. The user should specify the replica to be initialised and whether he would like to use the grid or the parameterisation via the input variables irep and imode respectively. Through imode it is also possible to select the Fourier transform of the TMD PDF, namely the distribution in transverse coordinate space (\(b_T\)distribution). For other uPDF/TMD sets, these options are not available and, if specified, they will be ignored.

Call to the distribution. The second step consists in calling the desired function. Specifically, the lightcone momentum fractions \(x^+\) and \(x^\) (often set \(x^=0\)) carried by the parton, the parton transverse momentum \(k_t\) (in GeV), the energy scale \(\mu \) (in GeV) and the flavour code identifying the target^{2} are the input variables. Returned is the momentum weighted parton distribution.

TMDutils: collection of methods used in TMDlib , including functions to get details about the initialised uPDF/TMD set (like \(\alpha _s\), \(\Lambda _{\mathrm {QCD}}\), number of flavours), see Table 4;

TMD_test: example program to handle uPDF/TMD distributions;

TMDplotter: ROOTbased script to plot uPDF/TMD distributions as obtained from TMDlib .
Available uPDF/TMD parton sets in TMDlib
Parton  uPDF/TMD set  Identifier  \(\Lambda ^{(4)}_{qcd}\)  \(k_t^{cut}\) (GeV)  \(Q_0\) (GeV)  Refs. 

Gluon  ccfmJS2001  101000  0.25  0.25  1.4  [79] 
ccfmsetA0  101010  0.25  1.3  1.3  [79]  
ccfmsetA0+  101011  0.25  1.3  1.3  [79]  
ccfmsetA0\(\)  101012  0.25  1.3  1.3  [79]  
ccfmsetA1  101013  0.25  1.3  1.3  [79]  
ccfmsetB0  101020  0.25  0.25  1.3  [79]  
ccfmsetB0+  101021  0.25  0.25  1.3  [79]  
ccfmsetB0\(\)  101022  0.25  0.25  1.3  [79]  
ccfmsetB1  101023  0.25  0.25  1.3  [79]  
ccfmJHset 1  101001  0.25  1.33  1.33  [85]  
ccfmJHset 2  101002  0.25  1.18  1.18  [85]  
ccfmJHset 3  101003  0.25  1.35  1.35  [85]  
ccfmJH2013set1  101201  0.2  2.2  2.2  [75]  
ccfmJH2013set2  101301  0.2  2.2  2.2  [75]  
GBWlight  200001  –  –  –  [80]  
GBWcharm  200002  –  –  –  [80]  
Quark  ccfmsetA0  –  0.25  1.3  1.3  
ccfmJH2013set1  –  0.2  2.2  2.2  [75]  
ccfmJH2013set2  –  0.2  2.2  2.2  [75]  
SBRS2013TMDPDFs  300001  –  –  1.55  [52] 
The function overload for TMDinit used to initialise uPDF/TMD parton sets
Method  Usage 

TMDinit(name)  To initialise the uPDF/TMD set specified by its name name. A complete list of uPDF/TMD sets available in the current version of TMDlib with the corresponding name is provided in Table 1 
TMDinit(name,irep)  To initialise a given irep replica in a Monte Carlo uPDF/TMD set specified by its name name 
TMDinit(name,irep,imode)  To initialise the uncertainty sets with irep or to initialise a given irep replica in a Monte Carlo uPDF/TMD set specified by its name name and imode: 
\(\bullet \) imode=0: the value obtained from the analytic form of the distribution is returned  
\(\bullet \) imode=1: the value obtained as a polynomial interpolation on a numerical grid is returned  
\(\bullet \) imode=2: the value obtained from the analytic form of the Fourier transform (bspace distribution) 
The function overload for TMDpdf used to call uPDF/TMD parton sets
Method  Usage 

TMDpdf(x,xbar,kt,mu, uval,dval,s,c,b,glu)  Voidtype function filling the variables uval, dval, s, c, b, glu with the values of \(xF(x,\bar{x},k_t,\mu )\) (\(F\) is the initialised uPDF/TMD) for valence uquarks uval, valence dquarks dval, light seaquarks s, charmquarks c, bottomquarks b, and gluons glu respectively for a proton target. The input variables x and xbar are the lightcome momentum fractions \(x^+\) and \(x^\) (in some parameterisations the latter is set to zero), kt is the parton transverse momentum \(k_t\), and mu is the energy scale \(\mu \) (in GeV) 
TMDpdf(kf,x,xbar,kt,mu, uval,dval,s,c,b,glu)  As the function above, but for hadron with flavour code kf (kf = 2212 for proton and kf = \(\)2212 for antiproton) 
TMDpdf(x,xbar,kt,mu)  Vector doubletype function returning an array of \(13\) variables with the values of \(xF(x,{\bar{x}},k_t,\mu )\) (\(F\) is the initialised uPDF/TMD): at index \(0,\dots ,6\) is \(\bar{t},\dots ,\bar{d}\), at index \(7\) is the gluon, and at index \(8,\dots ,13\) is \(d,\dots ,t\) densities for a proton target 
TMDpdf(kf,x,xbar,kt,mu)  As the function above, but for hadron with flavour code kf (kf = 2212 for proton and kf = \(\)2212 for antiproton) 
TMDpdf(x,xbar,kt,mu,xpq)  Voidtype function filling an array of \(13\) variables, xpq, with the values of \(xF(x,{\bar{x}},k_t,\mu )\) (\(F\) is the initialised uPDF/TMD): at index \(0,\dots ,6\) is \(\bar{t},\dots ,\bar{d}\), at index \(7\) is the gluon, and at index \(8,\dots ,13\) is \(d,\dots ,t\) densities for a proton target 
TMDpdf(kf,x,xbar,kt,mu, xpq)  As the function above, but for hadron with flavour code kf (kf = 2212 for proton and kf = \(\)2212 for antiproton) 
The list of methods included in the TMDutils.cc file
Method  Usage 

TMDalphas(mu)  Returns \(\alpha _\mathrm {s}\) used in the set initialised by TMDinit(name) 
TMDgetLam4( )  Returns the value of \(\Lambda _{QCD}\) at \(N_f=4\) 
TMDgetNf( )  Returns the number of flavours, \(N_f\), used for the computation of \(\Lambda _{QCD}\) 
TMDgetOrderAlphaS( )  Returns the perturbative order of \(\alpha _\mathrm {s}\) used in the evolution of the TMD/uPDF set initialised by TMDinit(name) 
TMDgetOrderPDF( )  Returns the perturbative order of the evolution of the TMD/uPDF set initialised by TMDinit(name) 
TMDgetXmin()  Returns the minimum value of the momentum fraction \(x\) for which the TMD/uPDF set initialised by TMDinit(name) was determined 
TMDgetXmax()  Returns the maximum value of the momentum fraction \(x\) for which the TMD/uPDF set initialised by TMDinit(name) was determined 
TMDgetQ2min()  Returns the minimum value of the energy scale \(\mu \) (in GeV) for which the TMD/uPDF set initialised by TMDinit(name) was determined 
TMDgetQ2max()  Returns the maximum value of the energy scale \(\mu \) (in GeV) for which the TMD/uPDF set initialised by TMDinit(name) was determined 
TMDnumberPDF(name)  Returns the identifier associated with the TMD/uPDF set initialised by TMDinit(name) 
5 Conclusions and feedback
The authors of this manual set up a collaboration to develop and maintain TMDlib and TMDplotter, respectively a C++ library for handling different parameterisations of uPDFs/TMDs and a corresponding online plotting tool. The redistribution of the fits has been agreed with the corresponding authors. The aim is to update these tools with more uPDF/TMD parton sets and new features, as they become available and are developed.
Footnotes
Notes
Acknowledgments
The work of A.S. is part of the program of the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). E.R.N. acknowledges the kind hospitality of the theory group at the Nationaal instituut voor subatomaire fysica (Nikhef) as part of the research activities within the ERC Advanced Grant program QWORK (Contract No. 320389). T.R. is supported by the US National Science Foundation under Grant No. PHY0969739.
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