Structure functions

Structure functions#

This documentation page lists all \(F_{2/L}(\xbj,Q^2)\) nucleon structure functions types modelled and embedded in the CepGen library. These modellings are intensively used in the photon fluxes computation, and each of these are tuned for a specific kinematics range.

All parameterisations derive from the following base class:

class Parameterisation : public NamedModule<Parameterisation>#

Subclassed by EvolutionStructureFunctions, EvolutionStructureFunctions, StructureFunctions, ALLM, BlockDurandHa, BodekKangXu, CLAS, CapellaEtAl, ChristyBosted, DonnachieLandshoff, FioreBrasse, KulaginBarinov, PartonicParameterisation, Schaefer, Shamov, SuriYennie, SzczurekUleshchenko, Grid

Detailed description

class Parameterisation : public NamedModule<Parameterisation>

Base object for the parameterisation of nucleon structure functions.

Subclassed by EvolutionStructureFunctions, EvolutionStructureFunctions, StructureFunctions, ALLM, BlockDurandHa, BodekKangXu, CLAS, CapellaEtAl, ChristyBosted, DonnachieLandshoff, FioreBrasse, KulaginBarinov, PartonicParameterisation, Schaefer, Shamov, SuriYennie, SzczurekUleshchenko, Grid

Public Functions

inline const sigrat::Parameterisation *sigmaRatio() const

Longitudinal/transverse cross section ratio parameterisation used to compute \(F_{1/L}\).

Parameterisation &operator()(double, double)

Compute all relevant structure functions for a given \((x_{\rm Bj},Q^2)\) couple.

Parameters:

  • xbj – [in] Bjorken’s x variable

  • q2 – [in] Squared 4-momentum transfer (in GeV^2)

double F2(double xbj, double q2)

Transverse structure function.

double FL(double xbj, double q2)

Longitudinal structure function.

double W1(double xbj, double q2)

Longitudinal form factor.

double FE(double xbj, double q2)

Electric proton form factor.

double FM(double xbj, double q2)

Magnetic proton form factor.

double F1(double xbj, double q2)

\(F_1\) structure function

inline const std::string &name() const

Module unique indexing name.

inline bool operator==(const SteeredObject &oth) const

Equality operator.

inline bool operator!=(const SteeredObject &oth) const

Inequality operator.

inline virtual const ParametersList &parameters() const override

Module user-defined parameters.

inline virtual void setParameters(const ParametersList &params) override

Set module parameters.

inline void setDescribedParameters(const ParametersList &params_orig)

Set (documented) module parameters.

Public Static Functions

static ParametersDescription description()

Generic description for the structure functions.

Friends

friend std::ostream &operator<<(std::ostream&, const Parameterisation&)

Human-readable dump of the SF parameterisation at this (xBj,Q^2) value.

struct Arguments
struct Values

Public Members

double f2 = {0.}

Last computed transverse structure function value.

double fl = {0.}

Last computed longitudinal structure function value.

double w1 = {0.}

Longitudinal form factor.

double fe = {0.}

Electric proton form factor.

double fm = {0.}

Magnetic proton form factor.

Note

All of these may be used and linked against any external code.

The parameterisation types handled in CepGen are listed in the cepgen::StructureFunctionsFactory.

Below, a semi-detailed review of a subset of the modellings handled in CepGen is presented.

Whenever not specified explicitely in the modelling, the \(F_L\) structure function can be computed from the \(R\) modelling-dependent relation:

\[F_L(\xbj,Q^2) = \left(1+\frac{4m_p^2\xbj^2}{Q^2}\right)\frac{R}{1+R}F_2(\xbj,Q^2).\]

Where this \(R\) ratio can be evaluated for any \((\xbj,Q^2)\) range of interest [A+99, B+12, SB13, WRB+90].

Hybrid models#

As the name suggests, this class of model combines multiple extrapolation models valid in multiple kinematic ranges into a set of uniform, continuous structure functions.

Shamov#

Note

This model is designed for soft, low-\(Q^2\) regimes under a broad range of \(x_{\rm Bj}\). Several operation modes are proposed, steered by the mode parameter:

  • SuriYennie, the standard, Suri and Yennie continuum (see below) ;

  • RealRes, using a linear grid interpolation of the real photon cross section for \(Q^2\to 0\) with resonances dependance as for \(\Delta(1232)\) ;

  • RealResAndNonRes, like the earlier, and using the Suri and Yennie non-resonant contribution ;

  • RealAndSuriYennieNonRes, using the Suri and Yennie non-resonant contribution ;

  • RealAndFitNonRes, like the RealResAndNonRes, but using a fit for the non-resonant contributions.

_images/shamov_f2.png

Kulagin-Barinov#

Note

Resonances are modelled through Breit-Wigner contributions from five states. For the DIS part, a higher twist correction is available from a global QCD fit.

_images/kulaginbarinov_f2.png _images/kulaginbarinov_fl.png

Bodek-Kang-Xu#

Note

_images/bodek_f2.png _images/bodek_fl.png

Continuum models#

Suri-Yennie#

Note

This set was used as a standard option in the LPAIR event generator. It provides a reasonable description of SLAC data in the resonance and continuum regions.

_images/suriyennie_f2.png _images/suriyennie_fl.png

Szczurek-Uleshchenko#

Note

This set puts an emphasis on the low-to-intermediate \(Q^2\) region and includes a smooth continuation to low-\(Q^2\).

Block-Durand-Ha#

Note

ALLM parameterisation#

Note

In this continuum region modelling the \(F_2\) proton structure function is parameterised as:

\[F_2(\xbj,Q^2) = \frac{Q^2}{Q^2+m_0^2}\left[F_2^{\Pom}(\xbj,Q^2)+F_2^{\Reg}(\xbj,Q^2)\right],\]

with \(m_0\) the effective photon mass. The pomeron/reggeon exchanges terms are parameterised as:

\[F_2^{\Pom,\Reg}(\xbj,Q^2) = c^{\Pom,\Reg}(t) x _ {\Pom,\Reg}^{a^{\Pom,\Reg}(t)} (1-\xbj)^{b^{\Pom,\Reg}(t)},\]

with the slowly-varying function \(t = t(Q^2)\) defined as:

\[t(Q^2) = \ln\left(\ln\frac{Q^2+Q_0^2}{\Lambda^2}\right)-\ln\left(\ln\frac{Q_0^2}{\Lambda^2}\right),\]

and the modified Bjorken-\(x\) functions:

\[x _ {\Pom,\Reg} = \left(1+\frac{w^2-m_p^2}{Q^2+m _ {\Pom,\Reg}}\right)^{-1}.\]

The six functionals \(a^{\Pom,\Reg}(t), b^{\Pom,\Reg}(t), c^{\Pom,\Reg}(t)\) are parameterised as:

\[\begin{split}a^{\Pom}(t) = a^{\Pom}_1+(a^{\Pom}_1-a^{\Pom}_2)\left[\frac{1}{1+t^{a^{\Pom}_3}}-1\right],\\ b^{\Pom}(t) = b^{\Pom}_1 + b^{\Pom}_2 t^{b^{\Pom}_3},\\ c^{\Pom}(t) = c^{\Pom}_1+(c^{\Pom}_1-c^{\Pom}_2)\left[\frac{1}{1+t^{c^{\Pom}_3}}-1\right]\end{split}\]

for the pomeron part, and

\[\begin{split}a^{\Reg}(t) = a^{\Reg}_1 + a^{\Reg}_2 t^{a^{\Reg}_3},\\ b^{\Reg}(t) = b^{\Reg}_1 + b^{\Reg}_2 t^{b^{\Reg}_3},\\ c^{\Reg}(t) = c^{\Reg}_1 + c^{\Reg}_2 t^{c^{\Reg}_3},\end{split}\]

for the reggeon subset.

Currently, four tunings of the 23 model parameters are embedded within CepGen:

Parameter

Units

ALLM91

ALLM97

GD07p

GD11p

\(m_0^2\)

GeV\(^2\)

0.30508

0.31985

0.454

0.5063

\(m _ {\Pom}^2\)

GeV\(^2\)

10.676

49.457

30.7

34.75

\(m _ {\Reg}^2\)

GeV\(^2\)

0.20623

0.15052

0.117

0.03190

\(Q_0^2\)

GeV\(^2\)

0.27799

0.52544

1.15

1.374

\(\Lambda_0^2\)

GeV\(^2\)

0.06527

0.06527

0.06527

0.06527

\(a^{\Pom}_1\)

-

-0.04503

-0.0808

-0.105

-0.11895

\(a^{\Pom}_2\)

-

-0.36407

-0.44812

-0.495

-0.4783

\(a^{\Pom}_3\)

-

8.17091

1.1709

1.29

1.353

\(b^{\Pom}_1\)

-

0.49222

0.36292

-1.42

1.0833

\(b^{\Pom}_2\)

-

0.52116

1.8917

4.51

2.656

\(b^{\Pom}_3\)

-

3.5515

1.8439

0.551

1.771

\(c^{\Pom}_1\)

-

0.26550

0.28067

0.339

0.3638

\(c^{\Pom}_2\)

-

0.04856

0.22291

0.127

0.1211

\(c^{\Pom}_3\)

-

1.04682

2.1979

1.16

1.166

\(a^{\Reg}_1\)

-

0.60408

0.584

0.374

0.3425

\(a^{\Reg}_2\)

-

0.17353

0.37888

0.998

1.0603

\(a^{\Reg}_3\)

-

1.61812

2.6063

0.775

0.5164

\(b^{\Reg}_1\)

-

1.26066

0.01147

2.71

-10.408

\(b^{\Reg}_2\)

-

1.83624

3.7582

1.83

14.857

\(b^{\Reg}_3\)

-

0.81141

0.49338

1.26

0.07739

\(c^{\Reg}_1\)

-

0.67639

0.80107

0.838

1.3633

\(c^{\Reg}_2\)

-

0.49027

0.97307

2.36

2.256

\(c^{\Reg}_3\)

-

2.66275

3.4942

1.77

2.209

The ALLM91 tuning is fitted from all pre-HERA data points available.

_images/allm91_f2.png _images/allm91_fl.png _images/allm97_f2.png _images/allm97_fl.png _images/gd07p_f2.png _images/gd07p_fl.png _images/gd11p_f2.png _images/gd11p_fl.png

Resonance models#

Fiore-Brasse#

Note

This parameterisation gives a very good description of photoabsorption in the resonance region from low to large \(Q^2\). It is designed to reproduce well JLAB and SLAC data.

_images/fiorebrasse_f2.png _images/fiorebrasse_fl.png

Christy-Bosted#

Note

The set developed by M.E. Christy and P.E. Bosted is emphasised on the very-low \(Q^2\) regime, with its particular use of JLAB’s Hall-C data on:

  • inclusive inelastic (up to \(Q^2simeq\) 7.5 GeV²),

  • photoproduction at \(Q^2\) = 0, and

  • DIS data at high-\((Q^2,W)\).

_images/christybosted_f2.png _images/christybosted_fl.png

CLAS#

Note

Perturbative models#

MSTW grid#

class Grid : public Parameterisation, private GridHandler<2, 2>#

External interfaces#

Several other models can also be interfaced through a base partonic structure functions interface allowing the conversion of PDFs into \(F_2\)/\(F_L\) structure functions. This object has the form:

class PartonicParameterisation : public Parameterisation#

Subclassed by PartonicStructureFunctions, LHAPDFPartonic

The conversion of quark/gluon PDF content into \(F_2\) structure function is computed as follows:

\[\begin{split}F_2^{\rm val}(\xbj,Q^2) = \sum_{i=1}^{n_q} e_i^2 \left[q_i(\xbj,Q^2)-\bar q_i(\xbj,Q^2)\right]\\ F_2^{\rm sea}(\xbj,Q^2) = 2 \sum_{i=1}^{n_q} e_i^2 \bar q_i(\xbj,Q^2)\\ F_2^{\rm tot}(\xbj,Q^2) = F_2^{\rm val}(\xbj,Q^2)+F_2^{\rm sea}(\xbj,Q^2)\end{split}\]

LHAPDF interface#

Note

  • Legacy code: 401 (“standard” parameterisation), or a more complex scheme: : The legacy-equivalent signature follows the convention 1MSSSSSS, where:

    • M specifies the set of partons included in the sum rule: : - 0: all partons,

      • 1: valence quarks only, and

      • 2: sea quarks only.

    • SSSSSS is the integer LHAPDF ID code for the selected PDF set.

  • Structure function modelled: \(F_2\)

  • Reference: [WBG05]

  • Implementation: cepgen::strfun::LHAPDFPartonic

  • Module parameters

APFEL++ interface#

Note

This interface to the APFEL++ C++ rewriting of the famous APFEL library covers the computation of order-0/1/2/3 perturbative \(F_{2,L}\) under several assumptions/modellings. In particular, two DIS processes are currently handled for the building of interpolation grids: charged currents and neutral currents.

Contents

By Laurent Forthomme

© Copyright 2013-2024, the CepGen Collaboration.