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>

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.

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

• Legacy code: 302

• Structure functions modelled: \(W_1\), \(W_2\), \(F_2\)

  class Shamov : public Parameterisation
  

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.

Kulagin-Barinov

Note

• Legacy code: 303

• Structure functions modelled: \(F_2\), \(F_L\)

• Reference: [KB22]

  class KulaginBarinov : public Parameterisation
  

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.

Bodek-Kang-Xu

Note

• Legacy code: 304

• Structure functions modelled: \(F_1\), \(F_2\)

• Reference: [BYX21]

  class BodekKangXu : public Parameterisation
  

Continuum models

Suri-Yennie

Note

• Legacy code: 11

• Structure functions modelled: \(F_E\), \(F_M\)

• Reference: [SY72]

  class SuriYennie : public Parameterisation
  

  Subclassed by SuriYennieAlt

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.

Szczurek-Uleshchenko

Note

• Legacy code: 12

• Structure function modelled: \(F_2\)

• Reference: [SU00]

  class SzczurekUleshchenko : public Parameterisation
  

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

• Legacy code: 13

• Structure function modelled: \(F_2\)

• Reference: [BDH14]

  class BlockDurandHa : public Parameterisation
  

ALLM parameterisation

Note

• Legacy codes: 201 (ALLM91), 202 (ALLM97), 203 (GD07p), 204 (GD11p)

• Structure function modelled: \(F_2\) [1]

• References:

  A full reference of this parameterisation by Abramowicz et al. can be found in [ALLM91] (ALLM91) and [AL97] (ALLM97). The HERMES Collaboration refits of this modelling, labelled GD07p and GD11p may be found in [A+11].

  class ALLM : public Parameterisation
  

  Subclassed by ALLM91, ALLM97, GD07p, GD11p, HHTALLM, HHTALLMFT

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.

Resonance models

Fiore-Brasse

Note

• Legacy code: 101

• Structure function modelled: \(F_2\)

• References: [BFG+76, FFJ+02]

  class FioreBrasse : public Parameterisation
  

  Subclassed by FioreBrasseAlt

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.

Christy-Bosted

Note

• Legacy code: 102

• Structure functions modelled: \(F_2\), \(F_L\)

• Reference: [BC08]

  class ChristyBosted : public Parameterisation
  

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-\((Q^2,W)\).

CLAS

Note

• Legacy code: 103

• Structure functions modelled: \(F_2\)

• Reference: [O+03]

  class CLAS : public Parameterisation
  

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

Among these is the LHAPDF interface:

Note

• Legacy code: : 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: \(F_2\) [1]

• Reference: [WBG05]

  class LHAPDFPartonic : public PartonicParameterisation
  

If linked to LHAPDF, CepGen may build the \(F_2\) structure function from quark/gluon PDF content.

It is computed as follows:

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

---

[1] (1,2)

This (set of) structure function(s) is/are only defining \(F_2\) and use 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 ratio can be evaluated for any \((\xbj,Q^2)\) range of interest [A+99, B+12, SB13, WRB+90].