(strfun)=

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:

:outline:

Detailed description

```{doxygenclass} cepgen::strfun::Parameterisation
:members:
:no-link:
```

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

The parameterisation types handled in CepGen are listed in the {code}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 {cite}Abe:1998ym,Beringer:1900zz,Sibirtsev:2013cga,Whitlow:1990gk.

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)=

Shamov

- Legacy code: `302`
- Structure functions modelled: \$W_1\$, \$W_2\$, \$F_2\$
- Implementation: {cepgen}`cepgen::strfun::Shamov`
- [Module parameters](/raw-modules.md#strfunShamov)

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.

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(kulaginbarinov)=

Kulagin-Barinov

- Legacy code: `303`
- Structure functions modelled: \$F_2\$, \$F_L\$
- Reference: {cite}`Kulagin:2021mee`
- Implementation: {cepgen}`cepgen::strfun::KulaginBarinov`
- [Module parameters](/raw-modules.md#strfunKulaginBarinov)

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.

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(luxlike)=

Bodek-Kang-Xu

- Legacy code: `304`
- Structure functions modelled: \$F_1\$, \$F_2\$
- Reference: {cite}`Bodek:2021bde`
- Implementation: {cepgen}`cepgen::strfun::BodekKangXu`
- [Module parameters](/raw-modules.md#strfunBodekKangXu)

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Continuum models

(suriyennie)=

Suri-Yennie

- Legacy code: `11` (and `12` for the alternative parameterisation)
- Structure functions modelled: \$F_E\$, \$F_M\$
- Reference: {cite}`Suri:1971yx`
- Implementations: {cepgen}`cepgen::strfun::SuriYennie` and {cepgen}`cepgen::strfun::SuriYennieAlt`
- Module parameters: [main](/raw-modules.md#strfunSuriYennie) and [alternative](/raw-modules.md#strfunSuriYennieAlt) parameterisations

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.

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(szczurekuleshchenko)=

Szczurek-Uleshchenko

- Legacy code: `12`
- Structure function modelled: \$F_2\$
- Reference: {cite}`Szczurek:1999wp`
- Implementation: {cepgen}`cepgen::strfun::SzczurekUleshchenko`, relying on the GRV Fortran interpolation subroutine
- [Module parameters](/raw-modules.md#strfunSzczurekUleshchenko)

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

(bdh)=

Block-Durand-Ha

- Legacy code: `13`
- Structure function modelled: \$F_2\$
- Reference: {cite}`Block:2014kza`
- Implementation: {cepgen}`cepgen::strfun::BlockDurandHa`
- [Module parameters](/raw-modules.md#strfunBlockDurandHa)

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

ALLM parameterisation

- Structure function modelled: \$F_2\$
- References:
  > A full reference of this parameterisation by *Abramowicz et al.* can be found in {cite}`Abramowicz:1991xz` (`ALLM91`) and {cite}`Abramowicz:1997ms` (`ALLM97`).
  > The HERMES Collaboration refits of this modelling, labelled `GD07p` and `GD11p` may be found in {cite}`Airapetian:2011nu`.
- Parameterisations:
  - ALLM91
    - Legacy code: `201`
    - Implementation: {cepgen}`cepgen::strfun::ALLM91`
    - [Module parameters](/raw-modules.md#strfunALLM91)
  - ALLM97
    - Legacy code: `202`
    - Implementation: {cepgen}`cepgen::strfun::ALLM97`
    - [Module parameters](/raw-modules.md#strfunALLM97)
  - GD07p
    - Legacy code: `203`
    - Implementation: {cepgen}`cepgen::strfun::GD07p`
    - [Module parameters](/raw-modules.md#strfunGD07p)
  - GD11p
    - Legacy code: `204`
    - Implementation: {cepgen}`cepgen::strfun::GD11p`
    - [Module parameters](/raw-modules.md#strfunGD11p)
  - HHTALLM
    - Legacy code: `206`
    - Implementation: {cepgen}`cepgen::strfun::HHTALLM`
    - [Module parameters](/raw-modules.md#strfunhhtALLM)
  - HHTALLMFT
    - Legacy code: `207`
    - Implementation: {cepgen}`cepgen::strfun::HHTALLMFT`
    - [Module parameters](/raw-modules.md#strfunhhtALLMft)

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:

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]

for the pomeron part, and

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},

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.

(allm91)=

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(allm97)=

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(gd07p)=

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(gd11p)=

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Resonance models

(fiorebrasse)=

Fiore-Brasse

- Legacy code: `101` (core), and `104` (alternative)
- Structure function modelled: \$F_2\$
- References: {cite}`Fiore:2002re,Brasse:1976bf`
- Implementation: {cepgen}`cepgen::strfun::FioreBrasse`, and {cepgen}`cepgen::strfun::FioreBrasseAlt`
- Modules parameters: [core](/raw-modules.md#strfunFioreBrasse) and [alternative](/raw-modules.md#strfunFioreBrasseAlt) parameterisations

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.

:width: 48%

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(christybosted)=

Christy-Bosted

- Legacy code: `102`
- Structure functions modelled: \$F_2\$, \$F_L\$
- Reference: {cite}`Bosted:2007xd`
- Implementation: {cepgen}`cepgen::strfun::ChristyBosted`
- [Module parameters](/raw-modules.md#strfunChristyBosted)

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-$(Q^2,W)$.

:width: 48%

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CLAS

- Legacy code: `103`
- Structure functions modelled: \$F_2\$
- Reference: {cite}`Osipenko:2003bu`
- Implementation: {cepgen}`cepgen::strfun::CLAS`
- [Module parameters](/raw-modules.md#strfunCLAS)

Perturbative models

MSTW grid

:outline:

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:

:outline:

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

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)

LHAPDF interface

- 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: {cite}`Whalley:2005nh`
- Implementation: {cepgen}`cepgen::strfun::LHAPDFPartonic`
- [Module parameters](/raw-modules.md#strfunlhapdf)

APFEL++ interface

- Legacy code: `405`
- Structure function modelled: \$F_2\$, \$F_L\$
- Reference: {cite}`Bertone:2017gds`
- Implementation: {cepgen}`cepgen::apfelpp::EvolutionStructureFunctions`
- [Module parameters](/raw-modules.md#strfunapfelppEvol)

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.

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structure-functions.md

structure-functions.md

Structure functions

Hybrid models

Shamov

Kulagin-Barinov

Bodek-Kang-Xu

Continuum models

Suri-Yennie

Szczurek-Uleshchenko

Block-Durand-Ha

ALLM parameterisation

Resonance models

Fiore-Brasse

Christy-Bosted

CLAS

Perturbative models

MSTW grid

External interfaces

LHAPDF interface

APFEL++ interface

Files

structure-functions.md

Latest commit

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structure-functions.md

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Structure functions

Hybrid models

Shamov

Kulagin-Barinov

Bodek-Kang-Xu

Continuum models

Suri-Yennie

Szczurek-Uleshchenko

Block-Durand-Ha

ALLM parameterisation

Resonance models

Fiore-Brasse

Christy-Bosted

CLAS

Perturbative models

MSTW grid

External interfaces

LHAPDF interface

APFEL++ interface