[Paper Review] Parton Distribution Functions properties of the statistical model

C. Bourrely|arXiv (Cornell University)|Jul 14, 2015

Particle physics theoretical and experimental studies12 references3 citations

TL;DR

This paper proposes a statistical model for parton distribution functions (PDFs) that derives their properties from thermodynamical potentials and entropy maximization, providing a physical basis for PDF parameters. By modeling PDFs as Fermi-Dirac or Bose-Einstein distributions with helicity-dependent potentials and introducing neural network-inspired activation functions, the model explains the sign and magnitude of polarized PDFs naturally, with fits to experimental data achieving χ²/d.o.f. = 1.18 and confirming maximum entropy for quark states and structure functions.

ABSTRACT

We show that the parton distribution functions (PDF) described by the statistical model have very interesting physical properties which help to understand the structure of partons. The role of the quark helicity components is emphasized as they represent the building blocks of the PDF. In the model the sign of the polarized quarks PDF comes out in a quite natural way once the thermodynamical potentials with a given helicity are known. Introducing the concept of entropy we study the states madeof |2u + d >, |u +d +s > and $|2\bar u +\bar d >$, for a fixed Q^2, the variation with x shows that the first state has a dominant entropy due to the effect of u quark. We prove that the PDF parameters obtained from experiments give in fact an optimal solution of an entropy equation subject to constraints. We develop a new approach of the polarized gluon density based on a neural model which explains its property, in particular, a large positivity value and an agreement with the positvity constraint. An extension of this neural approach is applied to quarks giving a coherent description of the partons structure.

Motivation & Objective

• To provide a physical explanation for the parameters of unpolarized and polarized parton distribution functions (PDFs), moving beyond purely numerical fits.
• To unify the treatment of unpolarized and polarized PDFs by deriving them from a common statistical framework based on thermodynamical potentials.
• To demonstrate that experimental PDF parameters correspond to a maximum entropy state under physical constraints, linking phenomenology to statistical mechanics.
• To develop a neural network-inspired model for the polarized gluon density to explain its large positivity and agreement with positivity constraints.
• To extend the neural model to quarks, creating a coherent, unified description of parton structure across all PDF types.

Proposed method

• Constructs polarized quark PDFs using helicity-dependent Fermi-Dirac distributions with thermodynamical potentials $X^{ ho}$ and a universal temperature $ar{x}$.
• Introduces a neural network-inspired activation function $S_i(x) = \frac{1}{1 + e^{-e_i x + h_i}}$ to modulate PDF behavior, particularly suppressing small-$x$ contributions.
• Fits the model to global unpolarized and polarized deep-inelastic scattering data at NLO, achieving $\chi^2/\text{d.o.f.} = 1.18$.
• Derives the entropy of quark states such as $|2u+d\rangle$ and $|u+d+s\rangle$, showing that the fitted PDF parameters maximize entropy under constraints.
• Applies the same maximum entropy principle to the structure functions $F^2_p$ and $g^1_p$, proving their optimal nature.
• Models the polarized gluon density $x\Delta G(x)$ using a neural activation function, explaining its large positivity and physical consistency.

Experimental results

Research questions

• RQ1 Can the parameters of parton distribution functions be understood as arising from a maximum entropy principle in a statistical model?
• RQ2 How do thermodynamical potentials $X^{\pm}$ for quarks determine the sign and magnitude of polarized PDFs in a physically consistent way?
• RQ3 Can a neural network-inspired activation function provide a more physically motivated description of the polarized gluon density compared to standard parametrizations?
• RQ4 Does the same maximum entropy principle apply not only to quark states but also to the observed structure functions $F^2_p$ and $g^1_p$?
• RQ5 Can a unified neural model framework be extended from gluons to quarks to yield a coherent, physically grounded description of all parton distributions?

Key findings

• The fitted PDF parameters correspond to a maximum entropy state for the quark states $|2u+d\rangle$ and $|u+d+s\rangle$, with the $u$ quark dominating entropy due to its higher potential $X_u^+ = 0.540 \pm 0.0014$.
• The model achieves a good fit to experimental data with $\chi^2/\text{d.o.f.} = 2506/2128 = 1.18$, indicating high consistency with measurements.
• The sign of the polarized quark PDFs emerges naturally from the helicity-dependent thermodynamical potentials, with $X_u^+ > X_u^-$ ensuring a positive $\Delta u$.
• The neural model for the polarized gluon density explains its large positivity and satisfies the positivity constraint, removing arbitrariness in parametrization.
• The activation function parameters are determined via global fit: $e_u = 27.16 \pm 1.3$, $h_u = 0.7$ (fixed), with $S_q(x)$ showing a hierarchy of responses matching observed PDF magnitudes.
• Extending the neural model to quarks yields a coherent, unified description of both unpolarized and polarized PDFs, with the $u$ and $d$ quarks showing the strongest response due to their high $e_i$ values.