An overview of scale invariance in proton structure with holographic insights

Abstract

The concept of self-similarity in the internal structure of the proton, rooted in scale invariance and fractal geometry, provides an intriguing framework for understanding the behaviour of parton distribution functions (PDFs), particularly in the small \textit{x} region probed in deep inelastic scattering (DIS). Phenomenological models based on self-similarity have been shown to reproduce key features of experimental data, suggesting that recursive scaling patterns may play an important role in partonic dynamics. In this work, we present an overview of scale-invariant descriptions of proton structure, focusing on self-similar models developed in earlier studies and their phenomenological implications for structure functions and parton distributions. We then explore possible conceptual connections between these fractal-inspired descriptions and modern holographic approaches to QCD, particularly within the framework of light-front holographic QCD. By comparing the scaling behaviour appearing in phenomenological models with the geometric structure underlying holographic QCD, we highlight qualitative correspondences that suggest a broader role of scale invariance in proton structure. Although the connection remains interpretive rather than derivational, it offers a complementary perspective of how fractal-like scaling observed in DIS may relate to geometric scaling in holographic descriptions of QCD.