The hierarchy problem in particle physics and gravity
Ismael Ahmed
The Standard Model, developed through the combination of electroweak theory and quantum chromodynamics, successfully achieves a unified theory for modern particle physics. 1 It provides a framework explaining the interactions between subatomic particles to extremely high precision, as evidenced by experimental results at the Large Hadron Collider. 1 However, the Standard Model is incomplete, confined to only three of the four fundamental forces; it describes the weak, electromagnetic and strong nuclear forces, yet it fails to incorporate gravity on this quantum scale. 2 This limitation underlines the problem that will be discussed throughout this essay, as this lack of explanation promotes several theories on why the force of gravity is so weak in comparison to the other fundamental forces. It introduces us to the root of the hierarchy problem, which involves the low, non- zero value of the Higgs field and Higgs boson mass, and their role in the relative strengths of the fundamental forces. 3 According to Einstein’s theory of general relativity, along with the Standard Model, at the subatomic scale, gravity is several orders of magnitude weaker than its closest force – the weak force – by approximately 10 27 times by a certain measure. 4 The reasoning behind its lack of visibility on larger scales is solely due to the non-existence of negative mass, as opposed to the existence of negative charge, which results in almost perfectly zero electric fields on larger scales, unlike gravity which maintains an inverse square relation to displacement from a body. 5 However, this effect can be ignored at quantum levels, and so it is at these scales that the vast difference in the magnitudes of these forces can be highlighted, and arrives in conjunction with the underlying problem explored throughout this essay: the inability of current quantum theories to account for gravity in the same framework as the other forces. To further contextualize this problem, we can explore the Planck mass: calculated from three of the four universal constants, the Planck mass (approximately 10 -8 kilograms) represents the theoretical mass of the smallest possible black hole at which the gravitational effects are strong enough for this phenomenon to occur. 6 Its calculation involves the gravitational constant as a divisor, and so the Planck mass is inversely proportional to the magnitude of the gravitational constant. 7 Notably, this mass is roughly 10^15 times greater than the mass of the W and Z bosons, which are the force mediators for the weak interaction. 8 A greater value of the gravitational constant would imply a stronger
1 Sutton, Standard Model . 2 CERN, The Standard Model . 3 Atlas 2023a . 4 Sutter 2022. 5 Baird 2013. 6 Tinelmis 2024b. 7 Ibid. 8 CERN, The Standard Model .
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