Solving the Hierarchy Problem without Supersymmetry or Extra Dimensions: An Alternative Approach

TL;DR

The paper proposes solving the gauge hierarchy and cosmological constant problems without supersymmetry or extra dimensions by exploiting string finiteness through misaligned supersymmetry and modular invariance. It develops a toy model with an infinite boson/fermion spectrum that cancels low-order mass supertraces and can yield $\Lambda=0$ at one loop, then connects these ideas to actual non-supersymmetric string constructions where modular invariance enforces cancellations across the full spectrum. The key idea is an 'all-scales' conspiracy among states at every mass level, regulated by modular symmetry, rather than integrating out heavy states. If realized in a stable non-supersymmetric string, this approach could embed the Standard Model within a finite, modular-invariant framework and relate hierarchy stabilization to vacuum stability, with potential implications for brane-world scenarios and beyond, though many open issues—especially higher-loop and open-string extensions—remain to be settled.

Abstract

In this paper, we propose a possible new approach towards solving the gauge hierarchy problem without supersymmetry and without extra spacetime dimensions. This approach relies on the finiteness of string theory and the conjectured stability of certain non-supersymmetric string vacua. One crucial ingredient in this approach is the idea of ``misaligned supersymmetry'', which explains how string theories may be finite even without exhibiting spacetime supersymmetry. This approach towards solving the gauge hierarchy problem is therefore complementary to recent proposals involving both large and small extra spacetime dimensions. This approach may also give a new perspective towards simultaneously solving the cosmological constant problem.

Figures (3)

• Figure 1: A sketch of a typical boson/fermion configuration. In this sketch, we have assumed a bosonic (fermionic) sector at even (odd) mass levels $n$, and plotted a typical configuration of degeneracies $g_n$ (black circles) versus $n$. The dashed lines connect these points in order of increasing mass $M_n=\sqrt{n}\mu$, and illustrate the regular bosonic/fermionic oscillations inherent in "misaligned supersymmetry". The cancellation of the supertraces arises as a result of the cancellation of the functional forms$\Phi(n)$ that separately govern the behavior of $g_n$ as a function of $n$ in each sector. Ordinary supersymmetry emerges as a special case when the bosonic and fermionic sectors have values of $n$ that coincide (no misalignment). We illustrate this for even values of $n$, with degeneracies (black circles) in the bosonic sector cancelling pairwise against degeneracies (white circles) in the fermionic sector. However, as the fermionic sector is shifted ("misaligned") by $n\to n+\Delta n$ relative to the bosonic sector, the white circles slide along the functional form $-\Phi(n)$ to their new locations $-\Phi(n+\Delta n)$ at which pairwise cancellations of states are no longer possible. Although supersymmetry is broken, finiteness is nevertheless maintained due to the cancellation of the functional forms, and the mass supertraces continue to vanish.
• Figure 2: Degeneracies $g_n$ as functions of $n$ in four-dimensional non-supersymmetric tachyon-free heterotic string models with gauge groups $SU_6 \times (SU_4)^3$ (top row) and $E_6\times SO_{10}$ (bottom row). In these figures we plot $\pm \log_{10}(|g_n|)$ where the sign chosen is the sign of $g_n$. In all cases, modular invariance causes cancellations to occur, leading to a misaligned supersymmetry and preserving finiteness.
• Figure 3: Two approaches to string phenomenology. In the traditional approach [paths (a) and (b)], the Standard Model is realized only after supersymmetry is broken in an effective theory ( e.g., the MSSM) which is itself derived as the low-energy limit of a supersymmetric string. In the alternative approach [paths (c) and (d)], supersymmetry is broken first in a manner that preserves the full string symmetries that underlie string finiteness. The light degrees of such a string theory would then constitute the Standard Model directly, and the gauge hierarchy would automatically be stabilized through a misaligned supersymmetry.