Number Theory in Quantum Physics: Minicharged Particles and the Prouhet-Tarry-Escott Problem

Abstract

In quantum gauge theories, anomaly cancellation severely restricts the allowed patterns of chiral charges. Here we show that, in a phenomenologically motivated framework for light minicharged particles, the anomaly cancellation conditions are equivalent to the degree $k=3$ Prouhet-Tarry-Escott problem in number theory. This correspondence immediately implies that the hidden sector must contain at least four minicharged states. For constructions based on minimal ideal solutions, the mass spectrum generically exhibits a near-degenerate doublet structure, so that the discovery of one minicharged particle would point to a partner state with the same minicharge and a nearby mass. Our results uncover an unexpected link between quantum consistency and number theory, with direct implications for model building and future searches.

Figures (6)

• Figure 1: Distribution of exponents in the mass hierarchy evaluated by solving a minimum-weight matching problem. The frequencies are normalized to unity. Bar colors distinguish solutions with a two-doublet spectrum, $e_1=e_2$ and $e_3=e_4$, which correspond to symmetric PTE solutions, from the non-symmetric solutions.
• Figure 2: Mass splitting between the two lightest eigenstates for degree $k=3$ PTE solutions with $n=4$. The grey and blue histograms show the distribution of $m_2/m_1$ with $m_2>m_1$ for coefficient magnitudes sampled in $[0,1]$ and $[0.1,1]$, respectively. The red dashed line shows the estimate based on the diagonal approximation for coefficients with magnitudes in $[0.1,1]$.
• Figure 3: Mass eigenvalue spectrum obtained by numerical singular value decomposition of the mass matrix. Random $\mathcal{O}(1)$ coefficients $y_{ij}$ are generated for each realization and we set $\epsilon = 10^{-3}$. The result is averaged over 400 realizations for each charge assignment. The bar color distinguishes solutions with a light-doublet spectrum, $e_1=e_2$. We confirmed that the result approaches the expectation from the minimum-weight matching problem as $\epsilon$ becomes smaller.
• Figure 4: The same as Fig. \ref{['fig: exponent_ideal']} but for $n=5$. The bar color distinguishes solutions with a light-doublet spectrum, $e_1=e_2$. Only about $24\%$ of the solutions exhibit a lightest doublet, which is much smaller than the case of $n=4$.
• Figure 5: The same as Fig. \ref{['fig: exponent_ideal']} but for $n=6$. The bar color distinguishes solutions with a light-doublet spectrum, $e_1=e_2$. About $64\%$ of the solutions exhibit a lightest doublet.