where ( |G\rangle ) is the glueball, ( |N\rangle = u\baru+d\bard ) and ( |S\rangle = s\bars ). The "mod" term appears when one imposes on the effective Lagrangian—specifically, requiring that the mixing angles be periodic under shifts of 1710 in a certain scalar potential.
More concretely, the has a principal congruence subgroup (\Gamma(19)) whose index is 1710. That is:
[ [\textPSL(2,\mathbbZ) : \Gamma(19)] = 1710 ]
[ M_i = 1710 \ \textMeV \times (1 + k_i \mod 3) ]
Introduction In the vast landscape of theoretical physics, few bridges are as tantalizing—and as technically challenging—as those connecting the discrete world of quarks to the elegant realm of number theory. The search term "quark mod 1710" might appear cryptic at first glance. Is it a new particle? A computational model? A resonance in a scattering matrix?