Abstract
What values of the Standard Model hypercharges result in a mathematically consistent quantum field theory? We show that the constraints imposed by the lack of gauge anomalies can be recast as the equation x^3 + y^3 = z^3x3+y3=z3. If hypercharge is quantised, then xx, yy and zz must be integers. The trivial (and only) solutions, with x=0x=0 or y=0y=0, reproduce the hypercharge assignments seen in Nature. This argument does not rely on the mixed gauge-gravitational anomaly, which is automatically vanishing if hypercharge is quantised and the gauge anomalies vanish.
Highlights
The delicate cancellation of gauge and mixed gauge-gravitational anomalies reveals the Standard Model to be a wonderfully elegant jigsaw, each piece interlocking perfectly with the others [1,2,3]
The third is a cubic equation arising from the Abelian triangle anomaly, 6q3 + 2l3 − 3u3 − 3d3 − x3 = 0
The first consistent 4d chiral gauge theory was constructed by Ramanujan from his hospital bed in Putney and suffers a mixed gaugegravitational anomaly 1
Summary
The delicate cancellation of gauge and mixed gauge-gravitational anomalies reveals the Standard Model to be a wonderfully elegant jigsaw, each piece interlocking perfectly with the others [1,2,3]. Are there other assignments of hypercharge that would result in a consistent theory? We could take the gauge group of the Standard Model to be, G = × SU(2) × SU(3) . The resulting quantum field theory is consistent only if the hypercharges {q, l, u, d, x}, each of which is a real number, are constrained to obey three anomaly conditions.
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