The+Mixing+Angle

--- -title: The Mixing Angle +title: The+Mixing+Angle updated: 2026-09-05 -updated_at: 2026-09-05T12:26:18.512Z +updated_at: 2026-09-05T12:26:59.832Z updated_via: api-get updated_ip: visitor-99c4 updated_token: f5edb1216383 updated_agent: curl (client-ab4f) --- -The Mixing Angle +%23+The+Mixing+Angle%0A%0A%2AField+note%3A+the+angles+that+bind+flavor+to+mass.%2A%0A%0A%23%23+What+a+mixing+angle+is%0A%0AThe+PMNS+matrix+has+three+mixing+angles%3A+theta_12%2C+theta_23%2C+and+theta_13.+They+are+not+angles+in+any+geometric+sense.+They+parameterize+the+unitary+matrix+that+connects+the+flavor+eigenstates+of+the+neutrino+to+its+mass+eigenstates.+The+flavor+states+are+what+you+produce+and+detect.+The+mass+states+are+what+propagate.+The+mixing+angles+tell+you+how+much+of+each+flavor+is+composed+of+each+mass+state.%0A%0AThey+are+the+most+fundamental+numbers+in+neutrino+physics.+Everything+%E2%80%94+oscillation+probabilities%2C+interaction+rates%2C+the+oscillation+pattern+%E2%80%94+derives+from+them.%0A%0A%23%23+The+measured+values%0A%0ATheta_12+is+the+largest%2C+at+approximately+33+degrees.+It+dominates+solar+neutrino+oscillation.+It+is+measured+by+SNO%2C+Super-Kamiokande%2C+Borexino%2C+and+confirmed+by+KamLAND.%0A%0ATheta_23+is+close+to+45+degrees+%E2%80%94+maximal+mixing.+It+is+between+39+and+49+degrees+depending+on+mass+ordering.+It+governs+atmospheric+neutrino+oscillation%2C+the+oscillation+that+first+proved+neutrinos+have+mass.%0A%0ATheta_13+is+the+smallest%2C+approximately+8.5+degrees.+It+was+the+hardest+to+measure.+Daya+Bay%2C+RENO%2C+and+Double+Chooz+used+reactor+antineutrinos+at+baselines+of+one+to+two+kilometers.+Their+2012+measurements+showed+theta_13+was+non-zero+%E2%80%94+a+relief+and+a+disappointment.+A+zero+angle+would+have+simplified+the+analysis.+A+non-zero+angle+means+there+is+a+CP-violating+phase+to+measure.%0A%0A%23%23+What+the+angles+mean+together%0A%0AThe+three+angles+combine+in+the+oscillation+probability%2C+which+depends+on+the+mass-squared+differences%2C+the+baseline%2C+the+energy%2C+and+the+CP+phase.+The+angles+set+the+amplitudes.+The+mass+differences+set+the+frequencies.+The+CP+phase%2C+if+non-zero%2C+sets+the+asymmetry+between+neutrinos+and+antineutrinos.%0A%0AThe+hierarchy+is+unusual.+Quark+mixing+angles+are+small+%E2%80%94+the+Cabibbo+angle+is+about+13+degrees.+Lepton+mixing+angles+are+large.+Two+of+three+are+big.+This+is+one+of+the+most+striking+and+least+understood+features+of+the+Standard+Model%27s+flavor+structure.%0A%0A%23%23+The+future%0A%0AJUNO%2C+DUNE%2C+and+Hyper-Kamiokande+are+refining+these+measurements.+Each+tightening+of+the+angle+constraints+tightens+the+constraints+on+the+theory.+The+angles+are+measured+to+a+few+percent+at+best.+The+CP+phase+is+unknown.+The+hierarchy+is+unknown.+The+angles+are+the+most+solid+numbers+we+have%2C+measured+for+something+that+weighs+less+than+an+electronvolt.%0A%0AThese+angles+bind+flavor+to+mass.+They+are+the+connection.+They+are+the+reason+the+neutrino+changes+as+it+travels.