The CKM matrix is nearly diagonal (small mixing angles), while the PMNS matrix has two large mixing angles (theta_12 ~ 34 degrees, theta_23 ~ 49 degrees). Only theta_13 ~ 8.5 degrees is small. Why is this difference significant?
ABecause it means the PMNS matrix is more accurately measured
BBecause the dramatically different mixing patterns in the quark and lepton sectors suggest that the mechanism generating neutrino masses may be fundamentally different from the Higgs-Yukawa mechanism that generates quark masses — models like the seesaw mechanism or discrete flavor symmetries attempt to explain the large leptonic mixing angles
CBecause large mixing angles make neutrino experiments easier
DBecause the PMNS matrix has more free parameters than the CKM matrix
In the quark sector, the mass hierarchy is steep (m_t/m_u ~ 10^5) and the mixing angles are small (Cabibbo angle ~13 degrees, others smaller). In the lepton sector, the mass-squared ratios are moderate (Delta m^2_{32}/Delta m^2_{21} ~ 30) and two mixing angles are large. This pattern is not explained by the Standard Model. Various models predict specific patterns: tri-bimaximal mixing (theta_12 = arctan(1/sqrt(2)), theta_23 = 45 degrees, theta_13 = 0) was a popular Ansatz that predicted large angles from discrete symmetries, but the observation of nonzero theta_13 ruled out the exact pattern while keeping the general idea alive.
Question 2 Short Answer
The PMNS matrix has one Dirac CP phase delta_CP, analogous to the CKM phase, plus two additional Majorana phases alpha_1 and alpha_2 that exist only if neutrinos are Majorana particles. Why don't the Majorana phases affect neutrino oscillation experiments?
Think about your answer, then reveal below.
Model answer: The oscillation probability involves the product U_{alpha i}* U_{beta i} U_{alpha j} U_{beta j}*, where U is the PMNS matrix. The Majorana phases appear in the matrix as diagonal phase factors: U -> U * diag(1, e^{i*alpha_1/2}, e^{i*alpha_2/2}). In the product U_{alpha i}* U_{beta i}, the Majorana phase from column i cancels between the U* and U factors. Therefore, oscillation probabilities are independent of the Majorana phases. These phases are physical and affect lepton-number-violating processes like neutrinoless double beta decay, where the amplitude is proportional to m_{ee} = |sum_i U_{ei}^2 m_i|, and the U_{ei}^2 (not |U_{ei}|^2) retains the Majorana phases.
This is why determining whether neutrinos are Dirac or Majorana requires experiments beyond oscillation, such as neutrinoless double beta decay (0nu-beta-beta). The observation of 0nu-beta-beta would simultaneously prove neutrinos are Majorana particles and provide information about the absolute mass scale and Majorana phases.
Question 3 Multiple Choice
The atmospheric mixing angle theta_23 is measured to be close to 45 degrees (maximal mixing). Current experiments cannot definitively determine whether theta_23 is slightly above or below 45 degrees (the 'octant' ambiguity). Why does the octant matter?
ABecause the sign determines whether the neutrino is a particle or antiparticle
BBecause the octant of theta_23 (whether nu_3 contains more nu_mu or more nu_tau) affects the interpretation of other measurements, particularly the sensitivity to the CP phase delta_CP in long-baseline experiments, and different theoretical models make distinct predictions about whether theta_23 is above or below maximal
CBecause maximal mixing would violate unitarity
DBecause the octant determines the neutrino mass ordering
If theta_23 = 45 degrees exactly, the nu_3 mass eigenstate is an equal mixture of nu_mu and nu_tau -- this would suggest an underlying mu-tau symmetry. If theta_23 differs from 45 degrees, the direction of the deviation (upper vs. lower octant) breaks this symmetry and constrains theoretical models. The octant also affects the oscillation probability through sub-leading terms involving theta_13 and delta_CP, so resolving it improves the measurement of CP violation. DUNE and Hyper-K aim to determine the octant definitively.