Domain: Physics — High-energy particle physics

Problem:
Within a theoretical framework attempting to explain fermion mass hierarchies, a \(U(1)\) flavor symmetry is introduced, under which the Standard Model fermions are charged. The Yukawa couplings are generated dynamically through the vacuum expectation value of a scalar field (a flavon) that breaks this symmetry.

The effective Yukawa matrices for charged leptons (\(Y_l\)) and Dirac neutrinos (\(Y_\nu\)) at a low energy scale are given by:

\[ (Y_l)_{ij} = c^l_{ij} \epsilon^{|X_{L_i} + X_{\bar{e}_j}|} \]

\[ (Y_\nu)_{ij} = c^\nu_{ij} \epsilon^{|X_{L_i} + X_{N_j}|} \]

where \(i, j \in \{1, 2, 3\}\) are generation indices, \(X_{L_i}, X_{\bar{e}_j}, X_{N_j}\) are the integer flavor charges for the left-handed lepton doublets, right-handed charged leptons, and right-handed neutrinos respectively. The parameter \(\epsilon\) represents the ratio of the flavon vacuum expectation value to the cutoff scale \(\Lambda\), and \(c^l, c^\nu\) are \(3 \times 3\) matrices with \(\mathcal{O}(1)\) complex coefficients.

The charged lepton mass matrix is \(M_l = \frac{v}{\sqrt{2}} Y_l\), where \(v\) is the Higgs vacuum expectation value. This matrix is diagonalized via a bi-unitary transformation: \(U_l^\dagger M_l W_l = \hat{M}_l\), where \(\hat{M}_l\) is the diagonal matrix of physical masses, and \(U_l, W_l\) are unitary matrices rotating the left- and right-handed fields, respectively.

This framework also predicts lepton flavor violating (LFV) processes mediated by higher-dimensional operators. The effective Lagrangian for the decay \(\mu \to e\gamma\) contains the dipole operator:

\[ \mathcal{L}_{eff} \supset \frac{e v}{\sqrt{2} \Lambda^2} (C_{e\gamma})_{ij} (\bar{\ell}_i \sigma^{\mu\nu} P_R \ell_j) F_{\mu\nu} + h.c. \]

where \(\ell_i\) are the charged lepton fields in the mass basis. The dimensionless Wilson coefficient matrix \(C_{e\gamma}\) in the mass basis is related to the corresponding matrix \(C'_{d}\) in the flavor (interaction) basis by \(C_{e\gamma} = U_l^\dagger C'_{d} W_l\). The flavor-basis matrix \(C'_{d}\) is constructed similarly to the Yukawa matrix:

\[ (C'_{d})_{ij} = c^d_{ij} \epsilon^{|X_{L_i} + X_{\bar{e}_j}|} \]

where \(c^d\) is another matrix of \(\mathcal{O}(1)\) complex coefficients.

The branching ratio for this decay is given by:

\[ BR(\mu \to e\gamma) = \frac{\alpha_{em} v^2 m_\mu^3}{8 \Gamma_\mu \Lambda^4} |(C_{e\gamma})_{12}|^2 \]

where the indices \(1\) and \(2\) correspond to the electron and muon, respectively.

The branching ratio for the given parameters can be expressed in the form \(BR(\mu \to e\gamma) = C \times 10^{-21}\).

Calculate the numerical pre-factor \(C\) using the following parameters.

**Flavor Charges:**
* \(X_L = \{6, 5, 5\}\)
* \(X_{\bar{e}} = \{3, 2, 0\}\)
* \(X_N = \{-9, -8, -8\}\)

**Model Parameters:**
* \(\epsilon = 0.10\)
* \(\Lambda = 1.0 \times 10^6 \text{ GeV}\)

**Coefficient Matrices:**
* \(c^l = \begin{pmatrix} 0.8+0.1i & 1.1-0.3i & 0.9+0.2i \\ 1.2+0.4i & 0.7-0.1i & 1.0+0.5i \\ 0.9-0.2i & 1.3+0.1i & 1.1-0.4i \end{pmatrix}\)
* \(c^d = \begin{pmatrix} 0.9+0.3i & 1.0-0.1i & 1.1+0.2i \\ 1.2-0.2i & 0.8+0.4i & 0.9-0.3i \\ 1.0+0.1i & 1.3-0.4i & 0.7+0.5i \end{pmatrix}\)

**Physical Constants:**
* Higgs VEV, \(v = 246.22 \text{ GeV}\)
* Muon mass, \(m_\mu = 0.10566 \text{ GeV}\)
* Fine-structure constant, \(\alpha_{em} = 1/137.036\)
* Total decay width of the muon, \(\Gamma_\mu = 2.996 \times 10^{-19} \text{ GeV}\)

Provide your final answer for the pre-factor \(C\) rounded to three decimal places.

Answer: 2.032

Solution code:
import numpy as np

# --- Given Parameters ---

# Flavor Charges
X_L = np.array([6, 5, 5])
X_e_bar = np.array([3, 2, 0])

# Model Parameters
epsilon = 0.10
Lambda = 1.0e6  # GeV

# Coefficient Matrices (complex)
c_l = np.array([
    [0.8+0.1j, 1.1-0.3j, 0.9+0.2j],
    [1.2+0.4j, 0.7-0.1j, 1.0+0.5j],
    [0.9-0.2j, 1.3+0.1j, 1.1-0.4j]
])

c_d = np.array([
    [0.9+0.3j, 1.0-0.1j, 1.1+0.2j],
    [1.2-0.2j, 0.8+0.4j, 0.9-0.3j],
    [1.0+0.1j, 1.3-0.4j, 0.7+0.5j]
])

# Physical Constants
v = 246.22        # GeV
m_mu = 0.10566    # GeV
alpha_em = 1 / 137.036
Gamma_mu = 2.996e-19 # GeV

# --- Step 1: Construct Flavor-Basis Matrices ---
# Initialize empty 3x3 complex matrices
Y_l = np.zeros((3, 3), dtype=complex)
C_prime_d = np.zeros((3, 3), dtype=complex)

# Populate matrices based on the FN mechanism formula
for i in range(3):
    for j in range(3):
        exponent = abs(X_L[i] + X_e_bar[j])
        Y_l[i, j] = c_l[i, j] * (epsilon ** exponent)
        C_prime_d[i, j] = c_d[i, j] * (epsilon ** exponent)

# --- Step 2: Diagonalize the Mass Matrix ---
# Calculate the charged lepton mass matrix
M_l = (v / np.sqrt(2)) * Y_l

# Perform Singular Value Decomposition to find the rotation matrices
# M_l = U @ Sigma @ Vh, where U is U_l and Vh is W_l_dagger
U, s, Vh = np.linalg.svd(M_l)

# W_l is the Hermitian conjugate of Vh (i.e., Vh_dagger)
W_l = Vh.conj().T

# --- Step 3: Transform to the Mass Basis ---
# C_e_gamma = U_l_dagger @ C_prime_d @ W_l
Ce_gamma = U.conj().T @ C_prime_d @ W_l

# --- Step 4: Calculate Branching Ratio and Pre-factor C ---
# Extract the (1, 2) element, which corresponds to the mu -> e transition
Ce_gamma_12 = Ce_gamma[0, 1]

# Calculate the squared magnitude of this element
abs_Ce_gamma_12_sq = np.abs(Ce_gamma_12)**2

# Calculate the full branching ratio
numerator = alpha_em * (v**2) * (m_mu**3) * abs_Ce_gamma_12_sq
denominator = 8 * Gamma_mu * (Lambda**4)
branching_ratio = numerator / denominator

# Calculate the pre-factor C where BR = C * 10^-21
C = branching_ratio / 1e-21

# --- Final Result ---
print(f"Calculated Branching Ratio (BR): {branching_ratio:.6e}")
print(f"Calculated Pre-factor (C): {C}")
print(f"Final Answer (C rounded to 3 decimal places): {C:.3f}")