Renormalization-group evolution of the fermion mass matrices

Punchline: the one-loop decoupling result is correct, but the paper overstates what it proves. Under the stated top/tau dominance approximation, the invariance of O_u and O_l is a clean repackaging of the standard RGE structure, and the new PMNS correction in the Babu–He tribimaximal model is genuinely new. The trouble is that the advertised stability of down-type texture zeros does not survive the paper's own Hermiticity convention.

What is new: Eqs. (10)–(17) correctly show that, with Y_d and Y_ν neglected, the diagonalizers of H_u and H_l do not run, and the solutions (18)–(33) follow. That is useful for model-builders. The first-order correction to the tribimaximal PMNS matrix, Eqs. (76)–(83), is not in the cited literature, and the paper honestly concedes overlap with Ref. [5] for the texture-zero running.

Where it wobbles: the texture-zero stability claim for M_d is a basis artifact. The paper itself admits that rotating right-handed fields to restore Hermiticity yields an M'_d(t) with no texture zero. Since the texture-zero ansatz is defined in the Hermitian basis, saying the zeros are stable while the Hermitian image fills the (1,1) zero is misleading. The same issue affects the five-zero RRR patterns. This is not a numerical quibble; it undercuts a central advertised result. In the tribimaximal section, there is also a genuine internal contradiction: the text reports Δ12 can reach 247.8 in the IO case, then concludes θ12 is stable against radiative corrections. Both cannot hold. The long expressions in Eqs. (82)–(83) look unverified; a referee should ask for a derivation or a symbolic check. Minor typos and dropped terms are secondary.

This paper is for flavor model-builders who care whether high-scale textures survive running. They will find the parametrization useful even if the stability claim needs to be weakened. It deserves serious peer review because the core derivation is sound and the tribimaximal result is worth checking.

Even granting the top/tau dominance approximation, the paper's advertised conclusion that texture zeros are stable under running is not supported for the down-type quark sector. In Section 3.1, the four-zero M_d(t0) is Hermitian (Eq. 45), but the evolved M_d(t) = γ_d O_u I_d O_u† M_d(t0) (Eq. 27) is non-Hermitian. Eq. (50) shows M_d(t) retains the first-row/column zeros in this fixed basis, and the text then admits that rotating right-handed fields to restore Hermiticity yields an M'_d(t) 'with no texture zero.' That admission is fatal to the stability claim: the texture-zero ansatz is defined in the Hermitian basis (Section 3, 'without loss of generality... Hermitian'), and the physically relevant Hermitian matrix is the polar factor H_d(t) = (M_d(t)M_d†(t))^{1/2}. Because M_d(t) has (1,2) nonzero, M_d(t)M_d†(t) has (1,1) = |M_d(t)_{12}|^2 ≠ 0, so H_d(t) has (1,1) ≠ 0. Thus the (1,1) zero is filled at first order in the running; the stability statement refers only to the unphysical non-Hermitian running matrix in a basis that was Hermitian only at t0. The same issue affects the five-zero RRR patterns for M_d. This is not a matter of numerical precision; it invalidates the claimed stability of down-type texture zeros in the basis the paper itself uses.

The central mathematical derivation of O_u and O_l invariance is a conditional theorem under the stated dominance approximation, and I do not dispute it. However, the paper's advertised payoff is that texture zeros of Hermitian quark mass matrices are stable under RGE evolution. The paper itself, in Section 3.1 after Eq. (50), concedes that restoring Hermiticity by a right-handed rotation yields an M'_d(t) with no texture zero. This is not a minor caveat: the texture-zero ansatz was defined on Hermitian matrices, and the physical low-energy Hermitian mass matrices are obtained by polar decomposition. Since M_d(t) has a nonzero (1,2) element, M_d(t)M_d†(t) automatically has a nonzero (1,1) element, so the polar Hermitian factor H_d(t) cannot have the advertised (1,1) zero. The paper's stability statement therefore applies only to the non-Hermitian matrix in the original weak basis, not to the Hermitian-basis textures that the quark-section analysis claims to preserve. This concern is more specific and more damaging than the general approximation caveat: it means the main application of the decoupling result to down-type texture zeros is not established even within the paper's own assumptions. I recommend keeping the verdict CONDITIONAL: the derivation should be accepted conditional on a major revision that either abandons the texture-zero stability claim for M_d or correctly derives the textures of the physical Hermitian mass matrices H_d(t).

The central derivation is internally consistent and provides a compact, useful method for evolving whole mass matrices rather than only masses and mixing angles. The decoupling argument in Eqs. (10)-(17) is valid under the stated assumptions, and the tribimaximal application yields explicit, falsifiable predictions about which mixing angle runs. The main advertised application, however, overstates the stability of down-type texture zeros: the paper's own non-Hermitian representative retains zeros that are not present in the physical Hermitian matrix. This affects a substantial part of the paper's conclusions and must be corrected before the results can be accepted as stated.

This paper aims to show that the one-loop renormalization-group evolution of the full $3\times 3$ fermion mass matrices can be solved without re-diagonalizing at every scale, as long as the nonlinear terms are dominated by the top-quark and tau-lepton Yukawa couplings. Its central result is that the unitary matrices $O_u$ and $O_l$ which diagonalize the up-type quark and charged-lepton mass matrices are scale-invariant; the running is carried entirely by overall factors and by diagonal matrices that rescale only the third row and column. If true, this makes texture-zero mass-matrix ansätze stable against radiative corrections and keeps models like tribimaximal mixing predictive after running.

What carries the argument. The central object is the pair of unitary matrices $O_u$ and $O_l$ fixed by diagonalizing the Hermitian matrices $H_u=M_uM_u^\dagger$ and $H_l=M_lM_l^\dagger$. Their scale invariance follows from the special structure of the truncated one-loop RGEs, in which the nonlinear term in each sector is proportional to an anticommutator of the same Hermitian matrix with itself; after diagonalization the off-diagonal components of the evolution equation reduce to $(D_i-D_j)T_{ij}=0$, and non-degeneracy forces $T_{ij}=0$. The diagonal matrices $I_u=\mathrm{diag}(1,1,\xi_t^{3b/2})$, $I_d=\mathrm{diag}(1,1,\xi_t^{3c/2})$, $I_l=\mathrm{diag}(1,1,\xi_\tau^{3b/2})$, and $I_\nu=\mathrm{diag}(1,1,\xi_\tau^{3c/2})$ then carry all the running, rescaling only the third row and column of each mass matrix, which is what protects the texture zeros.

Core claim. In a Hermitian basis and after neglecting the down-type and neutrino Yukawa matrices in the RGE coefficients, the paper derives that the unitary matrices diagonalizing $H_u = M_u M_u^\dagger$ and $H_l = M_l M_l^\dagger$ do not run with energy scale. Differentiating the diagonalized matrix $D_u = O_u^\dagger H_u O_u$ produces off-diagonal conditions $-T^u_{ij}D^u_j + D^u_i T^u_{ij}=0$ for $i\neq j$, which force $T^u_{ij}=0$ because the quark masses are non-degenerate; the same applies to the charged leptons. Therefore $O_u$ and $O_l$ are constants of the RGE flow, and the evolved mass matrices take the factorized form $M_f(t) = \gamma_f O_f I_f O_f^\dagger M_f(t_0)$ for Dirac fermions, with $I_u = \mathrm{diag}(1,1,\xi_t^{3b/2})$ and $I_l=\mathrm{diag}(1,1,\xi_\tau^{3b/2})$; the Majorana case adds a second factor $O_l^* I_\nu O_l^T$ for the neutrino matrix. Since $I_f$ is diagonal, only the third row and column of the mass matrices are rescaled, so texture zeros in the first two rows and columns are stable. The paper applies this to four- and five-texture-zero quark matrices and to a tribimaximal lepton mixing pattern, finding that only $\theta_{12}$ can receive a significant shift.

The premise it rests on. The load-bearing premise is that only the top-quark and tau-lepton Yukawa couplings matter in the RGE coefficients, so each sector's nonlinear term is a function of that sector's own Hermitian matrix; if down-type or neutrino Yukawas contribute appreciably, the diagonalizing matrices run and the conclusion does not hold.

For Hermitian quark ansätze with four texture zeros, the zero pattern of both $M_u$ and $M_d$ is preserved to leading order, so low-energy fits can be interpreted as high-scale textures.

Among the five phenomenologically allowed five-texture-zero quark matrices, several entries remain stable under running: $M_u$ and $M_d$ in Pattern I, $M_d$ in Pattern II, and $M_u$ in Patterns III, IV, and V.

In the seesaw model of Sec. 3.3, the charged-lepton and Dirac neutrino mass matrices keep their texture-zero forms approximately after one-loop evolution.

In the tribimaximal example, $\theta_{13}$ and $\theta_{23}$ stay close to their tribimaximal values while $\theta_{12}$ is the only angle that can shift substantially, especially in inverted neutrino mass ordering.

If $M_u$ and $M_l$ are diagonal at the initial scale, they remain diagonal at every scale, and texture zeros in $M_d$ and $M_\nu$ are essentially stable.

Read as a general criterion, the invariance of $O_u,O_l$ implies that any Hermitian texture whose zeros avoid the third row and column is radiatively stable under the same approximation; the specific examples are instances of that broader rule.

Because the approximation drops $Y_d$ and $Y_\nu$ entirely, the scale-invariance is parametric: in regions with large $\tan\beta$ or an order-one Dirac neutrino Yukawa, the diagonalizing bases are expected to run, and a numerical or iterative treatment would be needed.

The tribimaximal application suggests a testable target: if this model class is right, a non-zero $\theta_{13}$ could be radiatively generated with magnitude set by $\epsilon|x|$ and correlated with the deviation of $\theta_{12}$; future precision data on both angles would constrain the high-scale parameters $x$, $\phi$, and the lightest neutrino mass.

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