Power-aligned 2HDM: a correlative perspective on (g−2)_e,μ

The thing worth your time here is Section 2, not Section 3. If the Yukawa matrices are Hermitian — which the authors motivate from left-right symmetric models — the flavor symmetry collapses from SU(3)⁵ to SU(3)_q x SU(3)_l, and the MFV spurion expansion then admits Y2 = Y1ⁿ as a natural alignment. The commutator argument for rational n is correct, the diagonalization goes through, and the linear A2HDM of Pich-Tuzon drops out as n=1. That is a genuine generalization, it is derived rather than assumed, and it is the part I would cite. The observation that Hermiticity plus MFV gives you a one-parameter family is a nice piece of model building and it is done properly in a couple of pages without hand-waving. Credit there is earned. Section 3 is where I get off the bus. The B-physics bound from the 2019 Delta M_d,s predictions pushing M_H+ past 2 TeV is a real constraint and the paper's response — accept the heavy charged Higgs and go quasi-degenerate in the neutral sector — is coherent and, as far as I can tell, correctly computed. But the headline claim of a correlation between Delta a_e and Delta a_mu that distinguishes this from the gA2HDM is weaker than the abstract lets on. As the stress-test note says, n_l and M_S are two parameters fitting two anomalies, so the band in the right panel of Fig. 2 is a family of consistency curves indexed by M_S, not a prediction. The paper does have a real falsification handle — a future measurement off the band kills the single-power structure — but landing on the band does not confirm pA2HDM over gA2HDM unless M_S is independently known, and Section 3.2 fixes M_S by choosing lambda_5 small, which is a model-building choice. The abstract's 'distinguishable ... and hence testable' should be softened to say exactly this. Two smaller things. The sign pattern theta_e = 0, theta_mu = pi and the even-integer condition on 1/n_l are chosen to land on the observed signs; the authors are upfront about it and note the allowed region 'intriguingly' prefers an even integer, which is at least self-consistent, but it is post-hoc. And the Z2 vacuum with <H2> = 0 is a strong premise; the authors flag the basis-dependence worry in Section 3.2 and defend it, but the defense is a model-building assumption, not a derivation. Who should read it: anyone working on Yukawa alignment or 2HDM explanations of the two g-2 anomalies. The structural result is worth a referee's time even though the phenomenological framing needs tightening.

The paper's distinguishing claim (abstract:s1, and the closing of §3.3) is that the single lepton power n_l produces a correlation between Delta a_e and Delta a_mu that separates the pA2HDM from the gA2HDM and L2HDM. The mechanics are laid out in Eq. (3.11): delta a_e = a_0 + a_1 delta a_mu + a_2 delta a_mu^2, with coefficients that depend on M_S and n_l. The concern is not that Eq. (3.11) is wrong; it is that the two free parameters (M_S, n_l) are being used to fit two data points (Delta a_e, Delta a_mu), so the 'correlation' plotted in the right panel of Fig. 2 is a one-parameter family of solutions to a two-constraint system, not a prediction. For any M_S in [1.5, 3.5] TeV there exists an n_l that lands on the central values, and the shape of the curve is set by the same loop functions that were tuned to fit. The reader's rationale already flags this ('a consistency curve rather than an independent prediction unless M_S is independently fixed'), and I agree; the question is whether the paper supplies that independent fixing. It does not: §3.1 pushes M_H+ above ~2 TeV from Delta M_d,s, but §3.2 then sets M_H ~ M_A ~ M_H+ by choosing lambda_5 small, which is a model-building choice, not a measurement. So M_S is bounded but not fixed, and the 'correlation' is really a family of curves indexed by M_S. That is a weaker claim than the abstract's 'makes the model distinguishable ... and hence testable by future precise measurements' (abstract:s1). A future measurement that lands off the band would falsify the single-power structure, which is a real test; but a measurement that lands on the band does not confirm the pA2HDM over the gA2HDM unless M_S is known. The paper should say this explicitly rather than presenting the band as a prediction. This is a framing/strength-of-claim issue, not an error in the loop calculation, and it does not sink the structural result of §2.

The reader's weakest_assumption (the Z2-symmetric scalar potential with <H2> = 0) is a fair target, and the paper does acknowledge the basis-dependence subtlety in §3.2. I do not think that is the single most load-bearing concern, because the structural result of §2 (Hermitian Yukawa matrices plus MFV give Y_NP = (Y_SM)^n, Eq. (2.13)) stands independently of the vacuum choice; the Z2 assumption matters for the phenomenology of §3, not for the algebra of §2. The concern I would press is narrower and more specific: the 'correlation' between Delta a_e and Delta a_mu is presented in the abstract (abstract:s1) and in the closing paragraph of §3.3 as a distinguishing, testable feature, but the two anomalies are fitted with two parameters (M_S, n_l), so the curve in Fig. 2 (right) is a consistency band, not a prediction. The paper is honest that M_S is bounded rather than fixed (§3.1 gives M_H+ >~ 2 TeV; §3.2 sets M_H ~ M_A ~ M_H+ by a small lambda_5 choice), but it does not carry that honesty through to the abstract's 'distinguishable ... and hence testable' phrasing. This is a claim-strength issue, not a correctness error: the loop functions in Eqs. (3.7)-(3.10) are standard, the sign choice theta_e = 0, theta_mu = pi is stated, and the even-integer requirement on 1/n_l is flagged. The B-physics bound uses the 2019 Delta M_d,s predictions, which is the right benchmark at submission. The structural result of §2 is a genuine and new extension of the linear alignment of ref. [9], and the paper gives a clear derivation. I would keep the verdict at CONDITIONAL: the model-building proposal is defensible within its stated assumptions, but the abstract overstates the testability of the correlation, and that overstatement should be corrected before the paper is read as making a prediction rather than a fit. No change to the verdict is warranted on the basis of this concern alone; it is a framing fix, not a soundness fix.

The structural derivation of the power alignment in section 2 is clean and gives a genuine alternative to the linear alignment of the A2HDM, with a clear group-theoretic origin in the Hermitian-Yukawa limit. The observation that a single lepton power correlates the two g-2 anomalies is a real, falsifiable feature in principle, and the paper is careful to check B-physics, Z, tau, and LHC constraints rather than fitting the anomalies in isolation. If the correlation survives scrutiny as a prediction rather than a consistency curve, the model would be a useful benchmark for future g-2 measurements.

The paper asks whether a two-Higgs-doublet model can explain both the electron and muon magnetic-moment anomalies with one structural assumption rather than two independent sets of couplings. It shows that if the Yukawa matrices are Hermitian and obey minimal flavor violation, the two scalar doublets' Yukawa matrices commute and can be simultaneously diagonalized, leaving the new-physics Yukawa matrix as a power of the Standard Model one, $Y_f=(�hat Y_f)^{n_f}$. The same flavor-universal lepton power $n_�ell$ then controls both $Δa_e$ and $Δa_�mu$, so the two anomalies become correlated rather than adjustable separately. The paper argues that TeV-scale quasi-degenerate Higgs masses let this single-power structure fit both anomalies while staying safe under B-physics, $Z$ and $τ$ decay, and LHC constraints. If correct, the scheme makes a sharper prediction than generalized linear alignment or lepton-specific 2HDMs and can be tested by future magnetic-moment measurements.

What carries the argument. The load-bearing object is the power alignment $Y_f=(�hat Y_f)^{n_f}$, which follows because two Hermitian Yukawa matrices that are functions of one another commute, $[Y_1,Y_2]=0$, and can be diagonalized together; this is what removes tree-level flavor-changing neutral currents without introducing the three independent aligned parameters of the generalized linear alignment. The second piece is the $Z_2$-symmetric scalar potential with $�langle H_2�rangle=0$, which keeps the SM Higgs in $H_1$ and allows a TeV-scale spectrum with $M_H≃ M_A≃ M_{H^+}$ when $λ_5$ is small. The correlation between the two anomalies comes from using one flavor-universal lepton power $n_�ell$ for both the electron and muon entries, so that a fit to $Δa_�mu$ fixes the sign and approximate size of $Δa_e$.

Core claim. The central claim is that Hermiticity plus minimal flavor violation in an effective 2HDM does not merely suppress flavor-changing neutral currents; it forces the new-physics Yukawa couplings into the power-aligned form $Y_f=(�hat Y_f)^{n_f}$ (Eq. 2.13), with a separate integer or reciprocal-integer power for each fermion type. In the lepton sector a single power $n_�ell$ enhances the electron and muon Yukawa entries in a fixed ratio, and the paper shows that with a common TeV-scale mass $M_S≃ M_H≃ M_A≃ M_{H^+}$ the one-loop and Barr-Zee contributions can simultaneously reproduce $Δa_e=-(8.7±3.6)×10^{-13}$ and $Δa_�mu=(2.51±0.59)×10^{-9}$. The resulting $Δa_e$–$Δa_�mu$ relation is quasi-linear, not arbitrary, and the same parameter region survives the $ΔM_{d,s}$, $�bar B→ X_sγ$, $Z→�bar�ell�ell$, $τ$ universality, and LHC constraints discussed in the paper.

The premise it rests on. The load-bearing premise is that a discrete $Z_2$ symmetry, which forbids mixing between the two Higgs doublets, keeps the second doublet's vacuum expectation value at zero in the same basis where the Hermitian power alignment is defined, so the power $n_f$ is physical rather than an artifact of choosing a scalar basis.

A single lepton power $n_�ell$ and a common Higgs mass $M_S$ would replace the independent electron and muon couplings used in generalized linear alignment, so a future measurement of one anomaly would predict the other.

The allowed parameter region requires $M_{H^+}�gtrsim 1.7$ TeV and quasi-degenerate neutral Higgs bosons, so the model's collider signature is a TeV-scale Higgs spectrum rather than a light scalar.

Because the tau Yukawa grows more slowly under a power than under linear alignment, the same fit keeps $Z→�bar�ell�ell$ and $τ$ lepton-universality observables below current uncertainties.

The predicted quasi-linear $Δa_e$–$Δa_�mu$ correlation distinguishes the power-aligned 2HDM from the generalized aligned and lepton-specific 2HDMs, which can fit both anomalies only with independent couplings.

If the correlation survives improved measurements, the same single-power idea could be used to organize the charged-lepton mass hierarchy and Higgs coupling ratios, not just the two magnetic moments.

The basis-dependence issue noted in the paper suggests a UV test: any completion that produces Hermitian Yukawa matrices should leave a calculable pattern in Higgs coupling ratios, so comparing those ratios could distinguish a maintained $Z_2$ vacuum from a basis-transformed one.

Because the charged Higgs is pushed to a few TeV, a correlated deviation in $τ→�muν�barν$ versus $�mu→ eν�barν$ at the $10^{-4}$ level would be an additional signature of the same single-power structure.

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