Tree-unitarity bounds for THDM Higgs masses revisited

The one thing to take away: this paper adds the p1 < 8π inequality from charged scattering to the standard LQT set, and that alone tightens the charged and pseudoscalar mass bounds relative to Kanemura-Kubota-Takasugi [7]. The rest is a clean, explicit solution of the reduced system, with new closed forms for the lightest scalar (eqs. 50, 52) and the β-dependence of the Mh bound (eq. 53). The scattering-matrix eigenvalues are quoted from Akeroyd-Arhrib-Naimi [8], which is a published source and properly credited; that is not a soft spot, it is how the LQT literature works. The authors' own framing in the abstract — a modest extension and generalization [abstract:s1] — is accurate and I appreciate that they don't oversell it. The comparison table against [7] and [8] is the most useful page in the paper. On the reader's worry about Section 6: I don't think it lands. Dropping the positive square-root term makes (40) weaker than (38), so (38) implies (40) implies X ≤ 1 + λ'5, and that holds for every solution of the original constraint. The constructive step then exhibits a point that saturates (38) exactly — at β = π/4 the square-root argument vanishes and LHS = 1 = sin²2β — so the bound is tight, not merely admissible. The genuine gap is presentational: the sentence invoking (39) doesn't say β = π/4, and the parenthetical claim that 'for any β we were then able to find an appropriate value of α' is true for M± and MA but not for MH. Worth a footnote, not a referee objection. Real soft spots, in order: (i) the CP-violating scan in Section 8 is Matlab fmincon with no code shipped, and the authors themselves flag numerical error from the non-smooth positive-definiteness condition. Since the section's only job is to confirm the ξ = 0 bounds survive, that's acceptable but not reproducible. (ii) Eq. (19) has a typo — 'λ5 + λ5' in a± should be λ5 + λ6. (iii) The ν' ≤ 0.3 claim is a numerical observation, stated as such. Who it's for: model builders and collider-search phenomenologists who need a citable set of THDM unitarity bounds, and anyone comparing against [7] and [8]. It's a reference paper, not a conceptual one. Recommendation: send it out. The math is sound, the scope is honestly stated, and the comparison table earns its place. Expect minor revision only.

The reader's weakest_assumption targets Section 6: the authors solve a reduced inequality (40), obtained by dropping a positive term under the square root in (38), and then argue saturation for specific parameter values. The reader worries this is not a rigorous proof that no tighter bound exists. I read the logic the other way. Dropping a positive term makes (40) weaker than (38), so (38) implies (40) implies X ≤ 1 + λ'5; the bound is therefore valid for every solution of the original constraint. The constructive step then shows the bound is achieved, i.e. that it cannot be improved. Checking the stated point (M²h = 0, M²± = λ'5 + 1/2, M²A = 1 + λ'5, α = π − β) against (38) directly: at β = π/4 one has sin2β = 1, cos2β = 0, sin2α = −1, cos2α = 0, X = Y = 1 + λ'5, and the square-root argument vanishes identically, leaving LHS = 1 = sin²2β. So (38) is exactly saturated at β = π/4, and the bound is tight. The only genuine gap is presentational: the paper does not state β = π/4 in the sentence invoking eq. (39), and the parenthetical claim that 'for any β we were then able to find an appropriate value of α' is true for M²± and M²A but not for M²H, which requires β = π/4. That is a minor omission, not a load-bearing flaw in the central claim.

The paper's central claim is the set of analytic bounds M²H, M²A, M²± ≤ 1 + λ'5, M²h ≤ (1/2)(1 + λ'5), and M²lightest = 1/3 + (4/9)λ'5. I checked the algebra behind these. The reduction to (27) is asserted rather than proved, but any failure there would make the bounds conservative, not too tight, so it is not a correctness risk for the upper bounds. The hexagon vertex in Section 5 giving M²± ≤ 1 + λ'5 follows from combining p1 ≤ 8π with −f+ ≥ −8π, and I verified the vertex coordinates (34) and (36) are consistent with the stated parameter choices. The lightest-scalar formulae (50) and (52) follow correctly from (49) and (51) at β = π/4. The β-dependent bound (46) reduces to (44) at β = π/4, as claimed. The reader's concern about Section 6 is the one place I looked hardest, and I found the construction actually proves tightness rather than looseness: at β = π/4 the original constraint (38) is exactly saturated. The paper's own sentence 'for any β we were then able to find an appropriate value of α' is slightly overstated for M²H, since saturation there requires β = π/4, but this is a presentation issue. The numerical CP-violating scan is not shipped as code and the authors acknowledge numerical errors, which is a reproducibility caveat rather than a correctness concern for the analytic bounds. On balance the central claim is defensible and I would not move the verdict on the basis of the reader's stated weakest assumption. The paper would benefit from stating β = π/4 explicitly in the Section 6 saturation sentence and from clarifying the garbled comparison sentence with [8], but neither is load-bearing.

The paper is a careful and largely self-contained rederivation rather than a new physical result, and it says so. Its genuine additions are the inclusion of the p1 eigenvalue from charged channels, which does tighten the charged-Higgs bound relative to [7], the explicit β-dependence of the Mh limit in Eq. (53), and the lightest-scalar bound in Eq. (50). The algebraic reduction to the independent set (27) is clean and reproducible, and the analytic formulae are convenient for later phenomenological use. The improvement over earlier work is modest, and the CP-violating part is a numerical supplement rather than a proof.

In the two-Higgs-doublet model the scalar sector contains five particles — $h$, $H$, $A^0$ and $H^+H^-$ — and the question is how heavy any of them can be before the theory stops behaving as a perturbative quantum field theory. The authors take the standard tree-level unitarity condition, that no two-particle scattering amplitude in the scalar sector may exceed $8$ in units where the partial-wave bound is $|M|/8$ (concretely $|M| < 8$ in the paper's normalisation, corresponding to $|M| < 8$ with $M$ the tree amplitude), write down the whole matrix of such amplitudes, and reduce its many eigenvalue constraints to four independent inequalities. Solving those four in closed form gives explicit ceilings on each Higgs mass in terms of one parameter, $|M| < 8$ being the criterion and $|M| < 8$ the normalisation, with the answer expressed through $|M| < 8$ and the coupling $|M| < 8$. The point of the exercise is that a consistency requirement becomes a number: if the model is right and its couplings stay perturbative, none of its scalars can sit far above the electroweak scale, and the lightest of the five is pushed well below it.

What carries the argument. The carrying object is the matrix of high-energy two-particle scattering amplitudes among the states built from $w^+$, $z$, $h$ — the Goldstone and Higgs fields of the two doublets — whose leading terms come entirely from contact quartic interactions. Its eigenvalues, labelled $a_+$, $b_+$, $c_+$, $e_1$, $e_2$, $f_+$, $f_-$, $f_1$, $p_1$, must each obey $|eigenvalue| < 8$. Three linear identities, $3f_1 = p_1 + e_1 + f_+$, $3e_2 = 2p_1 + e_1$ and $3f_- = 2p_1 + f_+$, show that most of these are implied by the rest, and a further comparison shows $a_+$ dominates among $a$, $b$ and $c$. The whole constraint set therefore collapses to four inequalities, which are then solved as a hexagon in the $(M_+^2, M_A^2)$ plane and as a hyperbola-bounded region in the $(M_H^2 + M_h^2, M_H^2 - M_h^2)$ plane.

Core claim. The paper's central claim is that tree-level unitarity in the two-Higgs-doublet model, once the eigenvalue conditions on the two-particle scattering matrix are pared down to the four that actually bite, gives closed-form mass ceilings. Measuring masses in units of $m_{LQT} = 712$ GeV and writing $|M| < 8$ for the rescaled couplings, the authors obtain $M_+^2 < 1$, $M_A^2 < 1$, $M_H^2 < 1$, $M_h^2 < 1/2$, and for the lightest of the five scalars $M^2 < 1/3$, with all four inequalities shifted upward by terms linear in the quartic coupling $|M| < 8$ that controls the mixing between the two doublets. The charged and pseudoscalar ceilings come out stronger than earlier published ones because the authors retain an inequality, $|p_1| < 8$, that descends from charged-scalar scattering and that earlier treatments discarded. A numerical scan over the CP-violating version of the model finds no parameter point that lifts any of these ceilings.

The premise it rests on. The load-bearing premise is the saturation step: the full constraint on the heavy neutral mass is replaced by a weakened version obtained by discarding a positive term under a square root, the easier inequality is solved, and the resulting ceiling is then declared sharp on the strength of one exhibited parameter point rather than a general argument that no other corner of parameter space does better.

If the scalar sector is to stay perturbative, the five masses are bounded: with the extra quartic coupling set to zero, $H$, $A^0$ and $H^\pm$ each cap at 712 GeV when bounded one at a time, $h$ caps at 503 GeV once $h$ is required to be lighter than $H$, and the lightest of the five caps at 411 GeV.

The charged and pseudoscalar ceilings are tighter than the earlier published values of 872 GeV and 1233 GeV, precisely because the constraint from charged-scalar scattering is kept rather than dropped.

Every ceiling rises with the soft $Z_2$-breaking parameter, so any future observation of a heavier Higgs would itself translate into an upper limit on how much soft breaking the model can carry.

The bound on $h$ is the one place where a mixing angle enters: $M_h^2 < \frac{\sin^2 2\beta}{1+|\cos 2\beta|}\left(\frac12 - \frac{3}{32\pi}\lambda_5\right) + \frac{3\lambda_5}{16\pi}$, so measuring $m_h$ together with $\tan\beta$ constrains the quartic coupling.

Allowing CP violation in the scalar sector does not raise any ceiling for small breaking parameter; the extremes are still attained at the CP-conserving points.

The pattern that appears when one, two or three neutral scalars are bounded simultaneously — coefficients $1$, $1/2$, $1/3$ under the square root — looks like a general counting rule for the heaviest of $n$ nearly degenerate scalars; checking it in a three-doublet model would be a clean test of how far the rule extends.

A scalar discovered well above roughly 700 GeV with the others light would not merely pin down one coupling: since the analysis assumes no new states in the relevant energy range, such a discovery would point to physics entering below the unitarity scale rather than to a failure of the bound.

The same eigenvalue bookkeeping applies to the supersymmetric two-doublet sector, where the quartic couplings are fixed by gauge couplings; read that way, the result says the supersymmetric scalar sector already sits far inside the unitarity-allowed region at tree level.

Because the $\beta$-dependent $h$ ceiling is the only place a mixing angle survives, a future precision measurement of the lightest Higgs couplings could in principle be used to ask whether the scalar sector sits at its unitarity boundary or comfortably inside it.

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