Progress in lattice simulations for two Higgs doublet models

Alberto Ramos; Atsuki Hiraguchi; C.-J. David Lin; George W.-S Hou; Guilherme Catumba; Karl Jansen; Mugdha Sarkar; Ying-Jer Kao

REVIEW 3 major objections 5 minor 23 references

Progress in lattice simulations for two Higgs doublet models

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read On a line of constant Standard-Model physics, the custodial two-Higgs-doublet model with weak quartic couplings allows an extra neutral scalar H with mass about 0.2 times the W boson mass, and this ratio does not change as the lattice…

desk verdict First lattice line-of-constant-physics study of the inert/custodial 2HDM, with a striking low-mass claim that needs a finite-volume check before I'd trust it. read the letter →

arxiv 2412.13896 v1 pith:DS3D5PAO submitted 2024-12-18 hep-lat hep-ph

Guilherme Catumba , Atsuki Hiraguchi , George W.-S Hou , Karl Jansen , Ying-Jer Kao , C.-J. David Lin , Alberto Ramos , Mugdha Sarkar This is my paper · ORCID

classification hep-lathep-ph MSC 81T2581T13 PACS 11.15.Ha12.60.Fr

keywords twoHiggsdoubletmodellatticegaugetheorycustodialsymmetryinertlineofconstantphysicsgradientflowcouplingBSMscalarspectrumfinitetemperaturephasetransition

topics Yang-Mills Mass Gap

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks whether the custodial (inert) two-Higgs-doublet model can hide additional scalars far below the W boson mass while still reproducing Standard-Model physics. Using lattice simulations on a line of constant SM physics with cutoff from 300 to 630 GeV, the authors report that, for the small quartic couplings they choose, the neutral BSM state H reaches a mass of about $0.2\,m_W$ (near 16 GeV), and this ratio stays flat as the cutoff changes. They also compute the running of the weak gauge coupling and the finite-temperature transition, finding a smooth crossover at weak couplings. A sympathetic reader would take the result as a non-perturbative, cutoff-independent lower-bound estimate for a realizable H mass in this regime, and as a benchmark for later scans at larger couplings.

What carries the argument

The argument is carried by a line of partially constant physics (LPCP) in the bare parameter space of the lattice action. The SM sector is tuned at each bare gauge coupling $\beta$ by adjusting $\kappa_2,\eta_2$ so that $R\equiv m_h/m_W\approx1.5$ and the gradient-flow renormalized gauge coupling satisfies $g^2_{GF}(\mu=m_W)=0.5$ (equivalently $\sqrt{8t_0}\,m_W=1.0$); all BSM couplings $\kappa_1,\eta_1,\eta_3,\eta_4,\eta_5$ are held fixed at small values. The spectrum is read from two-point functions of composite operators that separate the SM Higgs and W boson from the BSM states $H,A,H^\pm$. The BSM scan then varies $\kappa_1$ within the $\mathbf{H}_2$ phase while monitoring $R$ and $S$ to check that SM physics is unchanged.

What would settle it

On the same five lattice spacings, re-tune $\kappa_2$ and $\eta_2$ for each $\kappa_1$ value so that $R$ and $S$ match the target values to much higher precision, then check whether $m_H/m_W$ still approaches $0.2$ and remains cutoff independent. If the plateau shifts or disappears, the reported lower bound is an artifact of the LPCP tuning rather than a property of the model with constant SM physics.

Watch

Extended reading notes

Core claim

The central discovery is that the custodial inert two-Higgs-doublet model on the lattice admits a BSM neutral scalar whose mass ratio $m_H/m_W$ saturates at roughly $0.2$ before the transition into the phase where both doublets condense, and this saturation is independent of the lattice cutoff over $300$--$630$ GeV. Along the same line of constant SM physics the heavier states $A$ and $H^\pm$ behave differently: $m_A/m_W = m_{H^\pm}/m_W$ keeps decreasing as the hopping parameter $\kappa_1$ is raised, so the mass gap between $H$ and $A/H^\pm$ widens. Because the SM conditions $R=m_h/m_W\approx1.5$ and $g^2_{GF}(m_W)=0.5$ are fixed at each cutoff, the light $H$ is a property of a theory that still looks like the SM in the Higgs-gauge sector.

Load-bearing premise

The whole construction rests on the assumption that the BSM sector is 'mostly insensitive' to the SM sector, so that scanning $\kappa_1$ while keeping all other BSM couplings fixed leaves the line of constant SM physics intact; the paper verifies only that $R$ and $S$ stay roughly constant, not that small mistunings would leave the BSM mass ratios unchanged.

Editorial extensions

If this is right

A neutral inert-doublet scalar $H$ with $m_H\simeq 0.2\,m_W$ can coexist with SM-like Higgs and W masses in a weakly coupled custodial 2HDM.
The cutoff independence of the ratio justifies calling $\sim0.2\,m_W$ a non-perturbative lower bound of the H mass for the chosen couplings.
$m_A/m_W=m_{H^\pm}/m_W$ keeps falling toward the W as $\kappa_1$ approaches the $\mathbf{H}_{12}$ transition, so the H--A mass splitting grows along the LPCP.
The gradient-flow running gauge coupling collapses onto one curve over a wide energy range, matching one-loop massless running at high energy and a screened Yukawa form at low energy.
At small quartic couplings the electroweak transition is a crossover, with no volume dependence in the susceptibility peak; the transition sits near $m_W/T_c\sim0.5$.
If the $\sim0.2\,m_W$ plateau survives a fully retuned LPCP, the model predicts a $\sim16$ GeV neutral scalar that lies within reach of low-energy collider and Higgs-decay searches, a target the paper does not discuss.
A natural next step is to repeat the $\kappa_1$ scan with larger $\eta_3$ or $\eta_4$, since the paper notes these couplings must be $O(1)$ for a strong first-order transition; the light-H plateau may move or disappear in that regime, which would change the phenomenology.
The partially constant physics line fixes only SM quantities; an extension that also matches a BSM renormalized quantity would test whether the H-mass bound is an accident of the tuning strategy.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Worth a look. This proceedings does something no lattice 2HDM paper has done: builds a line of constant SM physics (R = m_h/m_W ~ 1.5 and g^2_GF(m_W) = 0.5), measures gradient-flow running, and extracts BSM mass ratios across cutoffs from 300 to 630 GeV. The central observation is a plateau in m_H/m_W ~ 0.2 as kappa1 approaches the H12 transition, while m_A/m_W keeps falling. If true, a custodial/inert 2HDM with weak quartics can realize an extra neutral scalar near 16 GeV. That is a concrete, falsifiable map of parameter space, and the paper is honest that it is a proceedings with limited scope.

Credit where due: the LPCP construction is a genuine technical step; the gradient-flow running collapses onto one curve with a perturbative match at high energy and a Yukawa/screening fit in the IR; and the finite-T crossover argument, based on absence of volume dependence in the susceptibility peak, is reasonable even if limited to the two finest lattices. The authors also flag their own limitations: small BSM couplings are a choice, not a necessity, and only two beta values give a clean susceptibility peak.

The soft spots are real. First, the headline number has no quoted uncertainty anywhere; Fig. 4 clearly has errors, but the abstract and conclusion state m_H ~ 0.2 m_W as a result. A lattice number without an error bar is not yet a result. Second, the LPCP tuning is approximate: R ranges 1.46(5) to 1.53(8), a few sigma off nominal, and the BSM mass ratios could shift if the SM line were retuned more tightly. Third and most important, the stress-test concern lands: from Table 1, m_H L ranges from about 0.94 to 1.7. The lightest BSM state has a correlation length comparable to the box size at the finest point (beta = 8.56). Without a volume test, the apparent plateau in m_H/m_W could be the finite-volume floor: the extracted ground-state mass cannot fall much below 1/L. The observation that m_A/m_W keeps decreasing while m_H/m_W flattens is exactly what one would see if the lighter H hits the floor first. So the cutoff-independence claim is not established; it is an invitation to run a volume study. Finally, no code or data are released, so the numerics cannot be checked from the manuscript.

None of this is disqualifying for a proceedings. All the issues are addressable in a full paper: quote errors, document the correlator fits, add a volume scan of the BSM spectrum, and publish the data. The paper deserves a serious referee, but the referee should insist on the finite-volume check before the 0.2 m_W claim is accepted. I would not cite it as a result yet, but I would follow the full paper when it appears.

Referee Report

3 major / 5 minor

Summary. The manuscript reports exploratory lattice simulations of the custodial/inert two-Higgs-doublet model with SU(2) gauge fields. It defines a line of partially constant physics (LPCP) by fixing R = m_h/m_W ≈ 1.5 and g_GF^2(m_W) = 0.5, with cutoffs Λ ranging from about 300 to 630 GeV. Along this line the authors compute the SM spectrum, the gradient-flow running gauge coupling, the BSM mass ratios m_H/m_W and m_A/m_W as functions of the unbroken hopping parameter κ1, and the finite-temperature susceptibility of the Higgs angular variable. The central claim, stated in the abstract and Section 5, is that for small BSM quartic couplings the lighter BSM scalar has m_H ~ 0.2 m_W and that this value is independent of the cutoff.

Significance. If established, the main result would be a non-perturbative counterpoint to typical tree-level expectations: a weakly coupled 2HDM could contain a BSM scalar considerably lighter than the W boson, which is relevant for model building and for future searches. The paper also provides useful first lattice data on the running coupling and on the screening mass in this model. Strengths include the use of two independent SM quantities to define the LPCP, the gradient-flow definition of the coupling, and the clear description of the parameter space sectors. The principal limitations are the absence of a quoted uncertainty for the central m_H ~ 0.2 m_W value and the lack of any finite-volume test for the BSM spectrum; as a result the claim is plausible but not yet demonstrated to the standard needed for a quantitative mass bound.

major comments (3)

[Abstract; Section 4, Fig. 4 (right); Section 5] The central claim m_H ~ 0.2 m_W is quoted without any uncertainty in the abstract and conclusion, and Fig. 4 (right) is not accompanied by a table of the plateau values or fit errors. Since the LPCP tuning quantities R and S in Table 1 carry relative errors of several percent, and the mass ratios are derived from correlator fits on finite statistics, the statement that the value is 'independent of the cutoff' needs numerical values with errors, e.g. m_H/m_W = 0.21(3) for each β. Without an explicit error budget and a χ²/dof for the plateau, the abstract overstates the precision of the result.
[Section 4, Table 1 and Eq. (3)] No finite-volume study of the BSM two-point functions is reported. For the claimed m_H/m_W ~ 0.2, the five LPCP points have m_H L ~ 1.7, 1.5, 1.3, 0.94 and 1.2 (using m_W L from Table 1). These values are in a regime where finite-volume corrections to a ground-state mass extracted from a zero-momentum correlator can be substantial, and the volumes change together with β (L = 28, 28, 32, 32, 48). A volume dependence test at one or two β values, or at least an estimate of the expected finite-volume shift, is a prerequisite for the claim that the plateau in Fig. 4 (right) is a cutoff-independent physical mass rather than a finite-volume artifact.
[Section 3 and Section 4, Fig. 4 (left)] The LPCP fixes only R and g_GF, while the BSM bare couplings are held fixed, and the paper assumes that the BSM sector is 'mostly insensitive' to SM physics. The evidence shown is a coarse monitoring of R and S under the κ1 scan. Table 1 shows that R varies from 1.462(53) to 1.527(80), so the line is not tuned exactly to R = 1.5. If m_H/m_W depends on the residual deviations from the LPCP or on R itself, the plateau could be an artifact of an incompletely tuned constant-physics line. The authors should quantify the insensitivity, for example by varying η3 or R at one β and demonstrating that m_H/m_W shifts by less than the quoted statistical error.

minor comments (5)

[Abstract; Fig. 1; Conclusion] Several typographical errors should be corrected: 'bellow' in the abstract should be 'below', 'cutodial' in the Fig. 1 caption should be 'custodial', 'scaning' in the conclusion should be 'scanning', and 'LCPC' in the discussion of Fig. 4 should be 'LPCP'.
[References] Reference [12] has a formatting error: '1604 [1410.2740]' should be '1604 (2016) [arXiv:1410.2740]' or similar.
[Fig. 4 caption] The left panel caption should explicitly define which quantities are plotted (R and S) and explain the marker types and the meaning of the shaded bands, so that the figure is self-contained.
[Section 4, screening mass discussion] The sentence reporting m_screen ≈ 50 GeV should carry a caveat in the main text (not only in footnote 3) that the screening mass depends on the assumed Yukawa form and on the scheme, since the text later uses this value without further qualification.
[Section 2, Eq. (2)] The notation 2μ^2 Tr(Φ1†Φ2) for the bare mass-mixing term is easily confused with the renormalization scale μ used later; renaming this bare parameter, e.g. as μ12^2, and stating its relation to the parameters in Eq. (1) would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the BSM mass ratios are measured outputs of a scan after fixing the LPCP, and no fitted parameter is renamed as a prediction.

full rationale

The paper's central claim, m_H ~ 0.2 m_W independent of the cutoff, is a measured output rather than an input. The line of partially constant physics is defined by fixing R = m_h/m_W and g_GF^2(m_W) = 0.5 (Eq. 5), while the BSM bare couplings kappa1, eta1, eta3, eta4, eta5 are held fixed. The BSM spectrum is then extracted from two-point functions (Eq. 3) at different kappa1 values, and the resulting m_H/m_W and m_A/m_W ratios are reported as data in Fig. 4. Nothing in the defining equations forces m_H/m_W to equal 0.2; the plateau is an empirical feature of the data. The only calibrated step in the paper is matching the perturbative running coupling to the Lambda = 628 GeV curve at mu ~ 3.5 m_W, which is explicitly disclosed and is not used to set or infer m_H/m_W. The screening-mass fit from the Yukawa potential is also scheme-dependent and disclosed as such. The self-citations present in the paper (e.g., refs. [13] and [23]) are used for physics motivation or as methodological background, not as load-bearing support for the central spectrum result. Possible concerns about finite-volume floors or incomplete tuning of the line of constant physics are correctness or systematic-error issues, not circularity. The derivation chain is therefore self-contained in the relevant sense.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The LPCP and spectrum analysis use standard lattice methods, but the central claim rests on hand-chosen BSM couplings and on operator/phase identifications carried over from tree-level perturbation theory. No new particles, forces, or conserved quantities are introduced; the BSM scalars H, A, H+/- are part of the 2HDM Lagrangian, and the screening mass is a fitted parameter, not a new entity.

free parameters (4)

kappa1 (unbroken hopping parameter) = 0.1245 (fixed), scanned up to kappa_c1
Sets the mass scale of the BSM doublet; the claimed lower bound m_H ~ 0.2 m_W is obtained at the endpoint of this scan, so the result is conditional on this parameter choice.
BSM quartic couplings eta1, eta3, eta4, eta5 = 0.003, 0.002, 0.0001, 0.0001
Chosen small by hand so the BSM sector stays mostly insensitive to SM physics and no LPCP retuning is needed; the paper explicitly states there is no fundamental reason for this choice and O(1) couplings are needed for a strong first-order EWPT.
kappa2 and eta2 per beta (LPCP tuning) = Table 1: 0.13175-0.129985 and 0.00338-0.002737
Adjusted at each of five beta values to satisfy R ~ 1.5 and S = 1.0; these are tuned inputs, not predictions, and the imperfect tuning (R errors up to ~0.09) propagates into all derived mass ratios.
m_screen (Yukawa screening mass) = 0.6243(28) m_W
Obtained by fitting a Yukawa form e^{-mr}/r to the IR running coupling; the authors note the definition is scheme-dependent, so it is a fitted quantity, not a parameter-free prediction.

assumptions (5)

domain assumption The inert/custodial limit defined by mu12 = lambda6 = lambda7 = 0 and lambda4 = lambda5 has the same global symmetry structure as the SM.
Sec. 1 defines the model; all numerical results apply to this restricted subspace of the general 2HDM, not to the full model.
domain assumption The H12 phase, where both Higgs fields acquire vevs, is excluded from phenomenological consideration because it spontaneously breaks custodial symmetry and contains massless Goldstone bosons.
Sec. 2; this justifies restricting the LPCP scan to sector H2.
domain assumption The interpolators S22, W22, S12, W12 in Eq. (3) excite the states identified as h, W, H, A/H+/- with the corresponding perturbative quantum numbers.
Sec. 2; the mass assignments m_H = m_S12^4, m_A = m_H+/- = m_S12^j rely on this operator-state correspondence, which is standard but not proven non-perturbatively.
standard math The gradient-flow coupling defined in Eq. (5) with the perturbative prefactor 128 pi^2/9 provides a valid renormalized running coupling.
Sec. 3, following Luscher-Weisz; standard in lattice gauge theory, but the conversion uses perturbative matching.
domain assumption The confinement and Higgs phases of the single-doublet Higgs-gauge system are analytically connected.
Footnote 1; used to justify treating the finite-T transition in the 2HDM similarly to the single-Higgs case.

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Cite this review

Pith. "Pith review of Progress in lattice simulations for two Higgs doublet models." pith.science (2026). https://pith.science/paper/DS3D5PAO

@misc{pith2026241213896,
  author       = {Pith},
  title        = {Pith review of: Progress in lattice simulations for two Higgs doublet models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DS3D5PAO}},
  note         = {Machine review of arXiv:2412.13896}
}

abstract

The custodial Two-Higgs-Doublet-Model with SU(2) gauge fields is studied on the lattice. This model has the same global symmetry structure as the Standard Model but the additional Higgs field enlarges the scalar spectrum and opens the possibility for the occurrence of spontaneous symmetry breaking of the global symmetries. Both the spectrum and the running of the gauge coupling of the custodial 2HDM are studied on a line of constant Standard Model physics with cutoff ranging from 300 to 600 GeV. The lower bounds of the realizable masses for the additional BSM scalar states are found to be well bellow the W boson mass. In fact, for the choice of quartic couplings in this work the estimated lower mass for one of the BSM states is found to be about $\sim 0.2m_{W}$ and independent of the cutoff.

Figures

Figures reproduced from arXiv: 2412.13896 by the authors.

**Figure 1.** Summary of the cutodial 2HDM parameter space with the global symmetries and spectrum content for each of the sectors 𝐻0, 𝐻1, 𝐻2, 𝐻12. In the case of the inert models, it is possible to divide the parameter space in four different regions where none, one, or both scalar fields are in the Higgs phase [10]. In the perturbative formulation, using the vacuum expectation values, 𝑣𝑖 , for each scalar field this corresponds… view at source ↗

**Figure 2.** Data from table 1: physical conditions 𝑅 and 𝑆 for the selected points in the line of constant SM physics as a function of 𝑎𝑚𝑊 (left). For decreasing 𝑎𝑚𝑊, each points has a corresponding increasing 𝛽 value. The lattice cutoff, Λ = 1/𝑎, estimated from 𝑎 = 𝑚ˆ 𝑊/𝑚 phys 𝑊 is shown on the right as a function of 𝛽. The gradient flow running of the gauge coupling was computed for each point in table 1. The 5 [PITH_FULL_IM… view at source ↗

**Figure 3.** Running gauge coupling 𝑔 2 𝐺𝐹 (𝜇) as a function of 𝜇/𝑚𝑊 from lattice simulations along the LPCP in table 1. Only scales with √ 8𝑡/𝑎 > 2 are shown. The perturbative result is matched to the curve corresponding to Λ = 628 GeV. The gauge coupling from the Yukawa potential is fitted to the infrared region, with the estimated screening mass 𝑚screen = 0.6243(28)𝑚𝑊. In fig. 3 the perturbative result, 𝑔 2 PT is also shown f… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Left: Standard Model conditions for simulations at different 𝜅1 with the remaining couplings defined in table 1. Right: mass ratio between the BSM states, 𝑚𝑆 4 12 = 𝑚𝐻 , 𝑚𝑆 𝑗 12 = 𝑚𝐴 = 𝑚𝐻± , and the W boson mass, 𝑚𝑊, for the same points at different 𝜅1 below (𝐻12). The…

**Figure 5.** Figure 5: Finite temperature dependence of the ratio O (𝑇, 𝑔)/O (0, 𝑔) (left) and 𝜒O (𝑇, 𝑔)/𝜒O (0, 𝑔) (right) for O = 𝐿𝛼2 . The susceptibility is shown only for the two larges 𝛽 values and for different spatial volumes are shown. in the peak of the susceptibility indicates that,…

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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