Scalar–Higgs Mixing Angle

• Scalar–Higgs mixing angle is the rotation parameter that defines the admixture of Higgs doublet and additional scalar states into physical mass eigenstates after electroweak symmetry breaking.
• It controls the rescaling of couplings to Standard Model particles, thereby influencing production rates and decay patterns, which are key to interpreting collider search results.
• This parameter is central to theoretical models and experimental searches, impacting dark matter interactions, vacuum stability, and prospects for new physics beyond the Standard Model.

A scalar–Higgs mixing angle quantifies the rotation between two or more neutral scalar fields—most commonly a Standard Model-like Higgs and one or more additional scalar degrees of freedom—into the physical mass eigenstates observed after electroweak symmetry breaking. This angle (or, with more fields, a set of angles or a mixing matrix) determines the admixture of the “Higgs doublet” and “singlet” (or other sector) components in the observed scalar resonances. The mixing angle controls the couplings of the physical scalars to Standard Model fields, their production rates, decay patterns, and the phenomenology of direct searches, indirect constraints, and precision fits at colliders.

1. Theoretical Formulation and Diagonalization

The scalar–Higgs mixing angle, generically denoted $\alpha$ (or $\theta$ in some literature), arises from diagonalizing the (typically 2×2) mass-squared matrix of neutral scalars with symmetry-allowed bilinear couplings. In the simplest “Higgs portal” extension, where a real singlet $S$ is added to the Standard Model Higgs doublet $H$, the most general renormalizable scalar potential contains interaction terms,

$$V(H, S) = \mu_H^2 |H|^2 + \lambda_H |H|^4 + \mu_S^2 S^2 + \lambda_S S^4 + \kappa |H|^2 S^2 + \mu_{HS} |H|^2 S.$$

After spontaneous symmetry breaking $H^0=(v + h_{\text{SM}})/\sqrt{2}$, the quadratic part in $(h_{\text{SM}}, S)$ can be written as

$$\mathbf{M}^2 = \begin{pmatrix} m_h^2 & \Delta m^2 \ \Delta m^2 & m_S^2 \end{pmatrix}$$

with $\Delta m^2 \propto \mu_{HS} v + \kappa v^2$. Diagonalization is performed via an orthogonal rotation through the mixing angle $\alpha$: $$\begin{pmatrix} h_1 \ h_2 \end{pmatrix} = \begin{pmatrix} \cos \alpha & -\sin \alpha \ \sin \alpha & \cos \alpha \end{pmatrix} \begin{pmatrix} h_{\text{SM}} \ S \end{pmatrix}$$ yielding the mass eigenstates $h_1$, identified with the observed 125 GeV scalar, and $h_2$, a heavier (or lighter) scalar. The mixing angle is determined by

$\tan 2\alpha = \frac{2 \Delta m^2}{m_h^2 - m_S^2}$

or $$\alpha = \frac{1}{2} \arctan\left[\frac{2 \Delta m^2}{m_h^2 - m_S^2}\right]$$.

The same structure generalizes to Higgs doublet extensions, triplet models, or models allowing scalar–pseudoscalar mixing (see below).

2. Physical Consequences of Scalar–Higgs Mixing

2.1. Modification of Couplings

The mixing angle controls the relative content of the doublet (Higgs-like) and non-doublet sectors in each mass eigenstate. Crucially, all Standard Model–like couplings of $h_1$ and $h_2$ (i.e., to $WW$, $ZZ$, and SM fermions) are rescaled by

$$g_{h_1XX} = \cos\alpha \, g_{\mathrm{SM}}, \quad g_{h_2XX} = \sin\alpha \, g_{\mathrm{SM}}$$

for $X=W,\,Z,\,f$ [(Mou et al., 2015), (Cheung et al., 2015), (Bertolini et al., 2012), (Falkowski et al., 2015)]. This “universal rescaling” is a robust signature unless further scalar mixing or loop effects intervene.

2.2. Production and Decay Rates

For a heavy scalar $h_2$, both production cross sections and partial decay widths to SM particles scale as $\sin^2\!\alpha$ relative to a pure-SM Higgs of the same mass,

$$\sigma_{\text{VBF}}(pp \to h_2) \simeq \sin^2\alpha \, \sigma_{\text{SM}}(m = m_{h_2}),$$

$$\Gamma(h_2 \to VV) = \sin^2\alpha \, \Gamma_{\text{SM}}(m_{h_2} \to VV), \quad V = W, Z$$

(Mou et al., 2015). These scalings dominate the phenomenology of vector-boson fusion (VBF) searches and diboson final states.

2.3. Allowed Parameter Ranges

Global fits to LHC Higgs signal strengths place strong model-independent bounds on $\alpha$. For pure singlet–doublet mixing,

$$\cos\alpha \gtrsim 0.86 \quad \Rightarrow \quad |\sin\alpha| \lesssim 0.51 \quad \text{(95\% CL)},$$

and the non-SM contribution to the total width must satisfy $\Delta\Gamma_{\rm tot} \lesssim 1.9$ MeV (Cheung et al., 2015). Indirect constraints from electroweak precision observables (e.g., oblique $S,T,U$ parameters) and vacuum stability give comparable or (for heavy $h_2$) stronger limits, especially for $m_{h_2} \gtrsim 400$ GeV [(Mou et al., 2015), (Falkowski et al., 2015)].

3. Experimental Probes and Collider Reach

3.1. Direct Searches via VBF and Diboson Modes

The clearest direct probe of a nonzero scalar–Higgs mixing angle is the search for an extra Higgs-like scalar in VBF or diboson channels ($h_2 \to WW,\,ZZ$), where the signal rate is dictated by $\sin^2\alpha$ suppression. At 14 TeV LHC, projected $95\%$ CL exclusion with $300~\text{fb}^{-1}$ for $\sin^2\alpha = 0.04$ extends to

$$\begin{array}{ccc} & 300~\text{fb}^{-1} & 3000~\text{fb}^{-1} \ \text{WW}\to\ell\nu\ell\nu & 539~\text{GeV} & 937~\text{GeV} \ \text{ZZ}\to 2\ell 2\nu & 475~\text{GeV} & 790~\text{GeV} \end{array}$$

with exclusion reach lowering to $270$ ($459$) GeV for $\sin^2\alpha=0.01$ at $300$ ($3000$) fb$^{-1}$ (Mou et al., 2015).

3.2. Comparison to Indirect Constraints

Global fits to Higgs couplings (from Run I data) constrain $\sin^2\alpha \lesssim 0.20$ at $95\%$ CL; $S,T,U$ precision observables tighten this to $\lesssim 0.1$ for $m_{h_2} \gtrsim 400$ GeV (Mou et al., 2015). The direct VBF searches thus probe $\sin^2\alpha$ at levels far below indirect limits for $m_{h_2} \gtrsim 500$ GeV, extending sensitivity to multi-hundred-GeV scalars.

3.3. Effect of Systematics

Production cross section uncertainties (PDFs, scale variation) in $\sigma_\mathrm{VBF}^\mathrm{SM}(m)$ are $5$–$10\%$; NLO $k$-factors further modify the LO rates but are largely canceled in $S/B$ ratios. Omission of $h_2\to h_1h_1$ decays and neglect of systematic errors in $S,B$ may weaken the exclusion reach by $\mathcal{O}(20\%)$ (Mou et al., 2015).

4. Generalization to Multiple Scalars and CP-Mixing

4.1. Multi-Scalar Mixing

Models with more than one additional scalar (doublet, triplet, or singlet) require rotation matrices (e.g., $O(\theta_1,\theta_2,...)$ or $R(\alpha_1,\alpha_2,\alpha_3)$). For example, in the CP-even sector of the Higgs Triplet Model, the mixing angle $\alpha$ is determined by

$\tan 2\alpha = \frac{2 M_{12}}{M_{11} - M_{22}}$

where $M_{ij}$ are the basis-dependent mass matrix elements (Akeroyd et al., 2010). Maximal mixing ($|\alpha|=45^\circ$) is achieved when diagonal entries are degenerate, $(M_{11}=M_{22})$, and is phenomenologically motivated by $h_1, h_2$ near-degeneracy and enhanced discovery prospects.

4.2. CP-Violating Mixing

In CP-violating models, the scalar–Higgs mixing angle becomes part of a larger space of parameters. For instance, in models with scalar–pseudoscalar mixing, the physical mass eigenstate is

$h = \cos\alpha\, H + \sin\alpha\, A$

where $H$ is CP-even and $A$ is CP-odd (Freitas et al., 2012). In these models, $g_{hVV}$ is suppressed by $\cos\alpha$ (as $A$ does not couple at tree level), and the CP-odd component is restricted primarily by the observed $ZZ, WW$ production rates and angular distributions.

5. Phenomenological and Cosmological Implications

5.1. Impact on Dark Matter and Cosmology

The mixing angle is critical for Higgs-portal models explaining dark matter. The $h$–portal-mediated DM annihilation and direct-detection cross sections scale as $\cos^2\alpha$. Current LHC data force $\cos\alpha \gtrsim 0.86$, severely reducing allowed singlet–portal parameter space (Cheung et al., 2015). In Higgs-inflation models with a dark Higgs, the same parameter $\alpha$ controls the running of the Higgs quartic coupling, enabling inflection-point inflation and a large tensor-to-scalar ratio $r\sim0.08-0.1$ for $\alpha\sim0.02$–$0.04$ (Kim et al., 2014).

5.2. Vacuum Stability and RG-running

Nonzero scalar–Higgs mixing shifts the boundary value of the Higgs quartic $\lambda_H$ at low scale. In “restored” vacuum-stable regions, a tree-level uplift in $\lambda_H$ due to mixing can stabilize the electroweak vacuum up to the Planck scale, given the observed top mass [(Falkowski et al., 2015), (Kim et al., 2014)]. The allowed window in $(m_{h_2}, \sin\alpha)$ is consistent with both collider bounds and vacuum stability only for small to moderate mixing, $|\sin\alpha| \lesssim 0.2$–$0.4$ for $m_{h_2}\sim 200$–$400$ GeV.

6. Renormalization and Scheme Dependence

The scalar–Higgs mixing angle must be renormalized at one-loop and higher orders. Several renormalization schemes are in use:

• $\overline{\mathrm{MS}}$ scheme: Subtracts UV poles only; suffers from gauge dependence and instabilities near mass degeneracy.
• Physical on-shell (OS) schemes: Fix $\alpha$ from physical S-matrix ratios (e.g., $$M(H_1\to ZZ)/M(H_2\to ZZ)|_{\mathrm{NLO}} = \tan\alpha$$ or directly from the off-diagonal self-energies). The counterterm is then

$$\delta\alpha = \frac{\Sigma_{12}(M_2^2) + \Sigma_{12}(M_1^2)}{2(M_1^2 - M_2^2)}$$

ensuring UV-finiteness and numerical stability (Denner et al., 2018).

• Rigid-symmetry/BFM-inspired schemes: Enforce symmetry by relating mixing counterterms to background-field gauge-invariant quantities, yielding process-independent and well-behaved $\delta\alpha$.

For well-separated scalars, all schemes yield small differences at NLO for typical observables; near-degeneracy, the OS/BFM methods are preferred for stability and gauge invariance.

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Table: Summary of Scalar–Higgs Mixing Angle Parameterization and Key Phenomenological Impacts

Model class	Mass eigenstates, mixing parameter	Coupling scaling $h_1$	Collider constraints
Higgs + singlet	$h_1 = \cos\alpha\,h_\mathrm{SM} - \sin\alpha\, S$	$\cos\alpha$	$\cos\alpha \gtrsim 0.86$, $|\sin\alpha| \lesssim 0.51$ (Cheung et al., 2015); VBF: $\sin^2\alpha > 0.04$ excluded for $m_{h_2}\leq 939$ GeV (Mou et al., 2015)
Higgs + CP-odd admixture	$h = \cos\alpha\, H + \sin\alpha\, A$	$\cos\alpha$	Pure CP-odd ($\alpha=90^\circ$) excluded at %%%%90$H$91%%%%; 95% CL bounds $\alpha \lesssim$0.7 at 14 TeV LHC, projected (Freitas et al., 2012)
Higgs triplet model	$H_1 = \cos\alpha\, h^0 + \sin\alpha\, \Delta^0$	$\cos\alpha$	Unitarity+EW: $\alpha \sim 2v_t/v \ll 1$ (alignment) (Das et al., 2016)
Two Higgs doublets (CP-even basis)	$H_1, H_2$ via $\alpha$	$\cos\alpha, \sin\alpha$ (SM fields)	Fit-dependent; typically $|\alpha| \lesssim 0.1$–$0.2$ (Denner et al., 2018)

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7. Outlook and Future Prospects

High-luminosity $pp$ and $e^+e^-$ colliders will further cover the allowed region in $(\sin^2\alpha, m_{h_2})$. The HL-LHC is projected to exclude $h_2$ scalars up to nearly 1 TeV for $\sin^2\alpha \gtrsim 0.04$ in VBF+dilepton channels (Mou et al., 2015). The precision of $\alpha$ determination will reach the percent level with future coupling measurements, directly constraining or discovering singlet–like and triplet–like scalars, with clear implications for electroweak baryogenesis, Higgs inflation, and the structure of scalar extensions of the Standard Model.

Model-independent and robust renormalization prescriptions for $\alpha$ are established, with "OS" and BFM-symmetry schemes recommended for precision calculations (Denner et al., 2018). The scalar–Higgs mixing angle thus remains a central parameter bridging collider phenomenology, cosmology, and new physics searches.