Heavy Vector Triplet Framework

• The Heavy Vector Triplet (HVT) framework is a model-independent parametrization for TeV-scale spin-1 bosons that extend the Standard Model with both neutral and charged states.
• It employs a simplified effective Lagrangian with key parameters (gV, c_H, c_F) to map weakly-coupled and composite Higgs models through benchmark scenarios.
• The framework informs collider phenomenology by predicting production via Drell–Yan and vector boson fusion, guiding experimental strategies and limits at the LHC.

The Heavy Vector Triplet (HVT) framework provides a model-independent parametrization for new TeV-scale resonances transforming as an $\mathrm{SU(2)_L}$ triplet of spin-1 bosons with zero hypercharge. These states, denoted $V^a_\mu$ ($a=1,2,3$), encompass both neutral ($Z'$) and charged ($W'^\pm$) vector bosons, and arise naturally in both weakly-coupled and strongly-coupled extensions of the Standard Model (SM), including composite Higgs, extended gauge, and Higgless models. The HVT approach is characterized by a simplified effective Lagrangian, benchmark scenarios mapping to ultraviolet (UV) completions, analytic control over phenomenology, and tight connections to experimental searches at the LHC and future colliders.

1. HVT Simplified Model Lagrangian and Parametric Structure

The core of the HVT framework is the model-independent dimension-4 Lagrangian, which extends the SM by a real triplet $V^a_\mu$ with interactions to SM currents and Higgs doublet. In standard notation (Pappadopulo et al., 2014, Baker et al., 2022):

$$\mathcal{L}_V \supset -\frac{1}{4} D_{[\mu}V_{\nu]}^a D^{[\mu}V^{\nu]a} + \frac{1}{2} m_V^2 V_\mu^a V^{\mu a} + i\,g_V c_H V_\mu^a H^\dagger \tau^a \! \overset{\leftrightarrow}{D}{}^\mu\! H + \frac{g^2}{g_V} c_q V_\mu^a \sum_q \bar q_L \gamma^\mu \tau^a q_L + \frac{g^2}{g_V} c_\ell V_\mu^a \sum_\ell \bar \ell_L \gamma^\mu \tau^a \ell_L$$

where $H$ is the SM Higgs doublet, $\tau^a = \sigma^a/2$ are SU(2) generators, $g$ is the SM weak coupling, and $D_\mu$ incorporates mixing with SM $W$ bosons. The parameters are:

• $g_V$: overall strong-sector coupling (benchmark values $g_V=1$ for weakly-coupled, $g_V=3$ for composite models)
• $c_H$: controls $V$–Higgs–gauge mixing and bosonic partial widths ($V \to VV, Vh$)
• $c_q$, $c_\ell$: flavor-diagonal couplings to SM quarks and leptons; control fermionic partial widths ($V \to q\bar{q}, V \to \ell\ell$)
• $m_V$: physical heavy vector mass (typically $M_{V^\pm} \approx M_{V^0}$ up to small custodial-breaking effects)

For phenomenology, the combinations $g_V c_H$, $c_q / g_V$, $c_\ell / g_V$ determine the rates and branching ratios in various channels (Baker et al., 2022, Collaboration, 18 Jan 2026).

2. Mapping to UV Models and Benchmark Scenarios

Explicit embeddings of the HVT Lagrangian match to representative UV theories:

• Model A (gauge extensions): $g_V=1$, $c_F=1$, $c_H \simeq -g^2 / g_V^2$; moderate bosonic branching fraction.
• Model B (composite Higgs): $g_V=3$, $c_F=1$, $c_H \sim 1$; bosonic decay modes dominate, fermionic branching suppressed.
• Higgless/composite scenarios: $c_F \ll 1$, $c_H \sim 2$, $g_K = 1 / g_V$; nearly 100% diboson branching.

Relevant parameters and decay patterns for benchmark points are shown below (Pappadopulo et al., 2014, Obikhod et al., 2023):

Parameter	Model A (gauge)	Model B (composite)
$g_V$	$1$	$3$
$c_H$	$-g^2/g_V^2$	$1$
$c_F$	$1$	$1$

Distinct VBF-favored benchmarks (with $c_q = c_\ell = 0$ for purely bosonic, or with nonzero $c_\ell$ for di-lepton final states) are defined to optimize LHC sensitivity (Baker et al., 2022).

3. Production Mechanisms and Mass Dependence

HVT states are produced via:

• Drell–Yan (DY): $q\bar{q} \to V$; cross-section scales as $(c_q / g_V)^2$ and falls rapidly at high $M_V$ due to parton luminosity suppression.
• Vector Boson Fusion (VBF): $VV \to V$; cross-section scales as $(g_V c_H)^2$ and grows relative to DY at large $M_V$, eventually dominating for $M_V \gtrsim 1$–2 TeV in regions of parameter space with suppressed fermionic couplings.

Key relations:

$$\sigma_{\text{DY}} \propto \frac{\Gamma_{V \to q\bar{q}}}{M_V} \left. \frac{dL_{q\bar{q}}}{d\hat{s}} \right|_{\hat s = M_V^2}$$

$$\sigma_{\text{VBF}} \propto \frac{\Gamma_{V \to VV}}{M_V} \left. \frac{dL_{VV}}{d\hat{s}} \right|_{\hat s = M_V^2}$$

$$\frac{\Gamma_{V \to VV}}{\Gamma_{V \to q\bar{q}}} \sim \frac{g_V^4}{g^4} \frac{c_H^2}{c_q^2} \approx \frac{1}{12} \frac{M_V^4}{m_W^4} \quad \text{for } c_q \to 0$$

As $M_V$ increases, VBF becomes dominant: for $c_q / g_V \lesssim 0.1$ and large $g_V c_H$, $\sigma_{\rm VBF}/\sigma_{\rm DY}$ transitions from below unity ($M_V=1$ TeV) to above ($M_V=2$ TeV) (Baker et al., 2022, Obikhod et al., 2023).

4. Decay Channels, Branching Ratios, and Widths

HVT resonances exhibit decay patterns sharply dictated by $c_H$ and $c_F$:

• Fermionic widths: $\Gamma(V \to q\bar q),\, \Gamma(V\to \ell^+\ell^-)$, scale as $(g^2 c_{q,\ell}/g_V)^2 M_V$.
• Bosonic widths: $\Gamma(V \to VV),\, \Gamma(V\to Vh)$, scale as $(g_V c_H)^2 M_V$, with enhancement proportional to $M_V^4 / m_W^4$ at large masses.

Typical benchmarks yield nearly exclusive diboson branching for $c_q,\,c_\ell \ll 1$ (“VBF-DB”), or competitive di-lepton branching when $c_\ell \sim -3 g_V$ (“VBF-DL”) (Baker et al., 2022). Widths generally satisfy $\Gamma/M_V < 0.15$ for $M_V \gtrsim 1$ TeV. Finite-width effects are minimized by restricting analyses to the on-shell region (Pappadopulo et al., 2014).

5. Collider Phenomenology and Experimental Limits

Collider probes focus on di-boson ($WZ$, $WW$, $Wh$, $Zh$) and di-lepton channels, exploiting the unique HVT resonance topologies:

• Current limits (LHC, $\sim$140 fb$^{-1}$, 13 TeV): DY and VBF searches exclude up to $M_V \lesssim 1.2$–1.5 TeV (diboson, dilepton), with VBF sensitivity exceeding DY at high mass for VBF-favored points. Full exclusion contours in $(g_V c_H,\,c_\ell / g_V)$ show VBF as the only feasible search channel for $M_V \gtrsim 1.5$–2 TeV in large regions of parameter space (Baker et al., 2022, Collaboration, 18 Jan 2026).
• HL-LHC projections (14 TeV, 3 ab$^{-1}$): VBF reach will extend to $M_V \sim 2.5$–2.6 TeV, exceeding the DY sensitivity ($M_V \sim 1.7$–2 TeV) (Baker et al., 2022).
• CMS combination results (138 fb$^{-1}$): Model A (weak coupling) excludes $M_V < 5.5$ TeV, Model B (strong coupling) $M_V < 4.8$ TeV, with VBF-specific analyses excluding $M_V < 2.0$ TeV for pure bosonic coupling scenarios (Collaboration, 18 Jan 2026).

Experimental results are interpreted directly in terms of HVT parameter exclusions. Analytic mappings from $\sigma \times \mathrm{BR}$ limits to $(c_H,\,c_F)$ exclusion curves are implemented and public tools provided (Pappadopulo et al., 2014).

6. Model Variations, Theoretical Constraints, and Future Directions

• Perturbative unitarity and sum rules: Relations among couplings must be respected to ensure tree-level unitary high-energy behavior. For pure “SM+$V'$+$h$” setups, detailed sum rules limit $\mathrm{BR}(W' \to WZ)\lesssim 2\%$; adding CP-odd scalars relaxes the bound and allows order-one diboson branching (Abe et al., 2016).
• Composite/Higgless scenarios: In such models, the HVT triplet arises as a gauge or chiral adjoint of $SU(2)_{L+R}$, with couplings fixed by the demand of perturbative unitarity $\mathcal{A}(W_L W_L \to W_L W_L)$. Associated multi-lepton signals from cascade decays provide highly distinctive signatures (Hernández et al., 2010, Hernandez, 2010, Hernández, 2011).
• Resonant cross-section dependence: The cross-section is largely insensitive to $c_F$ for $c_H$ fixed except in extreme limits; mass scaling ($M_V$) is the principal controlling factor, falling steeply with $M_V$ due to luminosity suppression (Obikhod et al., 2023).

Future collider searches at 100 TeV will extend mass reach well beyond 10 TeV, probing VBF-dominated regions and mapping the full $(c_H,c_F)$ space (Obikhod et al., 2023). Precise measurements of Higgs and dilepton couplings will constrain $c_H$ and $c_F$ respectively.

7. Summary and Impact on LHC Searches

The HVT framework, with its minimal set of parameters $(g_V c_H, c_q/g_V, c_\ell/g_V)$, delivers a predictive and robust context for interpreting heavy vector searches. The interplay of DY and VBF production and their mass dependence fundamentally shape the strategy for discovery, with VBF analyses becoming pivotal for masses above $\sim$1.5–2 TeV and suppressed fermionic couplings. Stringent exclusion limits from recent CMS combinations have set the benchmark for new resonance searches in the multi-TeV domain, cementing HVT as the standard template for both experimental analyses and theory-to-data mapping in new heavy vector boson phenomenology (Baker et al., 2022, Pappadopulo et al., 2014, Collaboration, 18 Jan 2026).