Colored Scalar Mediator in BSM Physics

• Colored scalar mediators are hypothetical spin-0 particles in various SU(3) representations that bridge Standard Model fields and BSM sectors.
• They appear in models such as GUTs, dark matter portals, and extended Higgs sectors, impacting collider, flavor, and electroweak observables.
• Experimental searches and theoretical constraints, including nondecoupling effects and loop-induced interactions, set stringent limits on their masses and couplings.

A colored scalar mediator is a hypothetical spin-0 particle transforming under the color SU(3) gauge group, responsible for mediating new interactions between Standard Model (SM) fields (such as quarks) and/or beyond-SM particles (such as dark matter or extended Higgs sectors). These states are a central feature in a variety of extensions to the SM, arising in scenarios including Grand Unified Theories (GUTs), minimal flavor violation, dark matter models with $t$-channel exchange, and precision Higgs phenomenology. Colored scalars commonly appear in either the fundamental (triplet), adjoint (octet), or higher SU(3) representations (e.g., sextet), and can carry additional weak and hypercharge quantum numbers, leading to diverse phenomenological consequences at high energy colliders, in flavor observables, and in cosmology.

1. Quantum Numbers, Field Content, and Representative Models

Colored scalar mediators span a range of gauge quantum numbers, with the most frequently encountered representations including:

Model Context	SU(3) × SU(2) × U(1)	Components and Properties
GUT/composite Higgs, MFV (Chakrabarty et al., 2020, Dorsner et al., 2012)	(8,2,½)	Color-octet, weak doublet
Leptoquark/Fermion mass, $B$ anomalies (Dorsner et al., 2013)	(3,2,7/6)	Color-triplet, weak doublet
Dark Matter portals (Garny et al., 2014, Ko et al., 2016)	(3,1,Y), (3,2,1/6)	Color-triplet, singlets/doublets
Di-Higgs/h→γγ models (Dorsner et al., 2012)	(8,2,½), (6,3,1/3)	Color octet/sextet

For example, the SU(3)$_c$ octet weak doublet $S\sim(8,2,1/2)$ introduces charged and neutral colored scalars,

$$S^a = \begin{pmatrix} S^{a+} \ \frac{1}{\sqrt{2}}(S^a_R + i S^a_I) \end{pmatrix},$$

where $a=1,...,8$ runs over color.

Field content invariably determines both gauge and Yukawa couplings: colored mediators may couple to quarks, leptons, or exotic fermions depending on the scenario. For instance, minimal flavor-violating color-octet scalars can couple to SM Higgs doublets as well as to gluons and electroweak bosons via gauge-invariant kinetic terms, while leptoquark-like colored doublets ($\Delta\sim(3,2,7/6)$) couple directly to SM quark-lepton currents via dimension-4 Yukawa terms.

2. Theoretical Constraints and Scalar Potential Structure

Any realistic model containing colored scalar mediators must address theoretical consistency through the constraints imposed by perturbativity, vacuum stability, and unitarity. The scalar–mediator sector is described by an extended renormalizable potential, e.g., for an $S\sim(8,2,1/2)$ color-octet doublet plus two Higgs doublets ($\phi_1, \phi_2$) (Chakrabarty et al., 2020): $$\begin{aligned} V(\phi_1,\phi_2,S) = &\, V_{2HDM}(\phi_1,\phi_2) + m_S^2 \text{Tr}[S^\dagger S] + \lambda_1 \text{Tr}[S^\dagger S]^2 + \lambda_2 \text{Tr}[S^\dagger S S^\dagger S] \ &+ \lambda_3 \phi_1^\dagger\phi_1 \text{Tr}[S^\dagger S] + \lambda_4 \phi_2^\dagger\phi_2 \text{Tr}[S^\dagger S] \ &+ (\kappa_1 \phi_1^\dagger S S^\dagger \phi_1 + \kappa_2 \phi_2^\dagger S S^\dagger \phi_2 + \kappa_3 \phi_1^\dagger S \phi_2 + h.c.) \ &+ (\eta_1 \text{Tr}[S^\dagger S S^\dagger \phi_1] + \eta_2 \text{Tr}[S^\dagger S S^\dagger \phi_2] + h.c.) \end{aligned}$$ with "Tr" over color and $V_{2HDM}$ the conventional 2HDM scalar potential.

Theoretical constraints require:

• Perturbativity: $$|\lambda_i|, |\kappa_i|, |\eta_i| < \mathcal{O}(4\pi)$$.
• Vacuum stability: All quartic couplings must enforce boundedness from below via copositivity conditions, e.g., $\lambda_1 > 0$, $\lambda_2 > 0$, and combinations like $\sqrt{\lambda_1 \lambda_2} + \lambda_3 + ... > 0$.
• Tree-level unitarity: Linear inequalities from the 2-to-2 scalar scattering matrix, e.g., $|\lambda_3 \pm \lambda_4| < 8\pi$, $|\kappa_1| < 4\pi$.

In GUT embeddings, the colored scalar typically resides in large representations (e.g., $45_H$ or $50_H$ of SU(5)), and its couplings to fermion mass matrices and proton decay operators are tightly constrained by symmetry and experimental limits (Dorsner et al., 2013, Dorsner et al., 2012).

3. Loop-Induced and Effective Interactions

Colored scalar mediators induce a wide range of effective interactions at one loop or tree-level, depending on the process and model context.

Electroweak & Higgs physics: In color-octet–extended 2HDM, the mediators contribute at one-loop to anomalous vertices such as $H^+ \to W^+ Z$ and $H^+ \to W^+ \gamma$ (Chakrabarty et al., 2020). The generic Lorentz structure is: $$M = i g m_W V^{\mu\nu} \epsilon^*_\mu(W) \epsilon^*_\nu(V),$$

$$V^{\mu\nu} = F_V g^{\mu\nu} + G_V \frac{p_W^\mu p_V^\nu}{m_W^2} + i H_V \frac{\epsilon^{\mu\nu\rho\sigma} p_{W\rho} p_{V\sigma}}{m_W^2},$$

with $F_V, G_V, H_V$ further split into 2HDM (colorless) and colored–octet pieces.

In Higgs signal strength modifications, colored scalars can yield order-unity corrections to $h\to\gamma\gamma$ and $h\to Z\gamma$ widths, as well as substantial enhancements to di-Higgs ($gg\to h h$) production rates (Dorsner et al., 2012).

Dark matter and flavor: For dark matter models, colored scalar mediators induce $t$-channel interactions such as $\chi\chi\to q\bar q$ for dark matter $\chi$, generating both thermal relic annihilations and $s$- and $t$-channel collider signatures (Garny et al., 2014, Ko et al., 2016). The effective Lagrangian for Dirac $\chi$ can be schematically given by: $$\mathcal{L} \supset \lambda_L \tilde{q}_L^\dagger \bar\chi P_L q + \lambda_R \tilde{q}_R^\dagger \bar\chi P_R q + h.c.$$ Integrating out the mediator at tree-level yields four-fermion operators that map onto direct-detection-relevant nucleon currents.

In flavor physics, (3,2,7/6) colored doublets act as leptoquarks and can address anomalies in $B\to D^{(*)}\tau\nu$, provided they realize minimal-flavor-violating Yukawa structures and satisfy stringent constraints from $Z\to b\bar b$, $\tau\to\mu\gamma$, and $(g-2)_\mu$ (Dorsner et al., 2013).

4. Collider Phenomenology and Experimental Constraints

Colored scalar mediators are copiously produced at hadron colliders via QCD-driven processes such as $gg\to SS^\dagger$, $q\bar q\to SS^\dagger$ (pair production), and also via associated production or $t$-channel processes involving new physics particles (e.g., $\chi$-exchange for dark sector models) (Degrande et al., 2014, Garny et al., 2014). For benchmark “stop-like” triplet scalars of mass $m_3=500$ GeV, the NLO+PS cross section at $\sqrt{s}=13$ TeV LHC is $\sigma_{\mathrm{NLO}}(pp\to S S^\dagger)\simeq 0.5$ pb, with K-factors $K\sim1.2–1.6$, and PDF uncertainties at a few percent (Degrande et al., 2014).

Direct searches: LHC Run 2 and HL-LHC exclude colored scalar masses up to 700–800 GeV for states decaying to jets+MET, with higher reach for specific decay chains or larger couplings. Color-octet scalars in the (8,2,1/2) representation are strongly constrained by multi-jet and top-pair final states, typically requiring $M_S\gtrsim 700$–$800$ GeV (Chakrabarty et al., 2020).

Electroweak and flavor observables: Contributions to oblique parameters—especially $\Delta T$—arising from colored-scalar mass splittings are tightly constrained: e.g., $\Delta T$ must remain within $(0.07\pm0.12)$, forcing the splitting between neutral and charged scalar components to be moderate. Operators generated by colored mediators can induce proton decay, flavor-changing neutral currents, and electric dipole moments, demanding nontrivial alignment or suppression in model-building.

Dark matter and direct detection: Colored scalar mediators for dark matter induce nucleon-DM scattering via vector and scalar operators. Direct limits (e.g. from LUX/XENON) and the predicted relic density imply that, for Dirac or Majorana $\chi$, mediator and DM masses above 1–10 TeV are excluded unless the couplings are tuned or co-annihilation is invoked (Cahill-Rowley et al., 2015, Garny et al., 2014, Ko et al., 2016).

5. Nondecoupling, Signal Strengths, and Mass-Splitting Sensitivity

A salient property of colored scalar mediator contributions is their nondecoupling behavior in certain observables due to explicit dependence on mass splittings and trilinear scalar couplings. For example, the one-loop form factor $F_Z^{(8)}$ in $H^+\to W^+Z$ grows with the splitting $\Delta m_S = \sqrt{M_{S_I}^2-M_{S_R}^2}$ between the real and imaginary components of the octet doublet (Chakrabarty et al., 2020): $$F_Z^{(8)} \approx \frac{8}{16\pi^2 v \cos\theta_W}\left[ \lambda_{H^+S^-S_R} f_1(r) + \lambda_{H^+S^-S_I} f_2(r) \right],$$ with $r=M_{S_I}^2/M_{S_R}^2$. Thus, sizable mass splitting can enhance $F_Z^{(8)}$ by factors up to 2–3 compared to the minimal 2HDM value; as a result, the loop-induced $BR(H^+\to W^+Z)$ may reach order $1\%$ for $M_{H^+} \gtrsim 400$ GeV.

In Higgs physics, colored scalars with $O(1)$ couplings to $h$—as required by naturalness—can both accommodate $h\to\gamma\gamma$ signal strength excesses and predict correlated deviations in $h\to Z\gamma$ and $gg\to hh$ (Dorsner et al., 2012). Future precision measurements, particularly in $h\to Z\gamma$ and di-Higgs channels, will further test or constrain these scenarios.

6. Model Discrimination, Future Prospects, and Observability

Current and next-generation colliders (HL-LHC, FCC-hh) substantially extend sensitivity to colored scalar mediators.

• Discrimination: Kinematically sensitive observables (e.g., $M_{WZ}$, $\Delta\phi(\!W,Z)$ in $H^+\to W^+Z$), high-$p_T$ tails, and event rate scaling can distinguish loop-induced colored mediated signals from tree-level or colorless models. For example, the $G_Z$ term alters $WZ$ kinematics in ways not replicated by tree-level triplet scenarios (e.g., the Georgi–Machacek model).
• Discovery prospects: For $F_Z^{(8)}\sim 10^{-2}$, $O(10^2$–$10^3)$ VBF $H^+\to WZ$ events are expected at the HL-LHC for $3\,\text{ab}^{-1}$ before cuts, and up to $10^4$ with $30\,\text{ab}^{-1}$ at a 100 TeV FCC (Chakrabarty et al., 2020).
• Theoretical reach: Robust upper limits on both DM and mediator masses are set by unitarity plus relic density constraints; these scale as $$M^* \sim (4\pi/\lambda_{\text{max}})\sqrt{N_c N_f/3}$$ and, unless co-annihilation occurs, lie in the range $5$–$45$ TeV (Cahill-Rowley et al., 2015).

A plausible implication is that colored scalar mediators—through their nondecoupling behavior, sensitivity to the scalar potential structure, and strong collider signatures—serve as powerful probes of high-scale unification theories, minimal flavor violation, and new dark sectors.

7. Outlook and Theoretical Significance

Colored scalar mediators offer a rich testbed for experimental and theoretical progress in particle physics. They provide a direct link between high-scale symmetries (e.g., SU(5) or SO(10)), collider observables (e.g., dijet and $t\bar t$ resonances, Higgs precision channels, mono-X + MET), and precision flavor/dark matter constraints. Their paper has established the essential role of loop-induced processes, the importance of nondecoupling effects, and the need for a multi-pronged approach combining direct searches, indirect detection, and flavor observables. Ongoing and future experimental data, together with refinements in automated higher-order calculations (Degrande et al., 2014), will be crucial in either discovering these mediators or closing the remaining viable parameter space.