Ultraheavy Diquark Scalar

• Ultraheavy diquark scalars are hypothetical, ultraheavy color‐sextet particles that couple to symmetric pairs of right-handed up-type quarks with masses typically above 7 TeV.
• They are produced via strong QCD interactions at proton–proton colliders and exhibit unique resonance and cascade decay topologies, often involving vectorlike quarks.
• Nonperturbative QCD effects, advanced jet analysis, and machine learning techniques are essential for determining their mass shifts and enhancing signal detection at the LHC.

An ultraheavy diquark scalar is a hypothetical color-sextet ($\mathbf{6}$ of $\mathrm{SU}(3)_c$), weak-singlet, electric-charge $+4/3$ complex scalar particle that couples to a symmetric pair of right-handed up-type quarks. Such states arise in various extensions of the Standard Model, where they serve as mediators for resonant multi-jet or cascade signatures at high-energy hadron colliders. Their defining properties include ultraheavy masses, typically in the 7–10 TeV regime or beyond, a strong QCD production mechanism through valence up-quark fusion, and characteristic decay topologies often involving vectorlike quarks. Their theoretical underpinning spans QCD sum rule, Schwinger–Dyson, and effective field theory approaches, while experimental probes rely on advanced jet, multi-jet, and hadronic event analysis techniques at the LHC.

1. Quantum Numbers and Renormalizable Interactions

Ultraheavy diquark scalars, generically denoted $S_{uu}$ in the literature (Dobrescu, 2019, Dobrescu, 6 Nov 2024, Dobrescu et al., 2018, Duminica et al., 21 Mar 2025), transform as

$S_{uu} \sim (6,1,+4/3)$

under $SU(3)_c \times SU(2)_L \times U(1)_Y$, and carry baryon number $B=2/3$. The renormalizable gauge and flavor-conserving Lagrangian is

$$\mathcal{L} \supset \frac12\,y_{uu}\,K^n_{ij}\,S^n_{uu}\,\overline{u_{R\,i}}\,u^c_{R\,j} + \frac{1}{2}\,y_{\chi\chi}\,K^n_{ij}\,S_{uu}^n \overline{\chi_{R,i}}\,\chi^c_{R,j} + \mathrm{h.c.}$$

where $K^n_{ij}$ are the symmetric, normalized Clebsch–Gordan coefficients for the color sextet, and $y_{uu}$, $y_{\chi\chi}$ are dimensionless Yukawa couplings. The field $\chi$ denotes a vectorlike up-type quark (color triplet, weak singlet, charge $+2/3$), and flavor-diagonal and decay-stabilizing symmetries ($Z_2$, $B-L$) are imposed to suppress proton decay and minimize flavor violation.

2. Nonperturbative Mass and QCD Dynamics

The mass of an ultraheavy diquark scalar is not simply given by the sum of constituent quark pole masses. The nonperturbative QCD dressing of the scalar diquark is governed by the Schwinger–Dyson equation in Landau gauge (Imai et al., 2014): $$\Sigma^2(p^2) = m_0^2 + \Delta^{(4)}[p;\Sigma] - \Delta^{(3)}[p;\Sigma]$$ where $\Sigma(p^2)$ is the mass function, $m_0$ the bare mass, and $\Delta^{(4)}$, $\Delta^{(3)}$ are loop functionals containing the RG-improved QCD coupling and diquark size form factors,

$$f_\Lambda(p^2) = \left(\frac{\Lambda^2}{p^2 + \Lambda^2}\right)^2,\qquad \Lambda = 1/R.$$

In the ultraheavy regime ($m_0 \gg \Lambda_{\mathrm{QCD}}$), the resulting physical diquark mass function saturates as

$M(0) \simeq m_0 + (0.2-0.4)\,\mathrm{GeV},$

with the nonperturbative gluonic increment being highly sensitive to the diquark's spatial compactness ($R$): compact ($R\sim0.3$ fm) diquarks acquire larger shifts than extended ($R\sim1$ fm) ones. This universal, chiral-symmetry-independent QCD dressing is a defining property of colored bosonic composites in strongly interacting theories.

For heavy-light diquarks (e.g., $cq$, $bq$), QCD sum rule analyses (Wang, 2010) yield mass estimates of $M_S=1.77\pm0.08$ GeV for $cq(0^+)$ and $5.14\pm0.12$ GeV for $bq(0^+)$, with scalar diquark residues $\lambda_S$ in the range $0.43–0.91~\mathrm{GeV^2}$, but these are not ultraheavy in the sense relevant for collider searches.

3. Hadroproduction and Cross Sections at Colliders

Ultraheavy diquark scalars are predominantly produced via same-flavor up-quark fusion,

$u\,u \to S_{uu}$

at proton–proton colliders. The leading-order partonic cross section in the narrow-width approximation is (Dobrescu, 6 Nov 2024, Dobrescu, 2019, 0909.2666)

$$\hat\sigma_{uu\to S} = \frac{\pi}{6\,M_S^2}|y_{uu}|^2\,\delta(\hat s-M_S^2).$$

The total hadronic cross section, after PDF convolution, is

$$\sigma(pp\to S) = \frac{\pi}{6\,s\,M_S^2}|y_{uu}|^2\,\tau\,\frac{dL_{uu}}{d\tau}, \qquad \tau = \frac{M_S^2} s.$$

NLO QCD corrections yield a $K$-factor of $1.22$–$1.32$ (sextet) for $M_S=3$–10 TeV at $\sqrt{s}=14$ TeV (0909.2666), with residual PDF and scale uncertainties of $12$–$25\%$. Table of rates (for $y_{uu}=1$): | $M_S$ (TeV) | $\sigma_{\mathrm{NLO}}$ [fb], sextet, $14$ TeV | |-------------|-----------------------------------------------| | 3 | 3.7 | | 5 | 0.10 | | 8 | 0.01 | | 10 | $6.3\times 10^{-4}$ | All rates scale as $y_{uu}^2$.

The transverse momentum spectrum peaks at $p_T\approx 8$–$14$ GeV for masses $3$–$10$ TeV, with the distribution's width scaling with $M_S$ (0909.2666). For ultraheavy masses, the $u$-quark PDF suppression dominates the kinematic reach, making coupling choices sublinear with attainable cross sections.

4. Decays, Branching Ratios, and Cascade Topologies

Ultraheavy diquark scalars decay at tree-level to quark–quark pairs and, if lighter vectorlike quarks ($\chi$) are present, to pairs of such states: $\Gamma(S_{uu} \to uu) = \frac{y_{uu}^2}{32\pi}M_S.$ If the decay to $\chi \chi$ is open ($M_S > 2 m_\chi$),

$$\Gamma(S_{uu}\to\chi\chi) = \frac{y_{\chi\chi}^2}{32\pi} M_S \left(1-\frac{2m_\chi^2}{M_S^2}\right)\sqrt{1-\frac{4m_\chi^2}{M_S^2}}.$$

The branching ratio for $S_{uu} \to \chi\chi$, with equal couplings and $m_\chi\ll M_S/2$, can be $\sim 50\%$ (Dobrescu et al., 2018). For the benchmark $M_S=8.5$ TeV, $m_\chi=2.1$ TeV, $y_{uu}\approx 0.8$–$1$, the width is narrow ($\Gamma/M_S \lesssim 10\%$) (Dobrescu, 6 Nov 2024, Duminica et al., 21 Mar 2025, Costache et al., 4 Nov 2025).

The $\chi$ decays semi-promptly to $W^+ b$, $Z t$, or $h t$, with $B(\chi\to Wb)\simeq 50\%$, yielding characteristic $6j$ or multilepton final states. In cascade scenarios, S can decay via $S_{uu}\to\chi_2\chi_2$ or to a pair of lighter scalars $S_{2/3}$, yielding $4j$, $5j$, or $6j$ topologies, each with distinctive invariant-mass and jet-substructure properties (Dobrescu, 6 Nov 2024, Dobrescu, 2019).

5. Phenomenology and LHC Search Strategies

Ultraheavy diquark scalars are targeted at the LHC through both direct and cascade decay signatures. Key strategies include:

• Dijet Resonance: $pp\to S_{uu}\to jj$, with extremely high $p_T$ jets ($p_T\sim M_S/2$), targeted by imposing $p_T$ and $|\eta|$ cuts and tight mass window requirements ($\pm 5\%$) (Dobrescu, 2019).
• Paired-Dijet and Multijet Final States: $pp\to S_{uu}\to\chi\chi\to (Wb)(Wb)\to6j$, or $4j/6j$ via cascade decay, discriminated via jet-multiplicity, substructure tags for boosted objects, and multi-dijet mass combinations (Dobrescu, 6 Nov 2024, Duminica et al., 21 Mar 2025).
• High-Threshold Triggers: Large $H_T$ ($>4$–$8$ TeV) and leading jet $p_T$ ($>1$ TeV) triggers are essential for ultraheavy signal retention (Dobrescu, 2019).
• Advanced Classification: Machine learning (Random Forest, $k$-fold cross validation) is used for optimal signal–background separation, exploiting event-level observables: jet multiplicity, invariant-mass distributions, energy correlations, and event-shape variables (Duminica et al., 21 Mar 2025, Costache et al., 4 Nov 2025).
• Cascade Decay Tagging: For scenarios with $$S_{uu}\to\chi_2\chi_2\to (u\,A_\chi)(u\,A_\chi)\to 4j$$, dedicated selection of di-gluon jets, and tight pairwise resonance matching, enhances signal over complex QCD backgrounds (Dobrescu, 6 Nov 2024).
• Event Yield and Sensitivity: With $\sim 3\,\mathrm{ab}^{-1}$ at $14$ TeV and $y_{uu}\gtrsim0.2$, $S_{uu}$ with $M_S\sim8.0$–$8.5$ TeV can be discovered ($5\sigma$) or excluded at $95\%$ CL through the fully hadronic $6j$ channel (Duminica et al., 21 Mar 2025, Costache et al., 4 Nov 2025). Even for $M_S\sim10$ TeV and perturbative couplings, expected signal exceeds background in exclusive topologies (Dobrescu et al., 2018, Dobrescu, 2019).

6. Constraints, Hints, and Theoretical Implications

Recent CMS and ATLAS multi-jet search results include statistically significant excesses (4j events with $M_{4j}\sim8$ TeV) compatible with $S_{uu}$ benchmarks near $M_S=8.5$ TeV, $m_\chi=2.1$ TeV, and $y_{uu}\sim0.8$ (Dobrescu, 6 Nov 2024, Dobrescu et al., 2018). Nonresonant dijet and paired-dijet searches further constrain the $(M_S,y_{uu})$ parameter space, with exclusion of $M_S\lesssim8.3$–$8.4$ TeV for $y_{uu}\gtrsim0.2$ (Costache et al., 4 Nov 2025). The observed 3.9$\sigma$ local excess in the $4j$ CMS search and corresponding ATLAS events may be explained by $S_{uu}$ and cascade decay models, motivating dedicated 5- and 6-jet analyses (Dobrescu, 6 Nov 2024, Dobrescu et al., 2018).

A plausible implication is that multi-TeV diquark scalars, if realized in nature, can be both copiously produced and unambiguously identified at the HL-LHC, provided event selection includes high-multiplicity, high-mass, and substructure-aware strategies.

Ultraheavy diquark scalars also play a central conceptual role in models of exotic baryons, doubly-heavy baryons, and fully-heavy tetraquarks, providing calculable constituent masses and nonperturbative vertices for hadronic transitions (Agaev et al., 20 Jul 2024, Shi, 2020). The formalism is supported by effective heavy-diquark theories, path-integral hadronization, and QCD sum rule matching, linking the collider-accessible regime to the spectroscopy of multi-quark hadrons.

7. Summary Table: Key Properties and Predicted Signatures

Property	Value/Role	Reference
$S_{uu}$ quantum numbers	$(6,1,+4/3)$, $B=2/3$	(Dobrescu, 2019)
Perturbative coupling range ($y_{uu}$)	0.1–1 (LHC sensitivity for $M_S$ in 8–10 TeV)	(Duminica et al., 21 Mar 2025)
NLO K-factor (LHC 14 TeV, $M_S=$ 3–10 TeV)	Sextet: 1.22–1.32	(0909.2666)
Typical cross section ($M_S=8$ TeV, $y_{uu}$=1)	$\sim$0.01 fb (NLO, 14 TeV)	(0909.2666)
Benchmark decay $S_{uu}\to\chi\chi$	$m_\chi\sim2$ TeV, $B(S\to\chi\chi)\sim50\%$	(Costache et al., 4 Nov 2025)
ML-based $6j$ search sensitivity	$M_S=8.0$ TeV, $y_{uu} B_{\chi\to Wb}=0.1$: $>5\sigma$	(Duminica et al., 21 Mar 2025)
Gluonic mass dressing (Schwinger–Dyson)	$+0.2$–$0.4$ GeV, saturates at large $m_0$	(Imai et al., 2014)

Experimental confirmation or further exclusion will require continued multi-jet high-mass searches, advanced triggering, and event reconstruction at the high-luminosity LHC.