---
title: Vector Leptoquarks (vLQs)
url: https://www.emergentmind.com/topics/vector-leptoquarks-vlqs
type: topic
---

# Vector Leptoquarks (vLQs)

Vector leptoquarks (vLQs) are spin-1 color-triplet bosons that carry both baryon and lepton number and couple to quark–lepton currents. They occupy a distinctive place among extensions of the Standard Model because the same field can participate in collider signatures with leptons and jets, generate semileptonic four-fermion operators at low energy, and appear naturally in gauge-based constructions such as \(SU(4)\) or “4321” scenarios. Contemporary collider literature also emphasizes a terminological caution: vLQs are not vector-like quarks (VLQs), even though the acronyms are often conflated; the former are spin-1 bosons mediating quark–lepton transitions, whereas the latter are spin-\(\tfrac12\) colored fermions with SM-like gauge transformations for left- and right-handed components [2601.22425].

## 1. Definition, nomenclature, and representation content

The modern literature uses several closely related naming conventions for vector leptoquarks. The singlet state with Standard Model quantum numbers \((3,1,2/3)\) is denoted \(U_1\) in collider analyses and also appears as \(V_1\) in flavor-oriented simplified models; both notations refer to the same gauge representation, not to different particles [2210.04517]. Beyond this state, the most common representations in collider and EFT analyses include \(U_3\), \(V_2\), \(\tilde V_2\), and \(\tilde U_1\), each with distinct chiral couplings and electric-charge content [2507.18295].

| Label | SM representation | Typical role |
|---|---|---|
| \(U_1\) or \(V_1\) | \((3,1,2/3)\) | central benchmark for \(B\)-anomaly phenomenology |
| \(U_3\) | \((3,3,2/3)\) | left-handed triplet with correlated charged and neutral currents |
| \(V_2\) | \((\bar 3,2,5/6)\) | doublet with LR or RL structures |
| \(\tilde V_2\) | \((\bar 3,2,-1/6)\) | charge-shifted doublet |
| \(\tilde U_1\) | \((3,1,5/3)\) | singlet with larger electric charge |

This classification is not merely taxonomic. The electroweak representation fixes the allowed fermion bilinears, the component electric charges, and the pattern of interference with Standard Model amplitudes. For example, \(U_3(3,3,2/3)\) contains components with electric charges \(+5/3\), \(+2/3\), and \(-1/3\), while the singlet \(U_1(3,1,2/3)\) contains only the \(Q=2/3\) state [1912.03007]. The broader representation lists used in collider and CLFV studies also track the fermion number \(F=3B+L\), which is useful when organizing allowed operators and decay channels [2602.01690].

A recurrent theme is that \(U_1\) dominates phenomenological discussions because it can address both neutral-current and charged-current flavor anomalies, while still admitting direct collider searches in \(b\ell\), \(t\nu\), \(\tau b\), \(\tau s\), and related channels [2112.12129]. By contrast, other vLQ species are often introduced to illustrate the model dependence of interference patterns, electric-charge enhancements, or CLFV matching [2507.18295].

## 2. Interaction structure and effective descriptions

At the level of generic phenomenology, vLQs couple to quark–lepton currents through operators of the schematic form
\[
\mathcal L_{\rm int}\supset \lambda_{ij}\,\bar q_i\,\gamma^\mu\,\ell_j\,V_\mu + {\rm h.c.},
\]
with generation-dependent couplings \(\lambda_{ij}\) and model-dependent chiral projectors [2601.22425]. In the \(U_1\) benchmark used in polarized-collider studies, the interaction is written more explicitly as
\[
\mathcal L_{U_1} = \left(\lambda_{Lij}\,\bar Q_{Li}\gamma_\mu L_{Lj} + \lambda_{Rij}\,\bar d_{Ri}\gamma_\mu \ell_{Rj}\right)U_1^\mu + {\rm h.c.},
\]
which makes the left-handed and right-handed chiral structures manifest [2602.01690].

The gauge and kinetic sector is equally important because vector production depends on the vLQ–gluon interaction. ATLAS-oriented summaries frequently employ
\[
\mathcal L \supset -\tfrac12 V^{\dagger}_{\mu\nu}V^{\mu\nu} + M_V^2 V_\mu^\dagger V^\mu - i g_s\,\kappa\,V_\mu^\dagger G^{\mu\nu}V_\nu + \ldots,
\]
with \(\kappa=1\) identified as the Yang–Mills scenario and \(\kappa=0\) as minimal coupling [2601.22425]. A recent LHC reinterpretation instead writes
\[
\mathcal L \supset -\tfrac12 \chi_{\mu\nu}^\dagger \chi^{\mu\nu} + M^2\chi_\mu^\dagger\chi^\mu - i g_s(1-\kappa)\chi_\mu^\dagger T^a \chi_\nu G^{a\mu\nu},
\]
so the symbol \(\kappa\) enters with a shifted convention, but the physical point is the same: pair production is highly sensitive to the anomalous vector–gluon coupling [2507.18295].

Branching-fraction notation is standardized across much of the collider literature. One defines
\[
\beta = {\rm BR}({\rm LQ}\to \ell q), \qquad {\rm BR}({\rm LQ}\to \nu q)=1-\beta,
\]
with generation assignments usually grouped into first \((e/\nu_e)\), second \((\mu/\nu_\mu)\), and third \((\tau/\nu_\tau)\) families [2601.22425]. In the ATLAS pair-production search for \(U_1\), nominal samples were generated with purely left-handed couplings and \(\beta=0.5\), then reweighted to scan \({\rm BR}(V\to\) charged lepton\()\in[0,0.95]\) [2210.04517].

For indirect observables, integrating out vLQs generates SMEFT operators such as \(\mathcal O_{LQ}^{(1,3)}\), \(\mathcal O_{Qe}\), \(\mathcal O_{Lu}\), \(\mathcal O_{Ld}\), \(\mathcal O_{eu}\), \(\mathcal O_{ed}\), and \(\mathcal O_{LedQ}\). In the vector case, the CLFV analysis of tau processes gives explicit matching relations, for example
\[
x^\mathrm{LL}_1 x^\mathrm{LL}_{1,\tau}=-\frac12\big(C_{LQ}^{(1)}+3C_{LQ}^{(3)}\big), \qquad
x^\mathrm{RR}_1 x^\mathrm{RR}_{1,\tau}=-C_{ed},
\]
for the \(U_1\) singlet, and analogous relations for \(U_3\), \(V_2\), \(\tilde V_2\), and \(\tilde U_1\) [2111.06872].

## 3. Production mechanisms and characteristic kinematics

At hadron colliders, vLQ pair production is dominantly QCD-driven through \(gg\) and \(q\bar q\) initial states. For the ATLAS \(U_1\) search, this production was treated as largely insensitive to the leptoquark–lepton couplings and controlled instead by the vector–gluon interaction, with separate Yang–Mills and minimal-coupling scenarios giving enhanced or reduced cross sections, respectively [2210.04517]. Standard ATLAS interpretations also note that \(\sigma_{\rm pair}({\rm vLQ})\) is larger than for scalar leptoquarks at the same mass and depends strongly on the anomalous gluon coupling [2601.22425].

Single production probes a different part of parameter space because it scales with the fermionic couplings. Standard resonant topologies include \(qg\to V\ell\) or \(q\ell\to V\), and the salient observable is a lepton–jet resonance \(m_{\ell j}\) [2601.22425]. In flavorful \(U_1\) models with dominant \(\tau\) couplings, the decay pattern is fixed by \(SU(2)_L\): approximately,
\[
{\rm BR}(\Delta\to b\tau)\simeq {\rm BR}(\Delta\to t\nu_\tau), \qquad
{\rm BR}(\Delta\to s\tau)\simeq {\rm BR}(\Delta\to c\nu_\tau),
\]
with the ratio of second- to third-generation final states determined by \(\lambda_{s\tau}/\lambda_{b\tau}\) [2112.12129].

A more recent development is the systematic inclusion of nonresonant channels in dilepton tails. The LHC reinterpretation in “Fresh look at the LHC limits on vector leptoquarks” organizes the full signal into pair production, single production, indirect production from t/u-channel vLQ exchange, and indirect interference with Standard Model Drell–Yan. The interference term can be constructive or destructive depending on the vLQ species and chirality; notably, it is destructive for \(U_1\) with LL or RR couplings, negative for \(V_2\) with LR couplings, positive for \(V_2\) with RL couplings, and sign-dependent across \(U_3\) components [2507.18295]. This substantially alters the high-\(p_T\) dilepton spectrum.

Future lepton colliders access a different production regime. In the polarized \(e^-e^+\) study centered on \(U_1\), pair production proceeds primarily through \(s\)-channel \(\gamma/Z\) exchange, and helicity conservation favors RL and LR initial states. The polarized cross section is written as
\[
\sigma(P_{e^-},P_{e^+})=\frac14\Big[(1+P_{e^-})(1+P_{e^+})\sigma_{RR}+(1+P_{e^-})(1-P_{e^+})\sigma_{RL}
+(1-P_{e^-})(1+P_{e^+})\sigma_{LR}+(1-P_{e^-})(1-P_{e^+})\sigma_{LL}\Big],
\]
which makes beam polarization a direct handle on both signal enhancement and chirality diagnostics [2602.01690].

## 4. Collider searches and exclusion limits

The first dedicated ATLAS search for pair-produced scalar and vector leptoquarks decaying to third-generation quarks and first- or second-generation leptons used the full Run 2 dataset of \(139~{\rm fb}^{-1}\) at \(\sqrt s=13\) TeV and selected events with exactly one signal electron or muon, \(E_T^{\rm miss}>250\) GeV, at least four small-\(R\) jets, at least one \(b\)-tagged jet at the \(77\%\) working point, and a hadronic top candidate reconstructed by iterative reclustering. Separate NeuroBayes networks were trained for vector hypotheses at \({\rm BR}=0.0,0.25,0.5,0.9\), and no significant deviation from the Standard Model expectation was observed [2210.04517].

| Scenario at \({\rm BR}(V\to\) charged lepton\()=0.5\) | Decay mode | Observed 95% CL lower limit |
|---|---|---|
| Yang–Mills | \(t\nu/b\mu\) | \(1980\) GeV |
| Yang–Mills | \(t\nu/be\) | \(1900\) GeV |
| Minimal coupling | \(t\nu/b\mu\) | \(1710\) GeV |
| Minimal coupling | \(t\nu/be\) | \(1620\) GeV |

These limits are particularly relevant for \(B\)-anomaly-motivated \(U_1\) scenarios with \(\beta\simeq0.5\), where decays to charged and neutral second-generation leptons are comparable [2210.04517]. The same analysis also provided exclusion contours versus branching fraction, with sensitivity degrading as \({\rm BR}(V\to\) charged lepton\()\to0\) because the observed lepton increasingly originates from top decay and becomes more background-like.

A separate source of confusion is the 2026 ATLAS proceedings summary, which discusses leptoquark searches but does not present direct vLQ mass limits. The only vector-leptoquark appearance there is as an intermediate \(U_1\) state in a “4321” search for pair-produced vector-like leptons \(E\) and \(N\), where the excluded mass range is \(200\) to \(910\) GeV for the VLL cross section predicted by the model; no direct \(U_1\) mass, \(\kappa\), or \(\lambda\) constraint is extracted [2601.22425]. This is a common misreading of the proceedings.

Beyond dedicated resonance searches, the 2025 reinterpretation of LHC data shows that including QCD–QED mixed pair-production channels and Drell–Yan interference materially shifts exclusions. At \(13\) TeV with \(139~{\rm fb}^{-1}\), model-independent mass bounds in the \(\mu\mu jj\) mode extend, after including QCD+QED channels, from \(1.81\) TeV for \(U_1\) at \(\kappa=1\) to \(2.44\) TeV for \(U_3\) at \(\kappa=0\), with typical upward shifts of \(50\)–\(300\) GeV relative to QCD-only limits [2507.18295]. The same study emphasizes that destructive interference in dimuon tails can make coupling bounds stronger than direct pair- or single-production recasts for some species, especially \(U_1\).

For \(\tau\)-flavoured \(U_1\) states, collider sensitivity depends strongly on branching structure. In the simplified \(\Delta_\mu\sim(3,1,2/3)\) model, direct third-generation searches constrain the mass up to \(1500\)–\(1770\) GeV when the branching fraction is entirely to third-generation quarks, while jets-plus-missing-energy searches constrain it up to \(1150\)–\(1700\) GeV and are largely insensitive to the third-generation branching fraction [2112.12129].

## 5. Flavor anomalies, SMEFT matching, and indirect constraints

The theoretical prominence of vLQs derives from their ability to generate the two semileptonic operator patterns most often invoked in flavor anomalies. In the left-handed \(V_1\sim(3,1,2/3)\) simplified model, tree-level exchange yields
\[
C_{9,10}^{ij;\ell\ell'} = \mp\frac{\pi}{\sqrt{2}G_F\alpha_{\rm em}V_{3j}V_{3i}^*m_{V_1}^2}\left(K_L^{i\ell'}K_L^{j\ell}\right),
\]
so \(C_9=-C_{10}\) is generated in \(b\to s\ell\ell\), while charged-current \(b\to c\tau\nu\) transitions receive
\[
C_{V_L}^{jk;\ell i}=\frac{1}{2\sqrt{2}G_Fm_{V_1}^2}\frac{(V K_L U^P)_{ji}K_L^{k\ell}}{V_{jk}}.
\]
Global fits with \(m_{V_1}=1.5\), \(2.5\), and \(3.5\) TeV gave Standard-Model pulls of approximately \(5.78\sigma\), \(5.82\sigma\), and \(5.84\sigma\), respectively, while future discovery or exclusion power was identified in \(\tau\to\phi\mu\), \(B\to K^{(*)}\tau^+\tau^-\), LFV \(B\) decays, and \(\mu\)–\(e\) conversion [2012.05883].

Earlier \(U_1\) and \(U_3\) analyses expressed the same mechanism in the Wilson coefficients of the weak effective Hamiltonian. For \(b\to s\ell^+\ell^-\),
\[
C_9^{\rm NP}=-C_{10}^{\rm NP}
=\frac{\pi}{\sqrt2 G_F\alpha V_{tb}V_{ts}^*}\left[\frac{h_{1L}^{2\ell}h_{1L}^{3\ell *}}{M_{U_1}^2}+\frac{h_{3L}^{2\ell}h_{3L}^{3\ell *}}{M_{U_3}^2}\right],
\]
while \(b\to c\tau\nu\) receives \(C_{V_1}^\tau\) and, for \(U_1\), potentially \(C_{S_1}^\tau\) [1609.04367]. This explains why the singlet and triplet states remain the canonical flavor benchmarks.

Indirect constraints are often far stronger on coupling products than direct collider searches are on masses. In the CLFV analysis based on \(\tau\)-LFV hadronic decays and \(\ell\)–\(\tau\) conversion, Belle II projections imply \(M_V\gtrsim 25\)–\(28\) TeV for several LL, LR, RL, and RR coupling products with \(|xx'|\simeq1\), and up to \(M_V\gtrsim49\) TeV for the \(x_{1,2}\) combination entering \(\mathcal O_{LedQ}\); by contrast, current \(\ell\)–\(\tau\) conversion bounds are much weaker [2111.06872]. These are EFT bounds on coupling products, not direct resonance exclusions, but they strongly constrain off-diagonal flavor structure.

Loop effects can also be numerically important in gauge-based vLQ models. In an \(SU(4)\)-based framework with \(U_1\sim(3,1,2/3)\), one-loop corrections at fixed on-shell couplings enhance the left-handed SMEFT coefficient \(C_{LL}^U\) by approximately \(16\%\) and the right-handed scalar coefficient \(C_{LR}^U\) by approximately \(41\%\) for \(g_4=3\), while radiatively generating \(\mathcal O_{\ell q}^{(1)}\) at roughly \(11\%\) of the LO \(O_{LL}^U\) normalization [1910.13474]. This directly modifies the mapping between low-energy fits and collider constraints.

The same left-handed vector structure propagates into baryonic semileptonic decays. For \(U_3(3,3,2/3)\) in \(\Lambda_b\to\Lambda_c\ell\bar\nu_\ell\), the model induces only
\[
C_V=C_A=1+\frac{\sqrt2\,g_{b\tau}^*({\cal V}_g)_{c\tau}}{4G_FV_{cb}M_U^2},
\]
so normalized angular quantities remain largely SM-like while normalization-sensitive observables shift. In the reported fits, \(R(\Lambda_c)\) moved from the SM range \(0.314\)–\(0.339\) to \(0.410\)–\(0.421\) in one solution and \(0.335\)–\(0.445\) in another [1912.03007].

## 6. Future directions and unresolved issues

The next stage of vLQ phenomenology is shaped by both luminosity and methodology. ATLAS already frames the near-term collider outlook in terms of growing Run 3 luminosity and the HL-LHC, with improved sensitivity expected in pair-production channels because spin-1 cross sections are larger, and in single-production channels because they directly probe the flavor couplings \(\lambda_{ij}\) [2601.22425]. In \(\tau\)-flavoured \(U_1\) models, conservative HL-LHC extrapolations based on MET searches reach about \(1.4\) TeV for minimal couplings and \(2.0\) TeV for Yang–Mills-like couplings [2112.12129].

A parallel frontier is polarized lepton collisions. For \(e^-e^+\to U_1\bar U_1\) at \(\sqrt s=3\) TeV with \(M_{\rm VLQ}=1.2\) TeV, the beam configuration \(P_{e^-}=-0.8\), \(P_{e^+}=+0.6\) maximizes the cross section at approximately \(120\) fb, compared with an unpolarized baseline of approximately \(50\) fb. The same study finds \(A_{LR}\) up to \(0.16\) under full polarization and \(L_{\rm eff}/L\) up to approximately \(0.90\)–\(0.95\), while \(e^-e^+\) annihilation remains about \(10^3\) times larger than \(\gamma\gamma\) fusion for the same \(\sqrt s\) [2602.01690]. This makes polarization a genuine characterization tool, not merely a luminosity enhancement.

Several open issues remain model-dependent. First, direct reinterpretation of scalar-LQ limits as vector-LQ limits is not justified, because pair and single production have different kinematics and systematic structures, and the anomalous vector–gluon coupling plays a central role for vLQs [2601.22425]. Second, the fresh LHC recast shows that EFT descriptions reproduce the full theory reliably only once \(M_\chi\gtrsim3\) TeV for first-generation couplings, with higher thresholds for heavier flavors [2507.18295]. Third, widths are not always negligible: the ATLAS \(U_1\) search used \(g_U=3.0\), corresponding to a modeled width of approximately \(11\%\), and treated decays with MadSpin rather than a strict narrow-width approximation [2210.04517].

Taken together, these developments place vLQs in an unusual theoretical position. They are simultaneously direct-search targets, low-energy EFT mediators, and benchmarks for correlated flavor anomalies. This suggests that future progress will depend less on any single exclusion number than on consistent global treatments combining resonance production, nonresonant interference, chirality-sensitive observables, and flavor data across collider and non-collider experiments [2507.18295].

Source: https://www.emergentmind.com/topics/vector-leptoquarks-vlqs