Vector-Like Singlet Top Quarks (T)

• Vector-like singlet top quarks (T) are hypothetical color-triplet, weak-isospin singlet fermions with charge +2/3, introduced to address the gauge hierarchy problem.
• They decay predominantly via W, Z, and Higgs channels with branching ratios approximately 50%, 25%, and 25%, respectively, following patterns predicted by the Goldstone equivalence theorem.
• Collider searches utilize pair and single production channels with strategies targeting leptonic signatures and boosted jet topologies to reveal modified gauge currents from T–t mixing.

A vector-like singlet top quark, usually denoted as $T$, is a hypothetical color-triplet, weak-isospin singlet fermion with electric charge $+\tfrac{2}{3}$. Its left- and right-handed components transform identically under $SU(2)_L \times U(1)_Y$, allowing a gauge-invariant Dirac mass term distinct from the Standard Model (SM) chiral structure. Such particles are introduced in many theories addressing the gauge hierarchy problem, notably composite Higgs, Little Higgs, extra-dimensional, and extended scalar sector models. The $T$ quark mixes dominantly with the SM top via dimension-four Yukawa terms, altering its phenomenology compared to chiral quarks.

1. Theoretical Framework and Couplings

Vector-like singlet top partners have quantum numbers $T \sim (3,1,2/3)$. The leading interactions governing phenomenology are encapsulated by an effective Lagrangian after electroweak symmetry breaking: $$\mathcal{L}_{\rm eff} \supset \frac{g\,g^*}{2\sqrt2}\left[ \bar T_L \gamma^\mu W^+_\mu b_L + \frac{1}{c_W} \bar T_L \gamma^\mu Z_\mu t_L - \frac{m_T}{m_W} \bar T_R h t_L - \frac{m_t}{m_W} \bar T_L h t_R \right] + \mathrm{h.c.}$$ Here, $g$ is the $SU(2)_L$ gauge coupling, $c_W = \cos\theta_W$ is the weak mixing cosine, $h$ is the physical Higgs field, and $+\tfrac{2}{3}$0 parametrizes the electroweak strength of the $+\tfrac{2}{3}$1 couplings; it is related to the left-handed mixing angle $+\tfrac{2}{3}$2 by $+\tfrac{2}{3}$3 (Han et al., 2022, Girdhar et al., 2014).

The mass matrix for $+\tfrac{2}{3}$4--$+\tfrac{2}{3}$5 mixing is

$+\tfrac{2}{3}$6

with $+\tfrac{2}{3}$7 the SM top mass before mixing, $+\tfrac{2}{3}$8 the bare vector-like mass, and $+\tfrac{2}{3}$9 a Yukawa-induced mixing term (Girdhar et al., 2014, Benbrik et al., 2024). Diagonalization leads to physical mass eigenstates and modified gauge currents, inducing $SU(2)_L \times U(1)_Y$0--$SU(2)_L \times U(1)_Y$1--$SU(2)_L \times U(1)_Y$2 flavor-changing neutral currents (FCNCs) and $SU(2)_L \times U(1)_Y$3--$SU(2)_L \times U(1)_Y$4--$SU(2)_L \times U(1)_Y$5 interactions.

2. Decay Channels and Branching Ratios

The $SU(2)_L \times U(1)_Y$6 quark exhibits three principal two-body decays: $SU(2)_L \times U(1)_Y$7 with tree-level widths (neglecting subleading mass effects)

$SU(2)_L \times U(1)_Y$8

In the heavy mass limit $SU(2)_L \times U(1)_Y$9, the Goldstone equivalence theorem yields the approximate proportion (Han et al., 2022, Girdhar et al., 2014, Erdmann, 2018): $T$0 so that

$T$1

These ratios are robust for $T$2 TeV, and only weakly sensitive to the precise value of $T$3 as long as the total width remains small compared to $T$4 (Han et al., 2022, Erdmann, 2018). For "non-minimal" scenarios (e.g. with exotic singlet scalar/pseudoscalar or extended Higgs sectors), new decay topologies such as $T$5 can dominate, distorting standard BR patterns (Benbrik et al., 2019, Bhardwaj et al., 2022, Aguilar-Saavedra et al., 2019).

3. Collider Production and Phenomenology

Pair production is primarily via QCD and is independent of electroweak mixing: $T$6 with cross sections set by $T$7 and known up to NNLO+NNLL. For $T$8 TeV at $T$9 TeV, $T \sim (3,1,2/3)$0 fb (Girdhar et al., 2014, Erdmann, 2018).

Single production becomes dominant for higher $T \sim (3,1,2/3)$1, especially for moderate $T \sim (3,1,2/3)$2. At hadron colliders, $T \sim (3,1,2/3)$3 is produced via $T \sim (3,1,2/3)$4-channel exchange: $T \sim (3,1,2/3)$5 with $T \sim (3,1,2/3)$6 (Erdmann, 2018, Liu et al., 2017). At $T \sim (3,1,2/3)$7 colliders, the leading process is $T \sim (3,1,2/3)$8 via $T \sim (3,1,2/3)$9-channel $$\mathcal{L}_{\rm eff} \supset \frac{g\,g^*}{2\sqrt2}\left[ \bar T_L \gamma^\mu W^+_\mu b_L + \frac{1}{c_W} \bar T_L \gamma^\mu Z_\mu t_L - \frac{m_T}{m_W} \bar T_R h t_L - \frac{m_t}{m_W} \bar T_L h t_R \right] + \mathrm{h.c.}$$0 exchange, giving direct sensitivity to the $$\mathcal{L}_{\rm eff} \supset \frac{g\,g^*}{2\sqrt2}\left[ \bar T_L \gamma^\mu W^+_\mu b_L + \frac{1}{c_W} \bar T_L \gamma^\mu Z_\mu t_L - \frac{m_T}{m_W} \bar T_R h t_L - \frac{m_t}{m_W} \bar T_L h t_R \right] + \mathrm{h.c.}$$1--$$\mathcal{L}_{\rm eff} \supset \frac{g\,g^*}{2\sqrt2}\left[ \bar T_L \gamma^\mu W^+_\mu b_L + \frac{1}{c_W} \bar T_L \gamma^\mu Z_\mu t_L - \frac{m_T}{m_W} \bar T_R h t_L - \frac{m_t}{m_W} \bar T_L h t_R \right] + \mathrm{h.c.}$$2--$$\mathcal{L}_{\rm eff} \supset \frac{g\,g^*}{2\sqrt2}\left[ \bar T_L \gamma^\mu W^+_\mu b_L + \frac{1}{c_W} \bar T_L \gamma^\mu Z_\mu t_L - \frac{m_T}{m_W} \bar T_R h t_L - \frac{m_t}{m_W} \bar T_L h t_R \right] + \mathrm{h.c.}$$3 coupling (Han et al., 2022).

In $$\mathcal{L}_{\rm eff} \supset \frac{g\,g^*}{2\sqrt2}\left[ \bar T_L \gamma^\mu W^+_\mu b_L + \frac{1}{c_W} \bar T_L \gamma^\mu Z_\mu t_L - \frac{m_T}{m_W} \bar T_R h t_L - \frac{m_t}{m_W} \bar T_L h t_R \right] + \mathrm{h.c.}$$4 collisions at $$\mathcal{L}_{\rm eff} \supset \frac{g\,g^*}{2\sqrt2}\left[ \bar T_L \gamma^\mu W^+_\mu b_L + \frac{1}{c_W} \bar T_L \gamma^\mu Z_\mu t_L - \frac{m_T}{m_W} \bar T_R h t_L - \frac{m_t}{m_W} \bar T_L h t_R \right] + \mathrm{h.c.}$$5 TeV, current analysis strategies include:

• $$\mathcal{L}_{\rm eff} \supset \frac{g\,g^*}{2\sqrt2}\left[ \bar T_L \gamma^\mu W^+_\mu b_L + \frac{1}{c_W} \bar T_L \gamma^\mu Z_\mu t_L - \frac{m_T}{m_W} \bar T_R h t_L - \frac{m_t}{m_W} \bar T_L h t_R \right] + \mathrm{h.c.}$$6: isolated lepton + $$\mathcal{L}_{\rm eff} \supset \frac{g\,g^*}{2\sqrt2}\left[ \bar T_L \gamma^\mu W^+_\mu b_L + \frac{1}{c_W} \bar T_L \gamma^\mu Z_\mu t_L - \frac{m_T}{m_W} \bar T_R h t_L - \frac{m_t}{m_W} \bar T_L h t_R \right] + \mathrm{h.c.}$$7-jet + forward jet + missing $$\mathcal{L}_{\rm eff} \supset \frac{g\,g^*}{2\sqrt2}\left[ \bar T_L \gamma^\mu W^+_\mu b_L + \frac{1}{c_W} \bar T_L \gamma^\mu Z_\mu t_L - \frac{m_T}{m_W} \bar T_R h t_L - \frac{m_t}{m_W} \bar T_L h t_R \right] + \mathrm{h.c.}$$8; backgrounds: $$\mathcal{L}_{\rm eff} \supset \frac{g\,g^*}{2\sqrt2}\left[ \bar T_L \gamma^\mu W^+_\mu b_L + \frac{1}{c_W} \bar T_L \gamma^\mu Z_\mu t_L - \frac{m_T}{m_W} \bar T_R h t_L - \frac{m_t}{m_W} \bar T_L h t_R \right] + \mathrm{h.c.}$$9+jets, $g$0, single top (Erdmann, 2018, Collaboration, 18 Jun 2025).
• $g$1: dilepton or trilepton (from $g$2), top-tagged jet, $g$3-jet, forward jet; backgrounds: $g$4, $g$5+jets, $g$6+jets (Erdmann, 2018, Han et al., 2023, Spiezia, 2017).
• $g$7: highly boosted jets ($H\to b\bar