Excited fermions can be pair produced via their minimal gauge couplings. The couplings of excited leptons with Z are given by

$$ \frac{e}{2 \sin \theta_W \cos \theta_W} \left[ (-1 + 2 \sin^2 \theta_W) \bar{l}^* \gamma^\mu l^* Z_\mu + \bar{\nu}^* \gamma^\mu \nu^* Z_\mu \right] $$

in the homodoublet model. The corresponding couplings of excited quarks can be easily obtained. Although form factor effects can be present for the gauge couplings at $q^2 \neq 0$, they are usually neglected.

Excited fermions may also be produced via the contact interactions with ordinary quarks and leptons [46]

$$ \mathcal{L} = \frac{g_{\text{contact}}^2}{\Lambda^2} \left[ \eta'_{LL} (\bar{\psi}_L \gamma_\mu \psi_L) (\bar{\psi}_L^* \gamma^\mu \psi_L^*) + (\eta''_{LL} (\bar{\psi}_L \gamma_\mu \psi_L) (\bar{\psi}_L^* \gamma^\mu \psi_L) + \text{h.c.}) + \dots \right]. \quad (91.4) $$

Again, the coefficient is conventionally taken $g_{\text{contact}}^2 = 4\pi$. It is widely assumed $\eta'_{LL} = \eta''_{LL} = 1$, $\eta'_{LR} = \eta''_{LR} = \eta'_{RL} = \eta''_{RL} = \eta'_{RR} = \eta''_{RR} = 0$ in experimental analyses for simplicity.

In addition, transition-magnetic type couplings with a gauge boson are expected. These couplings can be generally parameterized as follows:

$$ \begin{aligned} \mathcal{L} = & \frac{\lambda_{\gamma}^{(\psi^*)} e}{2m_{\psi^*}} \bar{\psi}_* \sigma^{\mu\nu} (\eta_L \frac{1-\gamma_5}{2} + \eta_R \frac{1+\gamma_5}{2}) \psi F_{\mu\nu} \\ & + \frac{\lambda_Z^{(\psi^*)} e}{2m_{\psi^*}} \bar{\psi}_* \sigma^{\mu\nu} (\eta_L \frac{1-\gamma_5}{2} + \eta_R \frac{1+\gamma_5}{2}) \psi Z_{\mu\nu} \\ & + \frac{\lambda_W^{(l^*)} g}{2m_{l^*}} \bar{l}^* \sigma^{\mu\nu} \frac{1-\gamma_5}{2} \nu W_{\mu\nu} \\ & + \frac{\lambda_W^{(\nu^*)} g}{2m_{\nu^*}} \bar{\nu}^* \sigma^{\mu\nu} (\eta_L \frac{1-\gamma_5}{2} + \eta_R \frac{1+\gamma_5}{2}) l W_{\mu\nu}^\dagger \\ & + \text{h.c.}, \end{aligned} \quad (91.5) $$

where $g = e/\sin\theta_W$, $\psi = \nu$ or $l$, $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ is the photon field strength, $Z_{\mu\nu} = \partial_\mu Z_\nu - \partial_\nu Z_\mu$, etc. The normalization of the coupling is chosen such that

$$ \max(|\eta_L|, |\eta_R|) = 1. $$

Chirality conservation requires

$$ \eta_L \eta_R = 0. \quad (91.6) $$

These couplings in Eq. (91.5) can arise from SU(2) × U(1)-invariant higher-dimensional interactions. A well-studied model is the interaction of homodoublet type $l^*$ with the Lagrangian (see [47, 48])

$$ \mathcal{L} = \frac{1}{2\Lambda} \bar{L}^* \sigma^{\mu\nu} (g f \frac{\tau^a}{2} W_{\mu\nu}^a + g' f' Y B_{\mu\nu}) \frac{1-\gamma_5}{2} L + \text{h.c.}, \quad (91.7) $$

where L denotes the lepton doublet ($\nu, l$), $\Lambda$ is the compositeness scale, $g, g'$ are SU(2) and U(1)$_Y$ gauge couplings, and $W_{\mu\nu}^a$ and $B_{\mu\nu}$ are the field strengths for SU(2) and U(1)$_Y$ gauge fields. These couplings satisfy the relation

$$ \lambda_W = -\sqrt{2} \sin^2 \theta_W (\lambda_Z \cot \theta_W + \lambda_\gamma), \quad (91.8) $$

with $\lambda_{W,Z,\gamma}$ being defined in Eq. (91.5) with $\lambda_{W,Z,\gamma} = \lambda_{W,Z,\gamma}^{(\ell^*)}$ or $\lambda_{W,Z,\gamma} = \lambda_{W,Z,\gamma}^{(\nu^*)}$. Here $(\eta_L, \eta_R) = (1, 0)$ is assumed. It should be noted that the electromagnetic radiative decay of $l^*$ ($\nu^*$) is forbidden if $f = -f'$ ($f = f'$).

Additional coupling with gluons is possible for excited quarks:

$$ \mathcal{L} = \frac{1}{2\Lambda} \bar{Q}^* \sigma^{\mu\nu} \left( g_s f_s \frac{\tau^a}{2} G_{\mu\nu}^a + g f \frac{\tau^a}{2} W_{\mu\nu}^a + g' f' Y B_{\mu\nu} \right) \times \frac{1-\gamma_5}{2} Q + \text{h.c.}, \quad (91.9) $$

where Q denotes a quark doublet, $g_s$ is the QCD gauge coupling, and $G_{\mu\nu}^a$ the gluon field strength.

If leptons are made of color triplet and antitriplet constituents, we may expect their color-octet partners. Transitions between the octet leptons ($l_8$) and the ordinary lepton ($l$) may take place via the dimension-five interactions

$$ \mathcal{L} = \frac{1}{2\Lambda} \sum_l \left\{ \bar{l}_8^\alpha g_S F_{\mu\nu}^\alpha \sigma^{\mu\nu} (\eta_L l_L + \eta_R l_R) + \text{h.c.} \right\} \quad (91.10) $$

where the summation is over charged leptons and neutrinos. The leptonic chiral invariance implies $\eta_L \eta_R = 0$ as before.

Searches for the excited quarks and leptons have been performed over the last decades in experiments at the LEP [49–52], HERA [53, 54], Tevatron [55, 56], and LHC [57–89]. Most stringent constraints, which are described below at 95% confidence level, come from the LHC experiments.

The signature of excited quarks $q^*$ at hadron colliders is characterized by a narrow resonant peak in the reconstructed invariant mass distribution of the $q^*$ decay products. The decays via the transition-magnetic type operator in Eq. (91.9) are considered for excited quarks in LHC searches, and the final states to search for are dijet ($qg$) [57, 58, 70–73] or a jet in association with a photon ($q\gamma$) [59, 60, 74, 75, 89] or a weak gauge boson ($qW, qZ$) [76, 77]. All analyses consider only spin-1/2 excited states of first generation quarks ($u^*, d^*$) with degenerate masses, expected to be predominantly produced in proton-proton collisions except for the excited b quark searches described below. Only the minimal gauge interactions and the transition-magnetic couplings with the form given in Eq. (91.9) are considered in the production process, and hence the contact interactions in Eq. (91.4) are not considered. The compositeness scale $\Lambda$ is taken to be the same as the excited quark mass $m_{q^*}$. The transition-magnetic coupling coefficients $f_s, f$ and $f'$ are assumed to be equal to 1 (denoted by $f$).

<figure><img src="image_385.png" /><figcaption>**Figure 91.3:** Dijet mass distribution measured by CMS using wide jets reconstructed from two highest transverse momentum jets by adding nearby jets within $\Delta R = \sqrt{\Delta\eta^2 + \Delta\phi^2} < 1.1$. The data distribution is compared to a fit representing a smooth background spectrum (solid curve) and the background prediction obtained from the $\Delta\eta_{jj}$ control regions (open squares), denoted as Ratio method, at $m_{jj} > 2.4$ TeV. The excited quark signal with mass of 4.0 TeV (labeled as $qg$) is shown together with other benchmark signals. Shown at the bottom panel is the difference between the data and the fitted parametrization from the fit method or the data and the background prediction from the Ratio method, divided by the statistical uncertainty of the data. Figure adopted from [73].</figcaption></figure>
<figure><img src="image_386.png" /><figcaption>**Figure 91.4:** 95% C.L. lower mass limits for the excited quarks (left) and excited leptons (right) at ATLAS [58, 60, 66–68, 88] and CMS [73, 75, 77, 80–82, 89] experiments. Shown are the most stringent limits for each final state (denoted in parentheses) of the excited fermions from both experiments. Only first generation quarks (u, d) with transition-magnetic type interactions with $f_s = f = f' = 1$ are considered for the excited quarks. The excited lepton limits are given for the production via contact interactions with $\Lambda = m_{l^*}$.</figcaption></figure>

With proton-proton collision data recorded at $\sqrt{s} = 13$ TeV at the LHC, the excited quark masses are excluded in dijet resonance searches up to 6.7 TeV in ATLAS using $139 \text{ fb}^{-1}$ [58] and 6.3 TeV in CMS using $137 \text{ fb}^{-1}$ [73]. Figure 91.3 shows the dijet mass ($m_{jj}$) distribution measured in CMS [73] by using the two highest $p_T$ jets reconstructed with the anti-$k_T$ algorithm [90] of a distance parameter of 0.4, and by combining nearby jets within $\Delta R = \sqrt{\Delta\eta^2 + \Delta\phi^2} < 1.1$ around the leading two jets. The measured dijet mass spectrum is compared to a fit with smoothly falling background shape (solid curve) at $m_{jj} > 1.5$ TeV and the background prediction obtained from $\Delta\eta_{jj}$ control regions (open squares), denoted as Ratio method, at $m_{jj} > 2.4$ TeV to look for a narrow resonance. Here $\Delta\eta_{jj}$ is the pseudorapidity separation between the two jets. An excited quark signal with mass of 4.0 TeV is shown in the figure (denoted by $qg$) as one of the benchmark signals considered in the analysis.

The photon + jet resonance searches, targeting excited quarks decaying into a quark and a photon ($q^* \to q + \gamma$), have excluded $q^*$ masses up to 5.3 TeV in ATLAS [60] and 6.0 TeV in CMS [89] using collision data at $\sqrt{s} = 13$ TeV. The W/Z boson + jet final states are examined to look for the $q^* \to q + W$ and $q + Z$ signal in CMS [77], exploiting jet substructure technique designed to provide sensitivity for highly-boosted hadronically decaying W and Z bosons. The lower mass limit of 5.0 (4.8) TeV is obtained from the W + jet (Z + jet) search using dataset recorded at $\sqrt{s} = 13$ TeV.

The excited b quarks ($b^*$) are also considered in the present searches at the LHC. Assuming the similar production processes to the first-generation excited quarks, the $b^*$ has been searched for in final states containing at least one jet identified as originating from a b quark (b-tagging). The searches using two jets including at least one b-tagged jet have been performed at 8 and 13 TeV [58, 61, 62, 71, 85], resulting in $b^*$ lower mass limits of 3.2 TeV in ATLAS using $139 \text{ fb}^{-1}$ at $\sqrt{s} = 13$ TeV [58] and 4.0 TeV in CMS using $138 \text{ fb}^{-1}$ at $\sqrt{s} = 13$ TeV [85]. The CMS Collaboration also performed a search for $b^* \to b + \gamma$ in events with a b-tagged jet in association with a photon using data at $\sqrt{s} = 13$ TeV [89], and excluded $b^*$ masses up to 2.2 TeV. For the same coupling strength, the observed lower mass bounds on the excited bottom quarks extends to 3.8 TeV, when the production mode from contact interactions Eq. (91.4) is also considered. Excited b quarks with charged-current decay into a W-boson and a top quark ($b^* \to t + W$) were looked for in both ATLAS and CMS using the full 8 TeV data [63, 78] and in CMS using the full 13 TeV data [86, 87]. For the left(right)-handed couplings, CMS excluded the masses below 2.6(2.8) TeV in the fully hadronic final states [86] and 3.0(3.2) TeV in the lepton+jets final states [87].

The excited top quarks ($t^*$) have also been searched for at

the LHC. The CMS Collaboration performed the search for pair-produced $t^*$ quarks with the assumption of the $t^*$ quark decay into a top quark and a gluon, in events with a charged lepton, missing transverse momentum and multiple jets including b-tagged jets at $\sqrt{s} = 8$ TeV [84] and 13 TeV [83]. With $35.9 \text{ fb}^{-1}$ data at 13 TeV, CMS excluded the $t^*$ masses below 1.2 TeV for spin 3/2 $t^*$ quarks [83].

Searches for excited leptons $l^*$ are also performed at the LHC using proton-proton collision data recorded at $\sqrt{s} = 7$ and 8 TeV [65–69, 79, 80] as well as at 13 TeV [81]. Considering single $l^*$ production in contact interactions (Eq. (91.4)) and electromagnetic radiative decay to a SM lepton and a photon ($l^* \to l + \gamma$ where $l = e, \mu$), the excited electron and muon masses are excluded for $\Lambda = m_{l^*}$ up to 3.9 and 3.8 TeV, respectively, using $35.9 \text{ fb}^{-1}$ at $\sqrt{s} = 13$ TeV in CMS [81] and 2.2 TeV using $13 \text{ fb}^{-1}$ at $\sqrt{s} = 8$ TeV in ATLAS [66].

With the full $20.3 \text{ fb}^{-1}$ data at $\sqrt{s} = 8$ TeV, the inclusive search on multi-lepton signatures with 3 or more charged leptons in ATLAS [67] further constrains the excited charged leptons and neutrinos. Considering both the transition-magnetic (Eq. (91.7)) and contact interaction (Eq. (91.4)) processes, the lower mass limits for the $e^*, \mu^*, \tau^*$ and $\nu^*$ (for every excited neutrino flavor) are obtained to be 3.0, 3.0, 2.5 and 1.6 TeV, respectively, for $\Lambda = m_{e^*}$, $m_{\mu^*}$, $m_{\tau^*}$ and $m_{\nu^*}$. The rate of pair-produced excited leptons is independent of $\Lambda$ for the minimal gauge interaction processes, and it allows to improve search sensitivity with multi-lepton signatures at high $\Lambda$, especially for excited neutrinos because the predominant $\nu_l^* \to l + W$ decays result in a higher acceptance for $\ge 3$ charged lepton final states.

Both ATLAS and CMS Collaborations performed searches for singly produced excited leptons. The single excited leptons, both produced and decaying in contact interaction processes (Eq. (91.4)), were searched for in CMS with $77.4 \text{ fb}^{-1}$ at 13 TeV [82] and in ATLAS with $20.3 \text{ fb}^{-1}$ at 8 TeV [68] and $139 \text{ fb}^{-1}$ at 13 TeV [88] using the final states with two leptons and two jets ($q\bar{q} \to ll^* \to llq\bar{q}$). The CMS search [82] considered both excited electrons and excited muons, using the invariant mass of the combination of the two leptons and two jets as a discriminating variable to separate signal from background. The ATLAS search considered excited muons [68] and excited $\tau$ leptons [88]. The single excited electrons, produced in contact interactions and decaying either in contact interaction or charged-current processes, were considered in ATLAS using $36.1 \text{ fb}^{-1}$ at 13 TeV [69] in the final states with two electrons and two jets ($q\bar{q} \to ee^* \to eeq\bar{q}$) or an electron, missing transverse momentum and a hadronically decaying W-boson candidate ($q\bar{q} \to ee^* \to e\nu W$). The excited electron (muon) mass was excluded up to 5.6 (5.7) TeV in CMS [82] at $\Lambda = m_{e^*}$ ($\Lambda = m_{\mu^*}$), which is the best limit to date on the ex-
cited electrons (muons). The excited $\tau$ lepton mass was excluded up to 4.6 TeV in ATLAS [88] at $\Lambda = m_{\tau^*}$.

The CMS Collaboration also performed an excited lepton search in the final states containing a Z boson [80], probing the excited leptons produced in contact interactions and decayed in neutral-current processes ($l^* \to l + Z$) with $f = f' = 1$ or $f = -f' = 1$. The leptonic and hadronic decays of Z bosons have been considered in the search, and the most stringent limits are obtained from the hadronic Z decay to be 2.08 (2.34) TeV and 2.11 (2.37) TeV for the $e^*$ and $\mu^*$, respectively, with $f = f' = 1$ ($f = -f' = 1$) at $\Lambda = m_{l^*}$.

Figure 91.4 summarizes the most stringent 95% C.L. lower mass limits for excited quarks and leptons obtained from the LHC experiments.

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# 92. Dynamical Electroweak Symmetry Breaking:
Implications of the H⁰

Revised August 2023 by K.M. Black (Wisconsin U.), R. Sekhar Chivukula (UC San Diego) and A. Lister (U. of British Columbia).

## 92.1 Introduction and Phenomenology

In theories of dynamical electroweak symmetry breaking, the electroweak interactions are broken to electromagnetism by the vacuum expectation value of a composite operator, typically a fermion bilinear. In these theories, the longitudinal components of the massive weak bosons are identified with composite Nambu-Goldstone bosons arising from dynamical symmetry breaking in a strongly-coupled extension of the standard model. Viable theories of dynamical electroweak symmetry breaking must also explain (or at least accommodate) the presence of an additional composite scalar state to be identified with the H⁰ scalar boson [1,2] – a state unlike any other observed previously.

Theories of dynamical electroweak symmetry breaking can be classified by the nature of the composite singlet state to be associated with the H⁰ and the corresponding dimensional scales *f*, the analog of the pion decay-constant in QCD, and $\Lambda$, the scale of the underlying strong dynamics.<fnref n="1" /> Of particular importance is the ratio $v/f$, where $v^2 = 1/(\sqrt{2}G_F) \approx (246 \text{ GeV})^2$, since this ratio measures the expected size of the deviations of the couplings of a composite Higgs boson from those expected in the standard model. The basic possibilities, and the additional states that they predict, are described below.

### 92.1.1 Technicolor, $v/f \simeq 1$, $\Lambda \simeq 1$ TeV

Technicolor models [7–9] provided the first examples of theories of dynamical electroweak symmetry breaking. These theories incorporate a new asymptotically free gauge theory (“technicolor”) and additional massless fermions (“technifermions” transforming under a vectorial representation of the gauge group). The global chiral symmetry of the fermions is spontaneously broken by the formation of a technifermion condensate, just as the approximate chiral symmetry in QCD is broken down to isospin by the formation of a quark condensate. The SU(2)$_W \times U(1)_Y$ interactions are embedded in the global technifermion chiral symmetries in such a way that the only unbroken gauge symmetry after chiral symmetry breaking is U(1)$_{em}$.<fnref n="2" /> The theories naturally provide the Nambu-Goldstone bosons “eaten” by the W and Z bosons. There would also typically be additional heavy states (e.g. vector mesons, analogous to the $\rho$ and $\omega$ mesons in QCD) with TeV masses [13,14], and the WW and ZZ scattering amplitudes would be expected to be strong at energies of order 1 TeV.

There are various possibilities for the scalar H⁰ in technicolor models. First, the H⁰ could be identified as a singlet scalar resonance, analogous to the $\sigma$ particle expected in pion-scattering in QCD [15, 16]. Alternatively, the H⁰ could be identified as a dilaton, a (pseudo-)Goldstone boson of scale invariance in theories of “walking technicolor” [17–21].<fnref n="3" /> Finally, the H⁰ could be identified as an additional isosinglet state if the chiral symmetry breaking pattern of the technicolor theory provides for such a state.<fnref n="4" /> In all of these cases, however, one expects large deviations in the couplings of this particle from those of the standard model Higgs boson. Since the couplings observed for the H⁰ approximate those of the Higgs boson to the 10% level, models of this kind are very highly constrained.

### 92.1.2 The Higgs doublet as a pseudo-Nambu-Goldstone Boson, $v/f < 1$, $\Lambda > 1$ TeV

In technicolor models, the symmetry-breaking properties of the underlying strong dynamics necessarily breaks the electroweak gauge symmetries. An alternative possibility is that the underlying strong dynamics itself does not break the electroweak interactions, and that the entire quartet of bosons in the Higgs doublet (including the state associated with the H⁰) are composite (pseudo-) Nambu-Goldstone particles [23–25]. In this case, the underlying dynamics can occur at energies exceeding 1 TeV and additional interactions with the top-quark mass generating sector (and possibly with additional weakly-coupled gauge bosons) cause the vacuum energy to be minimized when the composite Higgs doublet gains a vacuum expectation value [26,27]. In these theories, the couplings of the remaining singlet scalar state would naturally be equal to that of the standard model Higgs boson up to corrections of order $(v/f)^2$ and, therefore, constraints on the size of deviations of the H⁰ couplings from that of the standard model Higgs [28] give rise to lower bounds on the scales $f$ and $\Lambda$.<fnref n="5" />

The electroweak gauge interactions, as well as the interactions responsible for the top-quark mass, explicitly break the chiral symmetries of the composite Higgs model and lead generically to sizable corrections to the mass-squared of the Higgs-doublet – the so-called “Little Hierarchy Problem” [29]. “Little Higgs” theories [30–33] are examples of composite Higgs models in which the (collective) symmetry-breaking structure is selected so as to suppress these contributions to the Higgs mass-squared.

Composite Higgs models typically require a larger global symmetry of the underlying theory, and hence additional relatively light (compared to $\Lambda$) scalar particles, extra electroweak vector bosons (e.g. an additional SU(2)×U(1) gauge group), and vector-like partners of the top-quark of charge +2/3 and possibly also +5/3 [34]. In addition to these states, one would expect the underlying dynamics to yield additional scalar and vector resonances with masses of order $\Lambda$. If the theory respects a custodial symmetry [35], the couplings of these additional states to the electroweak and Higgs boson will be related – and, for example, one might expect a charged vector resonance to have similar branching ratios to WZ and WH. Different composite Higgs models utilize different mechanisms for arranging for the hierarchy of scales $v < f$ and arranging for a scalar Higgs self-coupling small enough to produce an H⁰ of mass of order 125 GeV, for a review see [36]. If the additional states in these models carry color, they can provide additional contributions to Higgs production via gluon fusion [37]. The extent to which Higgs production at the LHC conforms with standard model predictions provides additional constraints (typically lower bounds on the masses of the additional colored states of order 0.7 TeV) on these models.

In addition, if the larger symmetry of the underlying composite Higgs theory does not commute with the standard model gauge group, then the additional states found in those models – especially those related to the top-quark, which tend to have the largest couplings to the electroweak sector – may be colorless. For example, in twin Higgs models [38], the top-partners carry no standard model charges. The phenomenology of the additional states in twin Higgs theories is rather different, since lacking color the production of these particles at the LHC will be suppressed – and, their decays may occur only via the electroweak symmetry breaking sector, leading to their being long-lived.

### 92.1.3 Top-Condensate, Top-Color, Top-Seesaw and related theories, $v/f < 1$, $\Lambda > 1$ TeV

A final alternative is to consider a strongly interacting theory with a high (compared to a TeV) underlying dynamical scale that would naturally break the electroweak interactions, but whose

<footnote n="1">¹In a strongly interacting theory “Naive Dimensional Analysis” [3, 4] implies that, in the absence of fine-tuning, $\Lambda \simeq g^*f$ where $g^* \simeq 4\pi$ is the typical size of a strong coupling in the low-energy theory [5, 6]. This estimate is modified in the presence of multiple flavors or colors [7].</footnote>

<footnote n="2">²For a review of technicolor models, see [10–12].</footnote>

<footnote n="3">³If both the electroweak symmetry and the approximate scale symmetry are broken only by electroweak doublet condensate(s), then the decay-constants for scale and electroweak symmetry breaking may be approximately equal – differing only by terms formally proportional to the amount of explicit scale-symmetry breaking.</footnote>

<footnote n="4">⁴In this case, however, the coupling strength of the singlet state to WW and ZZ pairs would be comparable to the couplings to gluon and photon pairs, and these would all arise from loop-level couplings in the underlying technicolor theory [22].</footnote>

<footnote n="5">⁵In these models $v/f$ is an adjustable parameter, and in the limit $v/f \to 1$ they reduce, essentially, to the technicolor models discussed in the previous subsection. Our discussion here is consistent with that given there, since we expect corrections to the SM Higgs couplings to be large for $v/f \simeq 1$. Current measurements constrain the couplings of the H⁰ to equal those predicted for the Higgs in the standard model to about the 10% level [28], suggesting that $f$ must have values of order a TeV or higher and, therefore, a dynamical scale $\Lambda$ of at least several TeV.</footnote>
strength is adjusted (“fine-tuned”) to produce electroweak symmetry breaking at 1 TeV. This alternative is possible if the electroweak (quantum) phase transition is continuous (second order) in the strength of the strong dynamics [39]. If the fine-tuning can be achieved, the underlying strong interactions will produce a light composite Higgs bound state with couplings equal to that of the standard model Higgs boson up to corrections of order $(1 \text{ TeV}/\Lambda)^2$. As in theories in which electroweak symmetry breaking occurs through vacuum alignment, therefore, constraints on the size of deviations of the $H^0$ couplings from that of the standard model Higgs give rise to lower bounds on the scale $\Lambda$. Formally, in the limit $\Lambda \to \infty$ (a limit which requires arbitrarily fine adjustment of the strength of the high-energy interactions), these theories are equivalent to a theory with a fundamental Higgs boson – and the fine adjustment of the coupling strength is a manifestation of the hierarchy problem of theories with a fundamental scalar particle.

In many of these theories the top-quark itself interacts strongly (at high energies), potentially through an extended color gauge sector [40, 40–44]. In these theories, top-quark condensation (or the condensation of an admixture of the top with additional vector-like quarks) is responsible for electroweak symmetry breaking, and the $H^0$ is identified with a bound state involving the third generation of quarks. These theories typically include an extra set of massive color-octet vector bosons (top-gluons), and an extra $U(1)$ interaction (giving rise to a top-color $Z'$) which couple preferentially to the third generation and whose masses define the scale $\Lambda$ of the underlying physics.

### 92.1.4 Flavor

In addition to the electroweak symmetry breaking dynamics described above, which gives rise to the masses of the $W$ and $Z$ particles, additional interactions must be introduced to produce the masses of the standard model fermions. Two general avenues have been suggested for these new interactions. In “extended technicolor” (ETC) theories [45, 46], the gauge interactions in the underlying strongly interacting theory are extended to incorporate flavor. This extended gauge symmetry is broken down (possibly sequentially, at several different mass scales) to the residual strong interaction responsible for electroweak symmetry breaking. The massive gauge-bosons corresponding to the broken symmetries then mediate interactions between mass operators for the quarks/leptons and the corresponding bilinears of the strongly-interacting fermions, giving rise to the masses of the ordinary fermions after electroweak symmetry breaking.

In the case of “partial compositeness” [47], the additional flavor-dependent interactions arise from mixing between the ordinary quarks and leptons and massive composite fermions in the strongly-interacting underlying theory. Theories incorporating partial compositeness include additional vector-like partners of the ordinary quarks and leptons, typically with masses of order a TeV or less.

In both cases, the effects of flavor interactions on the electroweak properties of the ordinary quarks and leptons are likely to be most pronounced in the third generation of fermions.<fnref n="6" /> The additional particles present in these theories, especially the additional scalars, often couple more strongly to heavier fermions.

Moreover, since the flavor interactions must give rise to quark mixing, we expect that a generic theory of this kind could give rise to large flavor-changing neutral-currents [46]. In ETC theories, these constraints are typically somewhat relaxed if the theory incorporates approximate generational flavor symmetries [48], the theory has a slowly running coupling constant or “walks” [17–21], or if $\Lambda > 1$ TeV [49]. In theories of partial compositeness, the masses of the ordinary fermions depend on the scaling-dimension of the operators corresponding to the composite fermions with which they mix. This leads to a new mechanism for generating the mass-hierarchy of the observed quarks and leptons that, potentially, ameliorates flavor-changing neutral current problems and can provide new contributions to the composite Higgs potential

which allow for $v/f < 1$ [50–54].

Alternatively, one can assume that the underlying flavor dynamics respect flavor symmetries (“minimal” [55, 56] or “next-to-minimal” [57] flavor violation) that suppress flavor-changing neutral currents in the two light generations. Additional considerations apply when extending these arguments to potential explanation of neutrino masses (see, for example, [58, 59]).

### 92.1.5 Theoretical Considerations

Since the underlying high-energy dynamics in these theories are strongly coupled, there are no reliable calculation techniques that can be applied to analyze their properties. Instead, most phenomenological studies depend on the construction of a “low-energy” effective theory describing additional scalar, fermion, or vector boson degrees of freedom, which incorporates the relevant symmetries and, when available, dynamical principles. In some cases, motivated by the AdS/CFT correspondence [60], the strongly-interacting theories described above have been investigated by analyzing a dual compactified five-dimensional gauge theory. In these cases, the AdS/CFT “dictionary” is used to map the features of the underlying strongly coupled high-energy dynamics onto the low-energy weakly coupled dual theory [61].

More recently, progress has been made in investigating strongly-coupled models using lattice gauge theory [62–65]. These calculations offer the prospect of establishing which strongly coupled theories of electroweak symmetry breaking have a particle with properties consistent with those observed for the $H^0$ – and for establishing concrete predictions for these theories at the LHC [66].

### 92.1.6 Summary

The theoretical ideas and models reviewed here motivate searches for a wide variety of new states. Heavy vector bound states decaying to dibosons and additional scalar states appear naturally in both technicolor (92.1.1) and composite higgs models (92.1.2). Composite Higgs models based on collective symmetry breaking naturally include $W'$, $Z'$, and vector-like quark states (especially for the third-generation, and potentially including charge +5/3 custodial top-partner quarks). Vector-like states are also natural for any theory including partial compositeness in order to address the flavor problem (92.1.4). Top-condensate and related models (92.1.3) predict the existence of colorons which preferentially couple to third-generation quarks. Finally, the new states discussed here also occur in extra-dimensional models [61], which is understandable given that many strongly-coupled theories can be viewed as compactified five-dimensional gauge theories via duality (92.1.5). We turn now to a review the status of experimental searches related to dynamical electroweak symmetry breaking.

## 92.2 Experimental Searches

As discussed above, the extent to which the couplings of the $H^0$ conform to the expectations for a standard model Higgs boson constrains the viability of each of these models. Measurements of the $H^0$ couplings, and their interpretation in terms of effective field theory, are summarized in the $H^0$ review in this volume. In what follows, we will focus on searches for the additional particles that might be expected to accompany the singlet scalar: extra scalars, fermions, and vector bosons. In some cases, detailed model-specific searches have been made for the particles described above (though generally not yet taking account of the demonstrated existence of the $H^0$ boson).

In most cases, however, generic searches (e.g. for extra $W'$ or $Z'$ particles, extra scalars in the context of multi-Higgs models, or for fourth-generation quarks) are quoted that can be used – when appropriately translated – to derive bounds on a specific model of interest.

The mass scale of the new particles implied by the interpretations of the low mass of $H^0$ discussed above, and existing studies from the Tevatron and lower-energy colliders, suggests that only the Large Hadron Collider has any real sensitivity. A number of analyses already carried out by ATLAS and CMS use relevant final states and might have been expected to observe a deviation from standard model expectations – in no case so far has any such deviation been reported. The detailed implications of these

<footnote n="6">⁶Indeed, from this point of view, the vector-like partners of the top-quark in top-seesaw and little Higgs models can be viewed as incorporating partial compositeness to explain the origin of the top quark's large mass.</footnote>
searches in various model frameworks are described below.

Except where otherwise noted, all limits in this section are quoted at a confidence level of 95%. The searches at $\sqrt{s} = 8$ TeV (Run 1) are based on 20.3 fb⁻¹ of data recorded by ATLAS, and an integrated luminosity of 19.7 fb⁻¹ analyzed by CMS. The datasets collected at $\sqrt{s} = 13$ TeV during Run 2 of the LHC since 2015 are based on analyses with varied integrated luminosities ranging from ~2-140 fb⁻¹.

### 92.2.1 Searches for $Z'$ or $W'$ Bosons

Massive vector bosons or particles with similar decay channels would be expected to arise in Little Higgs theories, in theories of Technicolor, or models involving a dilaton, adjusted to produce a light Higgs boson, consistent with the observed $H^0$. These particles would be expected to decay to pairs of vector bosons, or to third generation quarks, or to leptons. The generic searches for $W'$ and $Z'$ vector bosons listed below can, therefore, be used to constrain models incorporating a composite Higgs-like boson.

A general review of searches for $Z'$ and $W'$ bosons is also included in this volume [67, 68]. In the context of the dynamical electroweak symmetry breaking models, we emphasize their decays to third generation fermions by including a detailed overview, while also briefly summarizing the other searches.

$$Z' \to \ell\ell;$$

ATLAS [69] and CMS [70] have both searched for $Z'$ production with $Z' \to ee$ or $\mu\mu$. No deviation from the standard model prediction was seen in the dielectron and dimuon invariant mass spectra, by either the ATLAS or the CMS analysis, and lower limits on possible $Z'$ boson masses were set. A $Z'_{\text{SSM}}$ with couplings equal to the standard model $Z'$ (a "sequential standard model" $Z'$) and a mass below 5.1 TeV was excluded by ATLAS, while CMS set a lower mass limit of 5.15 TeV. The experiments also place limits on the parameters of extra dimension models and in the case of ATLAS on the parameters of a minimal walking technicolor model [17–21], consistent with a 125 GeV Higgs boson [71]. For a general review of searches in these channels see the PDG review of $Z$ prime in this volume [67].

In addition, both experiments have also searched for $Z'$ decaying to a ditau final state [72, 73]. An excess in $\tau^+\tau^-$ could have interesting implications for models in which lepton universality is not a requirement and enhanced couplings to the third generation are allowed. This analysis led to lower limits on the mass of a $Z'_{\text{SSM}}$ of 2.4 and 2.1 TeV from ATLAS and CMS respectively.
$$Z' \to q\bar{q};$$

The ability to relatively cleanly select $t\bar{t}$ pairs at the LHC together with the existence of enhanced couplings to the third generation in many models makes it worthwhile to search for new particles decaying in this channel. Both ATLAS [74] and CMS [75] have carried out searches for new particles decaying into $t\bar{t}$.

Both ATLAS and CMS searched for $t\bar{t}$ in the all hadronic mode [76] [77] in both the resolved and boosted regions. No evidence of resonance production were seen and limits were produced for various models including the $Z'$ boson in topcolor-assisted technicolor which excludes masses less than 3.1 to 3.6 TeV (ATLAS) depending on the details of the model and 3.3, 5.25, and 6.65 TeV for widths of 1, 10 and 30 percent relative to the mass of the resonance.

ATLAS also presented results on the lepton plus jets final state, where the top quark pair decays as $t\bar{t} \to WbWb$ with one W boson decaying leptonically and the other hadronically; CMS used final states where both, one or neither $W$ decays leptonically and then combined the results. The $t\bar{t}$ invariant mass spectrum was analyzed for any excess, and no evidence for any resonance was seen. ATLAS excluded a narrow ($\Gamma/m = 1.2\%$) leptophobic topcolor $Z'$ boson with masses between 0.7 and 2.1 TeV and with $\Gamma/m = 3\%$ between 0.7 and 3.2 TeV. CMS set limits on leptophobic $Z'$ bosons for three different assumed widths $\Gamma/m = 1.0\%$, $\Gamma/m = 10.0\%$, and $\Gamma/m = 30.0\%$ of 3.9 TeV to 4.0 TeV and exclude RS KK gluons up to 3.3 TeV.

Both ATLAS [78] and CMS [79] have also searched for resonances decaying into $q\bar{q}$, $qg$ or $gg$ using the dijet invariant mass spectrum. Excited quarks are excluded up to masses of 6.7 TeV and model-independent upper limits on cross sections with a

Gaussian signal shape were set. CMS excluded string resonances with masses below 7.9 TeV, scalar diquarks below 7.5 TeV, axigluons and colorons below 6.6 TeV; excited quarks below 6.3 TeV, color-octet scalars below 3.7 TeV, $W'$ bosons below 3.6 TeV, $Z'$ bosons with SM-like couplings below 2.9 TeV and between 3.1 TeV and 3.3 TeV, Randall-Sundrum Gravitons below 2.6 TeV.

$$W' \to \ell\nu;$$

Both LHC experiments have also searched for massive charged vector bosons. In this section we include a summary of the results, with emphasis on final states with third generation fermions, while the details on other decays are discussed in the mini-review of $W'$ [68]. ATLAS searched for a heavy $W'$ decaying to $e\nu$ or $\mu\nu$ and found no excess over the standard model expectation. A sequential standard model (SSM) $W'$ boson (assuming zero branching ratio to $WZ$) with mass less than 7 TeV was excluded [80] using the 139 fb⁻¹ dataset at $\sqrt{s} = 13$ TeV. Model independent cross-section limits as a function of mass were also set. Based on a similar dataset, the CMS experiment excluded a SSM $W'$ boson with mass up to 5.7 TeV [81] and presented upper limits on the production of generic $W'$ bosons decaying into this final state using a model-independent approach.

CMS [82] has carried out a complementary search in the $\tau\nu$ final state. As noted above, such searches place limits on models with enhanced couplings to the third generation. No excess was observed and limits between 2.0 and 2.7 TeV were set on the mass of a $W'$ decaying preferentially to the third generation; a $W'$ with universal fermion couplings was also excluded for masses less than 2.7 TeV.

$$W' \to t\bar{b};$$

Heavy new gauge bosons can couple to left-handed fermions like the SM W boson or to right-handed fermions. $W'$ bosons that couple only to right-handed fermions ($W'_R$) may not have leptonic decay modes, depending on the mass of the right-handed neutrino. For these $W'$ bosons, the $tb$ ($t\bar{b} + \bar{t}b$) decay mode is especially important because in many models the $W'$ boson is expected to have enhanced couplings to the third generation of quarks relative to those in the first and second generations. It is also the hadronic decay mode with the best signal-to-background. ATLAS and CMS have performed searches for $W'$ bosons via the $W' \to tb$ decay channel in the lepton+jets and all-hadronic final state.

The CMS lepton+jets search [83–86], $W' \to tb \to Wbb \to \ell\nu bb$, proceeded via selecting events with an isolated lepton (electron or muon), and at least two jets, one of which is identified to originate from a b-quark. The mass of the $W'$ boson ($M_{tb}$) was reconstructed using the four-momentum vectors of the final state objects ($bb\ell\nu$). The distribution of $M_{tb}$ is used as the search discriminant. A search [86] using 35.9 fb⁻¹ of data, collected at $\sqrt{s} = 13$ TeV, led to an exclusion of $W'_R$ bosons with masses below 3.4 TeV (3.6 TeV) if $M_{W'_R} \gg M_{\nu_R} (M_{W'_R} < M_{\nu_R})$, where $M_{\nu_R}$ is the mass of the right-handed neutrino.

The CMS search for $W' \to tb$ decays using the all-hadronic final state focused on $W'$ masses above 1 TeV [85, 87]. In this region, the top quark gets a large Lorentz boost and hence the three hadronic products from its decay merge into a single large-radius jet. Deep neural network(DNN) algorithms are used to identify the jet initiated by the bottom quark. Techniques including DNN, which rely on substructure information of the jets [88] are employed to identify boosted all hadronic W boson and top quark [87] decays. The $W'$ candidate mass was computed from back-to-back boosted top-tagged jet and a low mass b-tagged jet. From this all-hadronic search, $W'$ bosons were excluded for masses up to 3.4 TeV [87].

ATLAS has searched for $W'_R$ bosons in the $t\bar{b}$ final state both for lepton+jets [89] and all-hadronic [90] decays of the top. No significant deviations from the standard model were seen in either analysis and limits were set on the $W' \to t\bar{b}$ cross section times branching ratio and $W'$ bosons with both purely right and purely left-handed coupling and limits are set between 0.5 and 6 TeV for different values of the coupling strength.

In addition, the above studies also provided upper limits on the $W'$ effective couplings to right- and left-handed fermions.
In Fig. 92.1 (bottom) the upper limits on $W'$ couplings normalized to the SM W boson couplings derived by ATLAS [90] are shown. The top panel of Fig. 92.1 shows the upper limits for arbitrary combinations of left- and right-handed couplings of the $W'$ boson to fermions set using a model independent approach by CMS [86].

### 92.2.2 Searches for Resonances decaying to Vector Bosons and/or Higgs Bosons

Both ATLAS and CMS have used the data collected at $\sqrt{s} = 13$ TeV to search for resonances decaying to pairs of bosons. Overall no significant excesses were seen in the full datasets that were analyzed and the results are interpreted in models with heavy vector triplets (HVT) [92], models with strong gravity and extra spatial dimensions, and model independent limits as a function of mass are set. For a full review of models including extra spatial dimensions and the interpretation of many of these results in that context please see the review of extra dimensions in this volume [61].

Utilizing data collected at $\sqrt{s} = 13$ TeV, ATLAS [93] and CMS [94], have both looked for a resonant state decaying into $VV$ (with $V = W$ or $Z$), $VH$ (with $H$ representing the SM Higgs boson), and $HH$. ATLAS has searched in the $qqqq, \nu\nu qq, \nu lqq, llqq, \nu l\nu, ll\nu\nu, \nu l\nu, llll, q q b b \nu\nu b b, \nu l b b, \text{and } ll b b \text{ final states and combined the results. While CMS analyzed the } q q q q, \nu\nu q q, \nu l q q, llqq, \nu l\nu\nu, \nu\nu b b, \nu l b b, ll b b, b b b b, \tau\tau b b, \text{ and } q q \tau\tau \text{ final states.}$

The combined limits are expressed both as limits on the cross-section as a function resonance mass as well as constraints on the coupling of the heavy boson triplet to quarks, leptons, and the Higgs boson.

$$X \to WZ$$

ATLAS searched for new heavy resonances decaying into $WZ$ in the channels $WZ \to q q q q$ [95], $\nu l q q$ [96], and $\nu l \nu$ [97]. In the fully leptonic channel, the invariant mass of the $WZ$ pair is obtained by considering all possible four lepton permutations in each event. The dominant background is Standard Model continuum $WZ$ production, $ZZ$ production where one lepton is not identified or falls outside the detector acceptance, and top quark plus vector boson production. No resonant production is seen in data and lower limits on the mass of a HVT decaying into $WZ$ are set at 2260 (2460) GeV assuming a coupling constant of $g_V = 1$ ($g_V = 3$). In the $WZ \to \nu l q q$ mode, ATLAS searches in both the cases that the quarks are observed as individual jets (resolved) and where they merge into one jet in the detector (boosted) which probe the low and high $p_T$ regime of the $Z$ boson. No significant excess is seen in either channel and combined lower mass limits are placed at 2900 (3000) GeV for $g_V = 1$ ($g_V = 3$) in the HVT model. In the all hadronic decay mode, ATLAS searched for two high $p_T$ hadronically decaying vector bosons looking for a resonant structure. No excess is seen and limits are placed exclude 1200-3000 (1200-3300) GeV for $g_V = 1$ ($g_V = 3$) in the HVT model.

CMS searched for new heavy resonances in the $ZV$ final state using the semi-leptonic decay channels excluding $W'$ masses up to 1800 GeV in HVT two models with $g_V = 1$ and 3 and in the $ZZ$ channel excludes gravitons up to 1200 GeV. The CMS collaboration searched for $VV \to q q q q$ [98] in the large $R$ dijet search. The $W$ and $Z$ boson are identified through the mass of the large $R$ jet and substructure variables. No excess is seen and limits are set for charged HVT bosons with masses lower than 3200 (3800) GeV for $g_V = 1$ ($g_V = 3$). Cross-section limits as function of mass are reported for the charged spin-1 resonance interpretation and are placed at 44.4 fb at 1.4 TeV to 0.7 pb at 4 TeV. In the $\nu\nu q q$ final state [99], the CMS collaboration searched for a charged spin-1 resonance decaying into a $VZ$ final state with a $Z$ boson decaying into a pair of neutrinos and the other boson decaying into two collimated quarks reconstructed as a large $R$-jet. The transverse mass of the $VZ$ candidate is reconstructed and utilized to search for evidence of resonant $VZ$ production. No excess is seen and lower mass limits are placed on the charged resonance at 3100 (3400) GeV for $g_V = 1$ ($g_V = 3$). In the $2l2q$ final state, the CMS collaboration searched for a heavy resonance decaying into $ZV$ [100] looking for events with one large $R$-jet consistent with the hadronic decay of a vector boson and a $Z$ boson reconstructed in the charged lepton decay channel (e or $\mu$).

Limits are set for a HVT $W'$ with a lower mass of 2270 (2330) for $g_V = 1$ ($g_V = 3$).

$$X \to WW$$

ATLAS searched for a new heavy resonance decaying into $WW$ in the channels $WW \to q q q q$ [95], $\nu l q q$ [96], and $\nu l \nu$ [101]. In the case where both $W$s decay leptonically, ATLAS utilized the transverse mass of the two lepton and two neutrino final state and searched for an excess in this distribution between 200 GeV and 5 TeV. No excess is seen and a HVT is excluded for masses below 1300 GeV. Vector boson fusion is also considered and cross-section limits as a function of mass are placed ranging from 1.3 pb to 0.006 pb at 200 GeV to 3 TeV, respectively. In the $\nu l q q$ mode, ATLAS completed a companion analysis to the $WZ \to$ analysis discussed above and places lower mass limits of 2850 (3150) GeV for $g_V = 1$ ($g_V = 3$) in the HVT model. ATLAS also interprets the all hadronic mode analysis in the hypothesis that $WW \to q q q q$ and places limits on a HVT boson decaying into $WW$ in the all hadronic mode between 1200 and 2200 (1200 and 2800) GeV for $g_V = 1$ ($g_V = 3$).

The CMS collaboration searched for $VV \to q q q q$ [98] in the large $R$ dijet search. The $W$ and $Z$ boson are identified through the mass of the large $R$ jet and substructure variables. No excess is seen and limits are set for charged HVT bosons with masses lower than 2700 (2800) GeV for $g_V = 1$ ($g_V = 3$). Cross-section limits as function of mass are reported for the uncharged spin-1 resonance interpretation and are placed at 41.6 fb at 1.4 TeV to 0.6 pb at 4 TeV.

$$X \to VH$$

ATLAS searched for a new heavy resonance decaying into $WH$ and $ZH$ in the $q q b b$ ($WH$ and $ZH$) [102], $\nu l b b$ ($WH$), $\nu\nu b b$ ($ZH$), $ll b b$ ($ZH$) [103] modes. In the all hadronic mode, ATLAS searched for boosted $VH$ production looking for two large $R$ jets where the larger invariant mass large $R$ jet is interpreted as the Higgs boson decay products while the lesser invariant mass jet is taken to be the hadronically decaying vector boson requiring b-tagging on the Higgs boson subjets. The invariant mass is reconstructed and a search is performed for resonant production of $ZH$. None is found and limits from 1100 to 2500 (1300 to 3800 GeV) are placed for $g_V = 1$ ($g_V = 3$). ATLAS also searched for $XH$ where the $W$ or $Z$ boson decays into $\nu\nu, \nu l, \text{ and } ll$. The analysis searched for both resolved and merged (boosted) b-jets from the decay of the Higgs boson and defines the signal regions based on the number of reconstructed charged leptons (0,1,or 2). In the dilepton channel the invariant mass is explicitly reconstructed of the entire diboson system, the single lepton channel reconstructs the diboson final state constraining the lepton and missing transverse momentum utilizing the known W boson mass, while the 0 charged lepton channel reconstructs the transverse mass of the diboson system. No excess is seen in any channel and limits on the production of a HVT are placed at 2800 GeV (2930) GeV for $g_V = 1$ ($g_V = 3$).

The CMS Collaboration searched for a heavy resonance decaying into $VH$ [104] searching for a resonances decaying into a Higgs boson and a hadronically decaying $W$ or $Z$ boson. The search identifies events with two large-$R$ jets using substructure variables and requires one large-$R$ jets is tagged with a pair of b-hadrons clustered in a single jet. The invariant mass of the $VH$ bosons is reconstructed and evidence for resonance production is sought. No excess is seen and limits are placed. With $g_V = 1$ (3) a narrow $W'$ resonance with $m_{W'} < 2470(3150)$ GeV and $m_{Z'} < 1150(1190)$.

The CMS collaboration searched for a heavy resonance decaying into a pair of boosted Higgs boson with $HH \to b \bar b W W^*$ in the single and dilepton channel [105]. Events are categorized according to lepton flavor and multiplicity, along with discriminators that characterize the compatibility with having jets arising from the decay of boosted Higgs and $W$ bosons. The reconstructed Higgs and diHiggs mass distributions are fit and used as the final discriminators. No significant deviations in the 12 sub-categories are seen and 95% confidence limits on spin-0 bosons are set from 24.5 fb at 0.8 TeV to 0.78 fb at 4.5 TeV and spin-2 bosons from 16.7 fb at 0.8 TeV to 0.67 fb at 4.5 TeV. A similar search in the
<figure><img src="image_387.png" /><figcaption>**Figure 92.1:** Left panel: Observed limits on the W' boson mass as function of the left-handed ($a_L$) and right-handed ($a_R$) couplings. Black lines represent contours of equal W' boson mass [86]. Right panel: Observed and expected regions, on the g'/g vs mass of the W' boson plane, that are excluded at 95% CL, for right-handed W' bosons [91] showing the exclusion of the values of a hypothetical right-handed coupling of a heavy W particle.</figcaption></figure>

four b-quark final state is used to set limits range 9.74 to 0.29 fb and 4.94 to 0.19 fb in the spin-0 bosons and spin-2 gravitons (respectively) interpretation with masses of 1-3 TeV [106].

$X \to H\gamma$ or $V\gamma$:

Both CMS and ATLAS search for heavy resonances decaying into either a vector boson and a photon or a Higgs boson or a photon. These searched are motivated generically by the fact that new heavy gauge bosons are generically predicted to decay to either a Higgs boson or a vector boson and a photon by one loop decays [107]. ATLAS searched for a heavy resonances decaying into a Higgs boson and a photon and sets limits from 0.7 to 4 TeV from 11.6 fb to 0.11 fb [108]. ATLAS searched for a heavy resonance decaying into $XH$ where $XH \to qqbb$ [109] and sets limits from 1 to 4 TeV. CMS searched for a heavy resonance decaying into a W boson and a photon [110] setting limits for narrow resonances range between 0.17 fb at 6.0 TeV and 55 fb at 0.7 TeV.

### Summary of Searches with Diboson Final States:

Both ATLAS [93] and CMS [94] provide plots summarizing the various searched results and limits combining. The results are shown in the context of HVT models and models of strong gravity with extra spatial dimensions. No excess is seen in any search and limits on the 4.3 (4.5) TeV (ATLAS) and (CMS). Inclusion of decays directly to fermions increase these limits to 5.3 (5.5) TeV and 5.0 (5.2) TeV from the ATLAS and CMS combinations, respectively. Both collaborations also place varying limits on the coupling strength as a function of HVT boson mass as well.

### Searches for Triboson Resonances:

The CMS collaboration presents the first search for massive triboson resonances in both the semi-leptonic [111] and fully hadronic final state [112]. Three boson final states can be produced via the decay of Kaluza-Klein (KK) states decaying via $W_{KK} \to WR$ with $R \to WW$ where R is a scalar radion. The search utilizes novel deep neural networks to perform optimal separation between signal and background and searched for evidence of resonance structure in the triboson invariant mass-spectrum. No excess above SM backgrounds are seen and $W_{KK}$ and radion masses are excluded up to $M_{W_{KK}} = 3.4$ TeV and $M_R = 1$ TeV and up to $M_{W_{KK}} = 3.6$ TeV and $M_R = 0.35$ TeV. $M_{W_{KK}}$ below 3 TeV are excluded for $0.6 < \frac{M_R}{M_{W_{KK}}} < 0.7$ in the semi-leptonic final state and up to $M_{W_{KK}} = 3.0$ TeV and $M_R = 200$ GeV and up to $M_{W_{KK}} = 1.5$ TeV and $M_R = 1.5$ TeV in the fully hadronic final state.

### 92.2.3 Vector-like third generation quarks

Vector-like quarks (VLQ) have non-chiral couplings to W bosons, i.e. their left- and right-handed components couple in

the same way. They therefore have vectorial couplings to W bosons. Vector-like quarks arise in Little Higgs theories, top-coloron-models, and theories of a composite Higgs boson with partial compositeness. In the following, the notation T quark refers to a vector-like quark with charge 2/3 and the notation B quark refers to a vector-like quark with charge -1/3, the same charges as the SM top and b quarks respectively. The exotic vector-like quarks $X_{5/3}$ and $Y_{-4/3}$ have charges 5/3, and -4/3 respectively. Vector-like quarks couple with SM quarks with Yukawa interactions and may exist as SU(2) singlets ($T$, and $B$), doublets [(X$_{5/3}$, T), (T, B), (B, Y$_{-4/3}$)], or triplets [(X$_{5/3}$, T, B), (T, B, Y$_{-4/3}$)]. At the LHC, VLQs can be pair produced via the dominant gluon-gluon fusion process. VLQs can also be produced singly by their electroweak effective couplings to a weak boson and a standard model quark. The single production rate is expected to dominate over the rate of pair production at large VLQ masses. T quarks can decay to $bW$, $tZ$, or $tH^0$. Weak isospin singlets are expected to decay to all three final states with (asymptotic) branching fractions of 50%, 25%, 25%, respectively. Weak isospin doublets are expected to decay exclusively to $tZ$ and to $tH^0$ [113] with equal branching ratios. Analogously, B quarks can decay to $tW$, $bZ$, or $bH^0$. The $Y_{-4/3}$ and $X_{5/3}$ quarks decay exclusively to $bW$ and to $tW$. While these are taken as the benchmark scenarios, other representations and decays to exotic new particles are possible [114–116], and hence the final results are interpreted for many allowed branching fraction combinations.

Given the multiple decay modes of the VLQs, the final state signatures of both pair produced and the singly produced VLQs are fairly rich with leptons, jets, b-jets, and missing energy. Depending on the mass of the VLQ, the top quarks and $W/Z/H^0$ bosons may be Lorentz boosted and identified using jet substructure techniques. Thus the searches are performed using lepton+jets signatures, multi-lepton and all-hadronic decays. In addition, T or B quarks with their antiparticles can result in events with same-sign leptons, for example if the decay $T \to tH \to bWW^+W^-$ is present, followed by leptonic decays of two same-sign W bosons. In the following subsections, while we describe the searches for each of the decay modes of the VLQs, the same analysis can be re-interpreted to obtain the sensitivity to a combination with varied branching fractions to the different decay modes.

In the following sections, the results obtained for T (B) quarks assuming 100% branching ratio to $Wb$ ($Wt$) are also applicable to heavy vector-like $Y_{-4/3}$ ($X_{5/3}$) with charge 4/3 (5/3).
<figure><img src="image_388.png" /><figcaption>Figure 92.2: Left panel: Observed limits from $W'$ to diboson from CMS [94]. Right panel: Observed limits from $W'$ to diboson decays from ATLAS [93].</figcaption></figure>

### 92.2.3.1 Searches for $T$ quarks that decay to $W$, $Z$ and $H^0$ bosons

$$T/Y \to bW$$

CMS has searched for pair production of heavy $T$ quarks that decay exclusively to $bW$ [117–119]. The analysis selected events with exactly one charged lepton, assuming that the $W$ boson from the second $T$ quark decays hadronically. Under this hypothesis, a 2-constraint kinematic fit can be performed to reconstruct the mass of the $T$ quark as a narrow mass peak with a mass resolution of around 7%. In Refs. [118] and [119], the two-dimensional distribution of reconstructed mass vs $S_T$ was used to test for the signal, where $S_T$ is the scalar sum of the missing $p_T$ and the transverse momenta of the lepton and the leading four jets. This analysis, when combined with the search in the fully hadronic final state [120] excluded new quarks that decay 100% to $bW$ for masses below 0.89 TeV [119]. At times the hadronically-decaying $W$ boson is produced with a large Lorentz boost, leading to the $W$ decay products merged into a large-radius jet. Algorithms such as jet pruning [121] were used to remove contributions from soft, wide angle radiation, from large-radius jets, leading to better discrimination between jets from multijet events and those arising from decays of the heavy particles. If the mass of the boosted jet was compatible with the $W$ boson mass, then the $W$ boson candidate jet and its subjects were used in the kinematic reconstruction of the $T$ quark. No excess over standard model backgrounds was observed. Upper limits on the production cross section as a function of the mass of $T$ quarks were measured. By comparing them with the predicted cross section for vector like quark pair production, the strong pair production of $T$ quarks was excluded for masses below 1.30 TeV (1.28 TeV expected) [117].

Another “cut-based” search for pair produced $T$ quarks in the all-hadronic final state targeting the $Wb$ decay mode [122], relies on mass reconstruction of two highest $p_T$ $Wb$ combinations using boosted $W$ boson candidates with $p_T > 200$ GeV and b-tagged jets. $H_T$ is used as the signal discriminator, with selected events divided into nine categories based on the multiplicity of $W$ and b-jets in the event. From this search $T$ quarks with pure $Wb$ decays are excluded for masses below 1.04 TeV (1.07 TeV expected).

An analogous search has been carried out by ATLAS [123,124] for the pair production of heavy $T$ quarks. It used the lepton+jets final state with an isolated electron or muon and at least four jets, including a b-jet, and required reconstruction of the $T$ quark mass. Given that the mass range for the $T$ quark being explored was from a 0.4 TeV to a couple of TeV, the $W$ boson from the $T$ quark may fall in two categories: those with a high boost leading to merged decay products, and others where the two jets from the $W$ boson were resolved. In addition, the selection was optimized to require large angular separation between the high $p_T$ $W$ bosons and the b-jets.

The $T \to Wb$ candidates were constructed from both the leptonically and hadronically decaying $W$ bosons by pairing them with the two highest $p_T$ b-tagged jets in the event. The pairing of b-jets with $W$ bosons which minimizes the difference between the masses of leptonically decaying $T$ ($m_{lep}(T)$) and the hadronic $T$ ($m_{had}(T)$) was chosen. Finally, $m_{lep}(T)$ was used as the discriminating variable in a signal region defined by high $S'_T$ (here $S'_T$ is defined as the scalar sum of the missing $p_T$, the $p_T$ of the lepton and jets), and the opening angle between the lepton and the neutrino ($\Delta R(e, \nu)$). With the 36.1 fb$^{-1}$ data collected during Run 2 at $\sqrt{s} = 13$ TeV, assuming 100% branching ratio to the $Wb$ decay, the observed lower limit on the $T$ mass was 1.35 TeV, and in the SU(2) singlet scenario, the lower mass limit was obtained to be 1.17 TeV [123].

A targeted search for a $T$ quark, produced singly in association with a light flavor quark and a b quark and decaying into $bW$, was carried out by CMS at $\sqrt{s}=13$ TeV and a dataset corresponding to 2.3 fb$^{-1}$ [125]. The analysis used lepton+jets events, with at least one b-tagged jet with large transverse momentum, and a jet in the forward $\eta$ region. Selected events were required to have $S''_T > 500$ GeV, where $S''_T$ is defined as the scalar sum of the transverse momenta of the lepton, the leading central jet, and the missing transverse momentum. The invariant mass of the $T$ candidate was used as the discriminating variable and was reconstructed using the four-vectors of the leptonically decaying $W$ boson and the leading central jet. No excess over the standard model prediction was observed. As the VLQ width is proportional to the square of the coupling, upper limits were set on the production cross section assuming a narrow width VLQ with coupling greater than 0.5. For $Y/T$ quarks with a coupling of 0.5 and a 100% branching fraction for the decay to $bW$ the excluded masses were in the range from 0.85 to 1.40 TeV [125]. A similar search [126,127] was performed by ATLAS for a singly produced $T$ or $Y_{-4/3}$ quark decaying to $Wb$ using a dataset corresponding to 36.1 fb$^{-1}$. The search was performed using lepton+jets events with a high $p_T$ b-tagged jet, and at least one forward jet. The reconstructed mass of the $T/Y_{-4/3}$ quark, was used as the discriminating variable and showed no excess above the expectation from SM. Interference effects with the SM background are included in the study. This search led to 95% CL upper limits on the mixing angle $|\sin(\theta_L)|$ ($C_L^{Wb}$) in the range of 0.18–0.35 (0.25–0.49) for singlet $T$ quark mass between 0.8–1.2 TeV. This search also provided limits as a function of the $Y_{-4/3}$ quark mass, on the coupling of the $Y_{-4/3}$ quark to $bW$, and the mixing parameter $|\sin\theta_R|$ ($C_R^{Wb}$) for a $(B,Y_{-4/3})$ doublet model [126]. For VLQ masses between 0.08–1.8 TeV, the limits on $|\sin(\theta_R)|$ ($C_R^{Wb}$) are in the range 0.17–0.55 (0.24–0.77), and for $Y_{-4/3}$ quark mass between 0.9–1.25 TeV, the limits on $|\sin\theta_R|$ are around 0.18–0.19 and below the constraints from electroweak precision observables.
For $Y_{-4/3}$ quark in the triplets $(T, B, Y_{-4/3})$, limits on $|\sin(\theta_L)|$ ($C_L^{Wb}$) are between 0.16–0.39 (0.31–0.78) for masses between 0.8 GeV–1.6 TeV [126].

$T \to tH^0$:

ATLAS has performed a search for $t\bar{t}$ production with $T \to tH^0$ [124, 128]. Given the dominant decay mode $H^0 \to b\bar{b}$, these events are characterized by a large number of jets, many of which are b-jets. Thus the event selection required one isolated electron or muon and high jet multiplicity (including b-tagged jets). The sample is categorized by the jet multiplicity (5 and $\ge$6 jets in the 1-lepton channel; 6 and $\ge$7 jets in the 0-lepton channel), b tag multiplicity (2, 3 and $\ge$4) and mass-tagged jet multiplicity (0, 1 and $\ge$2). The distributions of $m_{eff}$, defined as the scalar sum of the lepton and jet $p_T$ and the missing $p_T$, for each category were used as the discriminant for the final signal and background separation. No excess of events was found. Weak isospin doublet $T$ quarks were excluded below 1.16 TeV.

A search by ATLAS for pair produced VLQs with an all-hadronic final state signature yields an exclusion of pure decays $T \to tH^0$ upto a $T$ quark mass of 2.3 TeV [129].

The CMS search for $t\bar{t}$ production, with $T \to tH^0$ decays has been performed in lepton+jets, multilepton and all-hadronic final states. The lepton+jets analysis [130] emphasizes the presence of a large number of b-tagged jets, and combines it with other kinematic variables in a Boosted Decision Tree (BDT) to enhance signal to background discrimination. The multilepton analysis [130] was optimized for the presence of b-jets and the large hadronic activity. For $B(T \to tH^0) = 1$, the combined lepton+jets and multilepton analyses led to a lower limit on $T$ quark masses of 0.71 TeV. A search for $T \to tH^0$ in all-hadronic decays [131], optimized for a high mass $T$ quark, and based on identifying boosted top quark jets has been carried out by CMS. This search aimed to resolve subjets within the jets arising from boosted top quark decays, including b tagging of the subjets. A likelihood discriminator was defined based on the distributions of $H_T$ and the invariant mass of the two b-jets in the events for signal and background. No excess above background expectations was observed. Assuming 100% branching ratio for $T \to tH^0$, this analysis led to a lower limit of 0.75 TeV on the mass of the $T$ quark.

Searches for $T$ quarks at $\sqrt{s}=13$ TeV, based on a $2.6 \text{ fb}^{-1}$ dataset [132] have been performed by CMS using the lepton+jets final state. This search has been optimized for high mass $T$ quarks by exploiting techniques to identify $W$ or Higgs bosons decaying hadronically with large transverse momenta. The boosted $W$ channel excluded $T$ quarks decaying only to $bW$ with masses below 0.91 TeV, and the boosted $tH$ channel excluded $T$ quarks decaying only to $tH$ for masses below 0.89 TeV.

A CMS search for $T \to tH^0$ with $H^0 \to \gamma\gamma$ decays has been performed [133] in pair production of $T$ quarks. To identify the Higgs boson produced in the decay of the heavy $T$ quark, and the subsequent $H^0 \to \gamma\gamma$ decay, the analysis focused on identification of two photons in events with one or more high $p_T$ lepton+jets or events with no leptons and large hadronic activity. A search for a resonance in the invariant mass distribution of the two photons in events with large hadronic activity defined by the $H_T$ variable showed no excess above the prediction from standard model processes. The analysis resulted in exclusion of $T$ quark masses below 0.54 TeV.

A search for electroweak single production of $T$ quarks decaying to $tH^0$ using boosted topologies in fully hadronic [134] and lepton+jets [135] in the final states has been performed by CMS. The electroweak couplings of the $T$ quarks to the SM third generation quarks are highly model dependent, and hence these couplings determine the rates of the single $T$ quark production. In both analyses, $T$ quark invariant mass was reconstructed using the boosted Higgs boson jet and the top quark. Higgs boson jets were identified using jet substructure techniques and subjet $b$ tagging. For the lepton+jets analysis, the top quark was reconstructed from the leptonically decaying $W$ and the $b$ jet, while in the all-hadronic analysis the top quark jet was tagged using substructure analysis. There was no excess of events observed above background. Exclusion limits on the product of the production cross

section and the branching fraction $(\sigma(pp \to Tqt/b) \times B(T \to tH^0))$ were derived for the $T$ quark masses in the range 0.70–1.8 TeV. From the lepton+jets analysis, for a mass of 1.0 TeV, values of $(\sigma(pp \to Tqt/b) \times B(T \to tH^0))$ greater than 0.8 and 0.7 pb were excluded assuming left- and right-handed coupling of the $T$ quark to standard model fermions, respectively [135]. For the all-hadronic analysis, upper limits between 0.31 and 0.93 pb were obtained on $(\sigma(pp \to Tqt/b) \times B(T \to tH^0))$ for $T$ quark masses in the range 1.0–1.8 TeV [134]. ATLAS performs a similar search [136] for single production of vector-like $T$ quarks decaying into a Higgs Boson and a top quark and places limits between 1 and 1.7 TeV depending on the coupling

$T \to tZ$:

Both ATLAS and CMS search for $T$ quarks that decay exclusively into $tZ$ in $pp$ collisions at $\sqrt{s} = 13$ TeV. No excesses were found in either search.

ATLAS performs a search [137] optimized for pair production of vector-like top quarks decaying into $tZ$ where the $Z$ boson subsequently decays into neutrino pairs, utilizing $139 \text{ fb}^{-1}$ of data. The search selected events with one lepton, multiple jets, and significant missing transverse momentum. No significant excesses were found and lower limits on the mass of a vector like top quark were placed, excluding masses below 1.59 TeV (weak-isospin doublet) and 1.47 TeV (pure $tZ$ mode).

Another search by ATLAS for pair produced $T$ decaying to $tZ$ has been carried out by reconstructing the high transverse momentum $Z$ boson from a pair of opposite-sign same-flavor leptons, using events with two or three charged leptons [138]. The final analysis is based on three final state signatures. In the dilepton events, at least one b-tagged jet is required and $H_T$, formed from all hadronic jets and charged leptons, is used as the discriminating variable. In events with two leptons, at least 2 b-jets are requested and events with zero or one high $p_T$ top-tagged jet, use $H_T$ as the discriminator. The second dilepton analysis with two top-tagged high $p_T$ jets focuses on hadronically decaying heavy resonances and the invariant mass of the $Z$ boson and the highest $p_T$ b-tagged jet is found to be a good discriminating variable. No excess was observed over the background expectations. Limits were placed at a heavy vector like top of 1.6 TeV.

ATLAS has subsequently carried out a search [139] for singly produced $T$ quarks decaying to $tZ$ where the $Z$ boson decays into neutrino pairs. The search is carried out using $36.1 \text{ fb}^{-1}$ of data in events with two different final state signatures: one with jets and significant missing $p_T$ (0L) and the other with a single lepton, jets and missing $p_T$ (1L). Events are divided into signal and dedicated $W$+jets and $t\bar{t}$ background control regions. The sensitivity to the $T$ quark signal is extracted using distributions of missing $p_T$ for the 1L and the distribution of $T$ quark transverse mass constructed from missing $p_T$ and the high $p_T$ large-radius top-tagged jet, for 0L analysis. There is no excess found over the expected background and lower limits on the production of $T$ singlets are obtained as a function of the left- and right-handed couplings $c_{L,W}$ and $c_{R,W}$ to top quarks and $W$ bosons, where $c_W$ above 0.7 is excluded for $T$ quark mass of 1.4 TeV. The limits on $c_W$ are also recasted into expected and observed 95% CL upper limits for the mixing angle $(\theta_L)$ of a singlet $T$ with the top quark.

CMS searched [140] for single production of $T$ quarks decaying into $tZ$ with the $Z$ boson decaying to pairs of charged leptons (electrons and muons) and the top quark decaying hadronically using $35.9 \text{ fb}^{-1}$ of data. Limits were placed on $T$ quarks with masses between 0.7 and 1.7 TeV excluding the product of cross section and branching fraction above values of 0.27 to 0.04 pb. Additionally, limits on the product of cross section and branching fractions for a $Z'$ boson decaying into $tZ$ were set between 0.13 and 0.06 pb for $Z'$ boson masses in the range from 1.5 to 2.5 TeV.

Similar searches by ATLAS for singly produced $T$ decaying to $Zt$ have been performed in final state signatures with two or three charged leptons [141]. The analysis relies on tagging b-jets and high $p_T$ large-radius leptons originating from top-quarks. Additional selections are devised to reduce the contributions from pair production of $T$ quarks. For events with dilepton analysis, the discriminating variable is the mass of the $T$ quark formed using the invariant mass of the $Z$ boson candidate and the highest $p_T$ top-
tagged jet, while for the trilepton analysis, the variable $S_T$ is used to search for an excess of data over the expected SM background. No excess above the SM expectations is observed. The two final states (dilepton and trileptons) are combined to obtain the final results. For the coupling parameter $\kappa_T$ between 0.1–1.6, the 95% CL upper limits on the production cross section times branching fraction into $Zt$ is between 0.16–0.18 (0.03–0.05) pb at T quark mass of 0.7 (2) TeV.

The search by the ATLAS experiment for VLQ pair production optimized to search for $T \to tZ$ decays [142] in a dataset with 139 fb⁻¹ of luminosity has been performed in final states with either two leptons or three or more leptons. A multi-class boosted object tagger based on DNN techniques for large-radius jets is used to categorize events according to the number of high $p_T$ boosted $H^0$, $Z/W$, and top quark jets. The discriminant in the signal to evaluate the sensitivity depends on the number of leptons in the event. For the two lepton analysis, the discriminant is the mass of the T quark which is constructed from b and Z candidates, while for the three or more lepton analysis, $H_T$ computed using the $p_T$ of the jets and leptons is used as the discriminant. This analysis excludes T quark masses up to 1.27 GeV (1.46 GeV), for the singlet (doublet) configuration [142].

Combination of $T \to tZ/tH^0$:

The search performed by the ATLAS experiment for electroweak single production of T quark decaying to $tH^0$ and $tZ$ uses a dataset corresponding to 139 fb⁻¹ of integrated luminosity, and events with a single lepton with multiple jets and b-jets in the final state [143]. The single production channel for VLQs probes $\kappa$, the universal coupling constant, which also controls the production cross section and the resonance width of the VLQ. The analysis uses techniques to tag boosted jets, and categorizes events by numbers of jets and b-jets. The event discriminant is the “effective mass” (meff) observable, defined as the scalar sum of the $p_T$ of all central jets, $p_T$ of leptons and the missing $p_T$ in the event. No significant excess is observed. Limits on the mass of T quark and universal coupling strength ($\kappa$) are obtained. For singlet T quarks, values of $\kappa$ above 0.5 are excluded for all masses below 1.8 TeV. For T quark mass of 1.6 TeV, $\kappa$ above 0.41 is excluded [143].

A CMS search published in 2020 concentrates on the electroweak production of the T quark with $T \to tH^0$, and $T \to tZ$ decays, where subsequently both $H^0$ and Z Bosons decay hadronically, leading to fully hadronic final states [144]. The search focuses on evidence of T quarks produced in association with a b quark ($qg \to Tbq'$) or a top quark ($qg \to Tiq$). The production cross sections are model dependent, as the electroweak production depends on the strength of the T quark coupling, $TbW$ ($TtZ$), for the charged-current (neutral-current) process, at the production vertex. The searches are split into two domains depending on the mass of the T quark and consider a wide range of widths of the T quark from a few percent to about 30% of the T quark mass. The event selection relies primarily on the large number of jets and on efficient identification of jets from b-quarks, in addition to double-b jet decays of the $H^0$ and Z bosons. The final sensitivity is derived by the search for a resonant peak in the $tH^0/Z$ invariant mass spectrum. For low mass T quark searches, three independent regions based on the b-tagged jet requirements are examined. For T quark masses above 1 TeV, which result in highly Lorentz-boosted top quarks and $H^0$ or Z boson, large-area jets are used to form the T quark. This search reports upper limits at 95% confidence level on $\sigma B(T \to tH)$ and $\sigma B(T \to tZ)$ between 2 pb and 20 fb for T masses between 0.6 to 2.6 TeV in the $Tbq$ and $Ttq$ production channels. The analysis also reports combined results for $T \to tH$ and $T \to tZ$ associated production with a bottom quark and provides constraints on T quarks in the T singlet model for masses below 1.00 TeV. For an expected fractional width of 30%, the expected sensitivity extends to 1.28 TeV, comparable to the most stringent results.

Combination of $T \to bW/tZ/tH^0$:

Most of the analyses described above target an individual decay mode of the T quark, with 100% branching ratio to either $bW$, $tZ$ or $tH^0$ and are optimized accordingly. However, they have

varied sensitivity to all three decay modes and the results can be interpreted as a function of branching ratios to each of the three decay modes, assuming the total adds up to unity ($B(tH) + B(tZ) + B(Wb) = 1$).

Combinations of analyses have been performed by both ATLAS and CMS. The limits set by ATLAS searches in $W(\ell\nu)b + X$, $H(bb)b + X$, $Z(\nu\bar{\nu})$, $Z(\ell\ell)t/b + X$, dileptons with same-sign charge, trileptons, all-hadronic final states have been combined and the results obtained for various sets of branching fractions for T quark decays to $bW$, $tH^0$ and $tZ$ are shown in Fig. 92.3 (left). In the combined analysis, ATLAS sets lower T quarks mass limit of 1.31 TeV for all possible values of the branching fractions to the three decay modes [123, 145, 146]. In Fig. 92.3, exclusion is shown in the plane of $B(T \to Ht)$ versus $B(T \to Wb)$, for different values of the T quark mass. The default branching ratio values for the weak-isospin singlet and doublet cases are also shown in Fig. 92.3 as yellow circle and star symbols respectively. Assuming a weak isospin ($T,B$) doublet and $|V_{Tb}| \ll |V_{tB}|$, T quark mass below 1.37 TeV is excluded.

A CMS analysis for pair production of T combines three channels with lepton final states: single lepton, two leptons with the same sign of the electric charge ($SS$), or at least three leptons (trilepton) [147]. For various combinations of branching fractions for T quark decays to $bW$, $tH^0$ and $tZ$, the combined results exclude T quarks with masses below 1.14–1.3 TeV and are shown in Figure 92.3 (right). Single lepton events are classified into 16 signal categories and 6 background control regions based on multiplicity of b-tagged, high $p_T$ $H^0$ and W-tagged jets. The discriminating variables are $H_T$ for $H^0$-tagged events and the minimum invariant mass constructed from the lepton and the b jet, $\min[M_{lb}]$, for zero $H^0$-tagged events. For the same-sign dilepton and trilepton analyses, the non-prompt backgrounds due to misidentified jets and leptons are derived from data control regions. In the trilepton analysis the $S_T$ variable is used as the signal discriminator binned in four categories based on the lepton flavor combinations ($eee$, $ee\mu$, $e\mu\mu$, $\mu\mu\mu$). The single lepton analysis is most sensitive for $tHbW$ and $WbWb$ decay modes, within the $SS$ dilepton analysis $tHtH$ and $tHtZ$ have the best efficiency and for trileptons the $tZtZ$ and $tHtZ$ decays modes have the highest efficiency. CMS excludes singlet (doublet) T quark masses below 1.2 (1.28) TeV. Masses below 800 GeV were excluded in previous searches. For T quark masses in the range 0.8–1.8 TeV, cross sections smaller than 30.4–9.4 fb (21.2–6.1 fb) are excluded for the singlet (doublet) scenario.

Another inclusive search for pair produced T in the all-hadronic final state [122] has been performed by CMS using the boosted event shape tagger (BEST NN) neural network technique to classify jets in six categories $W$, $Z$, $H^0$, $t$, $b$, and light. This search does not focus on a given VLQ mode, but on various combinations of the boson and quark jets in the final state. Anti-$k_T$ jets with a distance parameter of 0.8 are used. The BEST NN algorithm simultaneously classifies jets according to heavy object type. For each of the six particle hypotheses, it boosts each jet constituent into corresponding frame along the jet momentum direction, and calculates event shape and angular variables in the boosted frame, with the expectation that when boosting to the correct rest frame, jet constituents will be isotropic and show the expected N-prong structure of the decaying object in its rest frame. A neural network is trained using the event shape and angular variables in the boosted frame to classify jets according to one of those six possibilities ($W$, $Z$, $H$, $t$, $b$, or light). The analysis bins the events into 126 categories depending on the number of $W$, $Z$, $H$, $t$, $b$, or light jets in the final state with a maximum of four such objects. For each category $H_T^{AK8}$, the scalar sum of $p_T$ of all AK8 jets, is used as the signal discriminator. A scan over a combination of various branching fractions is also performed. This search excludes T quark masses in the range 0.74–1.37 TeV for the $tH$ decay mode in the NN analysis.

An inclusive search for VLQs has been carried out by CMS targeted at heavy T quarks decaying to any combination of $bW$, $tZ$, or $tH^0$ as described in [130]. Selected events have at least one isolated charged lepton. Events were categorized according to number and flavor of the leptons, the number of jets, and the
<figure><img src="image_389.png" /><figcaption>Figure 92.3: Left panel: observed limits on the mass of the T quark in the plane of $B(T \to tH^0)$ versus $B(T \to bW)$ from a combination [145] of all ATLAS searches for TT production. The markers indicate the default branching ratios for the SU(2) singlet and doublet scenarios. Right panel: the observed lower limits on the T quark mass (in GeV), from CMS searches after combining all lepton channels [147], for various branching fraction scenarios. $B(T \to Wb) + B(T \to tZ) + B(T \to tH^0) = 1$ is assumed.</figcaption></figure>

presence of hadronic vector boson and top quark decays that are merged into a single jet. The use of jet substructure to identify hadronic decays significantly increases the acceptance for high T quark masses. No excess above standard model backgrounds was observed. Limits on the pair production cross section of the new quarks are set, combining all event categories, for all combinations of branching fractions into the three final states. For T quarks that exclusively decay to $bW/tZ/tH^0$, masses below 0.70/0.78/0.71 TeV are excluded.

### 92.2.3.2 Searches for B quarks that decay to W, Z and H⁰ bosons

ATLAS and CMS have performed searches for pair production of heavy B quarks which subsequently decay to $Wt, bZ$ or $bH^0$. The searches have been carried out in final states with single leptons, dileptons (with same charge or opposite charge), multileptons, as well as in fully hadronic final states.

$$B \to bH^0;$$

A search for B pair-produced events has been performed by the CMS experiment [123] using 137 fb⁻¹ of data where the B decays into a $b\bar{b}$ quark and either a $H^0$ or a Z boson. This search concentrates on final states with fully hadronic signatures, and utilizes different jet multiplicity categories to account for two resolved jets or merged single jets from $H^0$ or Z boson to $b\bar{b}$ decays, depending on the $p_T$ of the $H^0$ or Z bosons. Nine final state categories are used, depending on the number of observed jets and the reconstructed event mode. No significant excess over expected background is observed. For the case $B(B \to bH) = 100\%(50\%)$, VLQs with masses below 1.57 (1.45) TeV are ruled out.

Using 36.1 fb⁻¹ of data, ATLAS has performed a search for pair produced VLQs with all-hadronic final state signature [148]. While this analysis provides exclusion limits for all third generation VLQs, it provides the strongest results for the $B \to bH^0$ decay mode and excludes decays with $B(B \to |H'|) = \infty, I$ scenario for B masses upto 1.01 TeV. The limits are also cast in a two-dimensional plane of branching ratio values of $B \to bH^0$ vs. $B \to Wb$. This analysis required the presence of high $p_T$ jets and multiple b tags. It used a multi-class DNN to classify jets arising from W, Z, $H^0$ bosons and top-quarks. In addition, the matrix element method was used to compute the likelihood for the event to arise from a particular VLQ final state and to construct the final discriminator. To increase the sensitivity of the analysis, processes with the same number of top quarks, W/Z bosons, and $H^0$ Higgs bosons are combined into a single hypothesis.

ATLAS has performed a search for single production of a vector-like B quark decaying to a b-quark and a $H^0$ boson, with $H^0 \to b\bar{b}$

decay, using a dataset corresponding to an integrated luminosity of 139 fb⁻¹ [149]. The all hadronic final state signature relies on identification of at least 3 b-jets, with an additional soft forward jet from the spectator quark. Large area jets are used to identify boosted $H^0$ decays. This analysis results in excluding the single production of a VLQ (B, Y) doublet with relative width larger than 5% between masses of 1.0 TeV and 2.0 TeV [150]. The CMS experiment has searched for a single vector-like quark decaying into tH or tZ in the all hadronic final state with 138 fb⁻¹ of data in the all hadronic channel reconstraining both the boson invariant mass and the tH or tZ invariant mass and searching for a resonance. No significant excess is found and cross-section limits are placed between 600 GeV and 1200 GeV between 600 and 68 fb [151].

$$B \to Wt;$$

A search for $B \to tW$ in B pair produced events has been performed by ATLAS [123] using lepton+jets events with one hadronically decaying W and one leptonically decaying W utilizing 36.1 fb⁻¹ of data at $\sqrt{s} = 13$ TeV. The search was optimized for T production decaying into $Wb$. Since the analysis was optimized for $T \to Wb$ rather than $Wt$ decays the analysis does not reconstruct the full B mass. As discussed earlier, the hadronically and leptonically decaying heavy quarks were required to have similar reconstructed masses (within 300 GeV). The interpretation of the $T \to Wb$ in the context of $B \to tW$ production led to the exclusion of heavy B like VLQs for masses less than 1.25 TeV and 1.08 TeV, assuming a 100% branching fraction to $tW$ or SU(2) singlet B scenario, respectively.

A similar search by CMS [152], using 19.8 fb⁻¹ of $\sqrt{s} = 8$ TeV data, selected events with one lepton and four or more jets, with at least one b-tagged jet, significant missing $p_T$, and further categorized them based on the number of jets tagged as arising from the decay of boosted W, Z or $H^0$ bosons. The $S_T$ distributions of the events in different categories showed no excess of events above the expected background and yielded a lower limit on the B quark mass of 0.73 TeV for $BR(B \to Wt) = 1$.

CMS [132] also searched for pair production of both TT and BB with collisions from 2.5 fb⁻¹ of $\sqrt{s} = 13$ TeV data. The analysis searched for events with one high $p_T$ lepton, multiple jets, and highly boosted W or Higgs bosons decaying hadronically. The analysis focused on pair production and selects events with either a boosted W or Higgs candidate and then proceeds to search for anomalous production in excess of standard model production. Seeing no significant excess CMS then proceeded to set limits in many different interpretations. The strongest was from the $B \to Wt$ interpretation leading to excluding heavy vector-like B quarks with mass less than 0.73 TeV.
The all-hadronic inclusive analysis [122] performed by CMS us-
ing the BEST NN technique to classify W/Z/H⁰/t/b/light jets
also gives exclusion limits on B quark production for various com-
binations of branching fractions for decays to tW, bZ, bH⁰. By
considering categories based on various combinations of the boson
and quark jets in the final state it excludes B quarks with masses
up to 1230 GeV, for B decays to tW with a 100% branching frac-
tion.

Electroweak production of single heavy B+b production has
been studied by CMS in the decay to tW with the lepton+jets
final state [153]. Single lepton events with hadronic jets, including
a forward jet, and missing pT are selected and divided into 10
different categories based on lepton flavor (e/μ), top-tagged, W-
tagged, and 0/1/2 b-tagged jets. The B quark mass mreco is
fully reconstructed from lepton, jets, and missing pT, where the
neutrino four-momentum is computed using the missing pT and
the W mass constraint. For events within the top-tagged category,
the high pT top-tagged hadronic jet and the leptonically decaying
W boson are used to compute mreco. The mreco distribution
are used as the signal discriminator. In the absence of an excess
over the expected SM background, the exclusion limits on the
production cross section for B quark masses between 0.7-2 TeV
varies between 0.3 to 0.03 pb. In addition, B quarks with left-
handed couplings and a relative width of 10, 20, and 30% are
excluded for masses below 1.49, 1.59, and 1.66 TeV respectively.

$B \to bZ:$

As mentioned above, a search for *B* pair produced events, with
final states with fully hadronic signature, using 137 fb⁻¹ of data,
has been performed by the CMS experiment [123]. In this search,
the *B* decays into a *b/b̄* quark and either a *H*⁰ or a *Z* boson. For
the case *B(B → bZ)* = 100%(50%), VLQs with masses below 1.39
(1.45) TeV are ruled out by this analysis.

A search by CMS [154] for the pair-production of a heavy B
quark and its antiparticle has been performed, where one of the
heavy B quark decays to bZ. Events with a Z boson decaying to
e⁺e⁻ or μ⁺μ⁻ and at least one b jet are selected. The signal
from B → bZ decays is expected to appear as a local enhance-
ment in the bZ mass distribution. No such enhancement was
found and B quarks that decay 100% into bZ are excluded be-
low 0.70 TeV. This analysis also set upper limits on the branching
fraction for B → bZ decays of 30-100% in the B quark mass range
0.45-0.70 TeV. A complementary search has been carried out by
ATLAS for new heavy quarks decaying into a Z boson and a b-
quark [155]. Selected dilepton events contain a high transverse
momentum Z boson that decays leptonically, together with two
b-jets. If the dilepton events have an extra lepton in addition
to those from the Z boson, then only one b-jet is required. No
significant excess of events above the standard model expectation
was observed, and mass limits were set depending on the assumed
branching ratios, as shown in Fig. 92.4. In a weak-isospin singlet
scenario, a B quark with mass lower than 0.65 TeV was excluded,
while for a particular weak-isospin doublet scenario, a B quark
with mass lower than 0.73 TeV was ruled out.

The search by ATLAS for $B \to bZ$ decays [142] in final states with either two leptons or three or more leptons using a multi-class boosted object tagger to categorize events according to the number of high $p_T$ boosted $H^0$, $Z/W$, and top quark jets excludes B quark masses up to 1.20 GeV (1.32 GeV), for the singlet (doublet) configuration [142].

In addition to pair production, ATLAS has also searched for the electroweak production of single B quarks, which is accompanied by a b-jet and a light jet [155]. The dilepton selection for double B production was modified for the single B production study by requiring the presence of an additional energetic jet in the forward region. An upper limit of 200 fb was obtained for the process $\sigma(pp \to B\bar{b}q) \times B(B \to Zb)$ with a heavy B quark mass at 0.70 TeV. This search indicated that the electroweak mixing parameter $X_{Bb}$ below 0.5 is neither expected nor observed to be excluded for any values of B quark mass.

Combination of $B \to tW/bZ/bH^0$:

ATLAS has combined the various analyses targeted for spe-
cific decay modes to obtain the most sensitive limit on the

pair production of B quarks [123, 124, 145]. Various searches
($W(\ell\nu)t + X$, $Z(\ell\ell)t/b + X$, same sign charge dilepton events,
trilepton events, and all-hadronic) are combined to obtain lower
limit on the mass of the B quark [in the plane of $BR(B \to Wt)$ vs
$BR(B \to bH)$]. The searches were optimized for 100% branching
fractions and hence are most sensitive at large $BR(B \to Wt)$, and
also at large $BR(B \to bH^0)$. For all possible values of branching
ratios in the three decay modes $tW$, $bZ$, or $bH^0$, the lower limits
on the B quark mass was found to be 1.03 TeV and shown in
Fig. 92.4 (left) as a function of the B quark branching ratios.

CMS combined three channels with lepton final states: sin-
gle lepton, two leptons with the same sign of the electric charge
(SS), or at least three leptons (trilepton) [147]. For various com-
binations of branching fractions for B quark decays to tW, bH⁰
and bZ, the combined results exclude b quarks with masses below
0.91–1.24 TeV and are shown in Figure 92.4 (right); the details are
provided earlier in subsection 92.2.3.1. The single lepton analysis
is most sensitive for tHbW and WbWb decay modes, within the
SS dilepton analysis tHtH and tHtZ have the best efficiency and
for trileptons the tZtZ and tHtZ decays modes have the highest
efficiency. CMS excludes singlet (doublet) B quark masses below
1.17 (0.94) TeV. Masses below 800 GeV were excluded in previ-
ous searches. For B quark masses in the range 0.8–1.8 TeV, cross
sections smaller than 40.6–9.4 fb (101–49.0 fb) are excluded for
the singlet (doublet) scenario.

92.2.3.3 Searches for top-partner quark $X_{5/3}$

Searches for a heavy top vector-like quark $X_{5/3}$, with exotic
charge $\pm 5/3$, such as that proposed in Refs. [156,157], have been
performed by both ATLAS and CMS [123,158].

The analyses assumed pair-production or single production of $X_{5/3}$ with $X_{5/3}$ decaying with 100% branching fraction to $tW$. Searches for $X_{5/3}$ have been performed using two final state signatures: same-sign leptons and lepton+jets.

The analysis based on searching for same-sign leptons, from
the two W bosons from one of the $X_{5/3}$, has smaller backgrounds
compared to the lepton+jets signature. Requiring same-sign lep-
tons eliminates most of the standard model background processes,
leaving those with smaller cross sections: $t\bar{t}W$, $t\bar{t}Z$, $WWW$, and
same-sign WW. In addition, backgrounds from instrumental ef-
fects due to charge misidentification were considered. Assuming
pair production of $X_{5/3}$, the analysis by CMS using $H_T$ as the dis-
criminating variable restrict the $X_{5/3}$ mass to be higher than 1.16
(1.10) TeV for a right (left) handed chirality particle [158–160].
The limits obtained by ATLAS, by classifying the signal region
by number of b-jets, $H_T$, and missing $p_T$ in the event, led to a
lower mass limit on $X_{5/3}$ of 1.19 TeV [161,162].

Searches for $X_{5/3}$ using leptons+jets final state signatures are
based on either full or partial reconstruction of the $T$ mass from
the lepton, jets (including b-jets) and missing $p_T$. The CMS
search [158,163] also utilized jet substructure techniques to iden-
tify boosted $X_{5/3}$ topologies. The discriminating variable used
was the mass constructed from the lepton and b-tagged jet, $M(\ell_b)$,
which corresponds to the visible mass of the leptonically decay-
ing top quark. To optimize the search sensitivity, the events were
further separated into categories based on lepton flavor (e, μ), the
number of b-tagged jets, the number of W-tagged jets, and the
number of t-tagged jets. In the absence of a signal, the CMS anal-
ysis excluded $X_{5/3}$ quark masses with right-handed (left-handed)
couplings below 1.32 (1.30) TeV [163]. Combining the lepton+jets
with the same-sign leptons analyses leads to a slight improvement
and excludes $X_{5/3}$ quark masses with right-handed (left-handed)
couplings below 1.33 (1.30) TeV.

The ATLAS lepton+jets search for $X_{5/3}$ utilized events with
high $p_T$ W bosons and b-jets. The search described earlier for $T$
pair production, with $T \to Wb$ decays, can be reinterpreted as
a search for $X \to tW$. This analysis excluded $X_{5/3}$ with masses
below 1.25 TeV [123].

The single $X_{5/3}$ production cross section depends on the cou-
pling constant $\lambda$ of the $tWX$ vertex. ATLAS has performed an
analysis of same-sign dileptons which includes both the single and
pair production. This analysis led to a lower limit on the mass of
the $X_{5/3}$ of 0.75 TeV for both values of $\lambda = 0.5$ and 1.0 [164].

Single heavy $X_{5/3}+t$ production has been studied by CMS in
<figure><img src="image_390.png" /><figcaption>**Figure 92.4:** Observed limits on the mass of the B quark in the plane of $BR(B \to bH^0)$ versus $BR(B \to tW)$ from ATLAS searches [145] on the left panel, and CMS searches [147] on the right panel, for $BB$ production. $B(B \to H^0) + B(B \to Z) + B(B \to W)$ = 1 is assumed. The yellow markers indicate the branching ratios for the $SU(2)$ singlet and doublet scenarios.</figcaption></figure>

the decay to $tW$ with the lepton+jets final state [153]. The description of the analysis is provided earlier in the discussion of $B \to WtX$ decays, where the reconstructed mass of $X_{5/3}$, $m_{reco}$ distribution is used as the signal discriminator. In the absence of an excess over the expected SM background, the exclusion limits on the production cross section for $X_{5/3}$ quark masses between 0.7–2 TeV varies between 0.3 to 0.03 pb, depending on the width of $X_{5/3}$ between 1–10%. In addition, $X_{5/3}$ quarks with left-handed couplings and a relative width of 10, 20, and 30% are excluded for masses below 0.92, 1.3, and 1.45 TeV respectively.

### 92.2.4 Heavy resonances decaying to VLQ

CMS has performed search for VLQ production in the decay of massive resonances such as $Z'$ and $W'$ bosons.

**$Z' \to tT$:** Specifically searches are presented by CMS in Refs. [165] and [166] for massive spin-1 $Z'$ resonances decaying to a top quark and a heavy VLQ top quark partner $T$. The results of this search for a heavy spin-1 resonance are interpreted in the context of two different models. In the $G^*$ model which predicts ten VLQs ($T, B, \tilde{T}, \tilde{B}, T_{5/3}, T_{2/3}, T', B', B_{-1/3}, B_{-4/3}$) with the mass relationship $M(T_{5/3}) = M(T_{2/3}) = M(T)\cos(\phi_L)$. For the benchmark scenario [167], $\cos(\phi_L)=0.84$ and the branching fractions $T \to tH^0$, $tZ$, $Wb$ are 0.25, 0.25 and 0.5 respectively. The $\rho^0$ model predicts a multiplet of four new VLQs $T, B, X_{2/3}, X_{5/3}$, and in the benchmark scenario [168], the branching fractions $T \to tH^0$, $tZ$, $Wb$ are 0.5, 0.5 and 0 respectively.

Two of the three decays of the $Z' \to tT$ with $T \to tH^0$, $tZ$, $Wb$ are characterized by the presence of two top quark decays and a boson ($H^0/Z$). A search [166] by CMS, optimized for $T \to tH/Zt$ decays was carried out in the lepton+jets final state using a dataset corresponding to an integrated luminosity of 35.9 fb$^{-1}$. Jet substructure techniques are used to identify (or tag) the high $p_T$ large-radius jets originating from $H^0$, $Z$ bosons and merged top quarks. The mass of the $Z'$ boson is used as the signal discriminator and constructed using $H^0$ or $Z$-tagged dijet, the hadronic and leptonic top quark four vectors. For the leptonic top quark reconstruction ($t \to b\ell\nu_\ell$), the neutrino four vector is obtained from the event missing $p_T$ using the W boson mass constraint. While the high $p_T$ hadronic top quark jets from decays of massive $Z'$ bosons are mostly merged and identified by top tagging techniques, those from $T$ decays maybe resolved. The reconstructed $Z'$ candidate events are classified into six different categories requiring the presence of either a $H^0$-tagged jet with 2 b-tagged subjets or one b-tagged subjet or a Z-tagged boson, each with either zero or one top-tagged jet. This search does not observe any significant deviation in data over the expectation from standard model backgrounds. Within the context of the $G^*$ model, for a $T$ mass of 1.2 (1.5) TeV, this search excludes $G^*$ [167] resonances

with masses between 1.5–2.3 (2.0–2.4) TeV.

The search in the all-hadronic final state is based on a 2.6 fb$^{-1}$ dataset [165], and optimized for $T \to Wb$ decays. Jet substructure techniques are deployed for tagging jets from high $p_T$ W boson and top quarks. Events are categorized into two groups based on the presence of b-tagged subjets in the top-tagged jet. The multijet background estimation is challenging and determined using side-bands defined by inverting the b-tagging requirement. Upper limits on the cross section for $Z' \to tT$ are obtained in the range of 0.13–10 pb.

$W' \to Tb/Bt$: $W'$ bosons are predicted to decay to VLQ third generation partners $T$, $B$ quarks within composite Higgs and warped extra dimensional models [169]. In the benchmark scenarios of this framework, $W'$ decays to $Tb$ or $Bt$ are equally distributed and the subsequent VLQ decays $T \to tH$ and $B \to bH^0$ each are assumed to have a branching fraction of 0.5. The search for $W' \to Tb/bH^0 \to tbH^0$ is performed using a sample of 35.9 fb$^{-1}$ by CMS [170] in the final state with all-hadronic decays of both the Higgs boson ($H^0 \to b\bar{b}$) and the top quark. Both the $H^0$ boson and the top quark are expected to be boosted in the decay of a heavy $W'$, and hence jet substructure techniques, including subjet b-tagging and double b-tagging are deployed to identify the $H^0$-tagged and the top-tagged jets. The three particle mass $m_{t b H^0}$, is used as the signal discriminant to observe the $W'$ resonance. There is no excess observed in data above the expected SM background. This search excludes $W'$ production cross section above 0.01–0.43 pb for masses between 1.5–4.0 TeV.

### 92.2.5 Colorons and Colored Scalars

These particles are associated with top-condensate and top-seesaw models, which involve an enlarged color gauge group. The new particles decay to dijets, $t\bar{t}$, and $b\bar{b}$.

Direct searches for colorons, color-octect scalars and other heavy objects decaying to $q\bar{q}$, $q\bar{q}$, $qq$, or $gg$ have been performed using LHC data from pp collisions at $\sqrt{s}=7$, 8 and 13 TeV. Based on the analysis of dijet events from a data sample corresponding to a luminosity of 19.6 fb$^{-1}$, at $\sqrt{s}=8$ TeV, the CMS experiment excluded pair production of colorons with mass between 1.20–3.60 and 3.90–4.08 TeV [171]. Analyses of inclusive 8- and 10-jet final states with low missing transverse momentum by CMS [172], set limits in several benchmark models. Colorons (axigluons) with masses between 0.6 and 0.75 (up to 1.15) TeV were excluded, and gluinos in R-parity violating supersymmetric scenarios were ruled out from 0.6 up to 1.1 TeV.

A search for pair-produced colorons based on an integrated luminosity of 5.0 fb$^{-1}$ at $\sqrt{s}=7$ TeV by CMS excluded colorons with masses between 0.25 TeV and 0.74 TeV, assuming colorons decay 100% into $q\bar{q}$ [174]. This analysis was based on events with
<figure><img src="image_391.png" /><figcaption>**Figure 92.5:** Observed 95% C.L. limits on $\sigma \times B \times A$ for string resonances, excited quarks, axigluons, colorons, E6 diquarks, s8 resonances, W' and Z' bosons, and Randall–Sundrum Gravitons $g_{KK}$ from [173].</figcaption></figure>

at least four jets and two dijet combinations with similar dijet mass. Color-octet scalars (s8) with masses between 1.20–2.79 TeV were excluded by CMS [171], and below 2.7 TeV by ATLAS [175].

These studies have now been extended to take advantage of the increased center-of-mass energy during Run 2 of the LHC. Using $35.9 \text{ fb}^{-1}$ of data collected at $\sqrt{s}=13$ TeV, searches for narrow resonances have been performed by CMS. An analysis of the dijet invariant mass spectrum formed using wide jets [173, 176, 177], separated by $\Delta\eta_{jj} \le 1.3$, led to limits on new particles decaying to parton pairs (qq, qq, gg). Specific exclusions on the masses of colorons and color-octet scalars were obtained and are shown in Fig. 92.5. Exclusions have been obtained for axigluons and colorons below 6.1 TeV, and color-octet scalars below 3.4 TeV.

Additional searches for dijet resonances have been performed by both ATLAS [178–180] and CMS [181, 182], though they have not been interpreted in the context of coloron production.

## 92.3 Conclusions

As the above analyses have demonstrated, there is already substantial sensitivity to possible new particles predicted to accompany the $H^0$ in dynamical frameworks of electroweak symmetry breaking. No significant hints of any deviations from the standard model have been observed, and limits typically at the scale of a few hundred GeV to a few TeV are set.

Given the need to better understand the $H^0$ and to determine in detail how it behaves, such analyses continue to be a major theme of Run 3 of the LHC, and we look forward to increased sensitivity as a result of the higher luminosity at the increased center of mass energy of collisions.

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## 93. Grand Unified Theories

Revised August 2023 by A. Hebecker (Heidelberg U.), J. Hisano (Nagoya U.; KMI, Nagoya U.) and N. Nagata (UTokyo).

### 93.1 The standard model

The Standard Model (SM) may be defined as the renormalizable field theory with gauge group $G_{SM} = SU(3)_C \times SU(2)_L \times U(1)_Y$, with 3 generations of fermions in the representation

$$ (\mathbf{3}, \mathbf{2})_{1/3} + (\bar{\mathbf{3}}, \mathbf{1})_{-4/3} + (\bar{\mathbf{3}}, \mathbf{1})_{2/3} + (\mathbf{1}, \mathbf{2})_{-1} + (\mathbf{1}, \mathbf{1})_2, \quad (93.1) $$

and a scalar Higgs doublet $H$ transforming as $(\mathbf{1}, \mathbf{2})_1$. Here and below we use boldface numbers to specify the dimension of representations of non-Abelian groups (in this case fundamental and antifundamental) and lower indices for $U(1)$ charges. The fields of Eq. (93.1) should also be familiar as $[Q, u^c, d^c, L, e^c]$, with $Q = (u, d)$ and $L = (\nu, e)$ being the quark and lepton $SU(2)$-doublets and $u^c, d^c, e^c$ charge conjugate $SU(2)$-singlets.<fnref n="1" /> Especially after the discovery of the Higgs, this model is remarkably complete and consistent with almost all experimental data.

A notable exception are neutrino masses, which are known to be non-zero but are absent in the SM even after the Higgs acquires its vacuum expectation value (VEV). The minimalist attitude is to allow for the dimension-five operator $(HL)^2$ [1], which induces (Majorana) neutrino masses. In the seesaw mechanism [2-4] this operator is generated by integrating out heavy singlet fermions (right-handed (r.h.) neutrinos). Alternatively, neutrinos can have Dirac masses if light singlet neutrinos are added to the SM spectrum.

Conceptual problems of the SM include the absence of a Dark Matter candidate, of a mechanism for generating the baryon asymmetry of the Universe, and of any reason for the observed smallness of the $\theta$ parameter of QCD ($\theta_{QCD}$). In addition, the apparently rather complex group-theoretic data of Eq. (93.1) remains unexplained. Together with the abundance of seemingly arbitrary coupling constants, this disfavors the SM as a candidate fundamental theory, even before quantum gravity problems arise at energies near the Planck mass $M_P$.

To be precise, there are 19 SM parameters which have to be fitted to data: Three gauge couplings$^2 g_3$, $g_2$ and $g_1$, 13 parameters associated with the Yukawa couplings (9 charged fermion masses, three mixing angles and one CP phase in the CKM matrix.), the Higgs mass and quartic coupling, and $\theta_{QCD}$. In addition, Majorana neutrinos introduce 3 more masses and 6 mixing angles and phases. As we will see, the paradigm of grand unification addresses mainly the group theoretic data of Eq. (93.1) and the values of the three gauge couplings. In many concrete realizations, it then impacts also the other mentioned issues of the SM, such as the family structure and fermion mass hierarchy.

More specifically, after precision measurements of the Weinberg angle $\theta_W$ in the LEP experiments, supersymmetric GUTs (SUSY GUTs) have become the leading candidates in the search for 'Physics beyond the SM'. Supersymmetry (SUSY) is a symmetry between bosons and fermions which requires the addition of superpartners to the SM spectrum. If SUSY is motivated as a solution to the gauge hierarchy problem (i.e. to the naturalness or fine-tuning problem of the electroweak scale) [5], superpartners have to be present near the weak scale. Assuming a minimal Higgs sector, SUSY GUTs [6] then lead to the prediction of $\theta_W$, in good agreement with subsequent observations [7]. However, the non-discovery of new particles at the LHC puts into question the presence of new physics at the TeV scale in general and in particular of low-scale supersymmetry. Still, SUSY may be present just outside the presently explored energy domain.

The measured Higgs mass (125 GeV) is in principle consistent with this picture, assuming superpartners in the region of roughly 10 TeV. Such heavy superpartners then induce radiative corrections raising the Higgs mass above the Z boson mass $m_Z$ [8, 9]. However, from the vantage point of the hierarchy problem, heavy

superpartners are problematic: They also contribute to SUSY-breaking Higgs mass parameters and thereby to the Higgs potential, tending to raise the Z mass. As a result, the incarnation of SUSY in terms of the minimal supersymmetric SM (MSSM) is becoming questionable. Turning the logic around, one may say that compared to expectations based on the MSSM with superpartner masses below about 1 TeV, the measured Higgs mass value of 125 GeV is somewhat too high [10]. Independently, the LHC has disfavored light colored superpartners (which does not imply that all superpartners are heavy). These facts represent new hints for future work on SUSY GUTs or on GUTs without TeV-scale supersymmetry.

### 93.2 Basic group theory and charge quantization

#### 93.2.1 $SU(4)_C \times SU(2)_L \times SU(2)_R$

Historically, the first attempt at unification was the Pati-Salam model with gauge group $G_{PS} = SU(4)_C \times SU(2)_L \times SU(2)_R$ [11]. It unifies SM fermions in the sense that one generation (plus an extra SM singlet) now comes from the $(\mathbf{4}, \mathbf{2}, \mathbf{1}) + (\bar{\mathbf{4}}, \mathbf{1}, \mathbf{2})$ of $G_{PS}$. This is easy to verify from the breaking pattern $SU(4)_C \to SU(3)_C \times U(1)_{B-L}$ together with the identification of SM hypercharge as a linear combination between $B-L$ (baryon minus lepton number) and the $T_3$ generator of $SU(2)_R$. This model explains charge quantization, that is, why all electric charges are integer multiples of some smallest charge in the SM. Concretely, the $\mathbf{4}$ and $\bar{\mathbf{4}}$ of $SU(4)_C$ identify lepton number as the 4th colour and the tracelessness of the diagonal generator implies that quark charges are expressed in terms of $1/N_c$ fractions of lepton charges. However, $G_{PS}$ is not simple (containing three simple factors), and thus it does not predict gauge coupling unification.

#### 93.2.2 $SU(5)$

Since $G_{SM}$ has rank four (two for $SU(3)_C$ and one for $SU(2)_L$ and $U(1)_Y$, respectively), the rank-four group $SU(5)$ is the minimal choice for unification in a simple group [12]. The three SM gauge coupling constants derive from a universal coupling $\alpha_G$ at the GUT scale $M_G$. Explicitly embedding $G_{SM}$ in $SU(5)$ is straightforward, with $SU(3)_C$ and $SU(2)_L$ corresponding e.g. to the upper-left 3×3 and lower-right 2×2 blocks, respectively, in traceless 5×5 matrices for $SU(5)$ generators of the fundamental representation. The $U(1)_Y$ corresponds to matrices generated by $\text{diag}(-2/3, -2/3, -2/3, 1, 1)$ and hence commutes with $SU(3)_C \times SU(2)_L \subset SU(5)$. It is then easy to derive how one SM generation precisely comes from the $\mathbf{10} + \bar{\mathbf{5}}$ of $SU(5)$ (where $\mathbf{10}$ is the antisymmetric rank-2 tensor):

$$ \mathbf{10}: \begin{pmatrix} 0 & u_b^c & -u_g^c & u_r & d_r \\ -u_b^c & 0 & u_r^c & u_g & d_g \\ u_g^c & -u_r^c & 0 & u_b & d_b \\ -u_r & -u_g & -u_b & 0 & e^c \\ -d_r & -d_g & -d_b & -e^c & 0 \end{pmatrix} \quad \text{and} \quad \bar{\mathbf{5}}: \begin{pmatrix} d_r^c \\ d_g^c \\ d_b^c \\ e \\ -\nu_e \end{pmatrix}. \qquad (93.2) $$

In addition to charge quantization this structure explains why the l.h. quark and lepton states fall in $SU(2)_L$ doublets while the r.h. states are singlets.

Since $SU(5)$ has 24 generators, $SU(5)$ GUTs have 12 new gauge bosons known as X bosons (or X/Y bosons) in addition to the SM. X bosons form an $SU(3)_C$-triplet and $SU(2)_L$-doublet. Their interaction connects quarks and leptons such that baryon and lepton numbers are not conserved and nucleon decay is predicted. Furthermore, $U(1)_Y$ hypercharge is automatically quantized since it is embedded in $SU(5)$.

In order to break the electroweak symmetry at the weak scale and give mass to quarks and leptons, Higgs doublets are needed. In the minimal $SU(5)$ model, they can sit in either a $\mathbf{5}_H$ or $\bar{\mathbf{5}}_H$. The three additional states are referred to as color-triplet Higgs scalars. Their couplings also violate baryon and lepton numbers, inducing nucleon decay. In order not to violently disagree with the non-observation of nucleon decay, the triplet mass must be greater than $\sim 10^{11}$ GeV [13]. Moreover, in SUSY GUTs [6], in order to cancel anomalies as well as give mass to both up and down quarks, both Higgs multiplets $\mathbf{5}_H$ and $\bar{\mathbf{5}}_H$ are required. As we shall discuss later, nucleon decay now constrains the Higgs triplets to have

<footnote n="1">¹ In our convention the electric charge is $Q = T_3 + Y/2$ and all our spinor fields are left-handed (l.h.).</footnote>

<footnote n="2">² Equivalently, the $SU(2)_L$ and $U(1)_Y$ couplings are denoted as $g = g_2$ and $g' = \sqrt{3/5} g_1$. One also uses $\alpha_s = \alpha_3 = (g_3^2/4\pi)$, $\alpha_{EM} = (e^2/4\pi)$ with $e = g \sin \theta_W$ and $\sin^2 \theta_W = (g')^2/(g^2 + (g')^2)$.</footnote>
mass significantly greater than $M_G$ in the minimal SUSY $SU(5)$ GUT since integrating out the Higgs triplets generates dimension-five baryon-number-violating operators [14]. The mass splitting between doublet and triplet in the $5_H$ (and $\bar{5}_H$) comes from their interaction with the $SU(5)$ breaking sector.

### 93.2.3 $SO(10)$

While $SU(5)$ allows for the minimal GUT models, unification is not complete: Two independent representations, **10** and $\bar{5}$, are required for one SM generation. A further representation, an $SU(5)$ singlet, has to be added to serve as r.h. neutrino in the seesaw mechanism. In this case, the r.h. neutrino masses are not necessarily related to the GUT scale. By contrast, a single **16**-dimensional spinor representation of $SO(10)$ accommodates a full SM generation together with an extra singlet, potentially providing a r.h. neutrino [15]. This is most easily understood from the breaking pattern $SO(10) \to SU(5) \times U(1)_X$ and the associated branching rule<fnref n="3" /> **16** = $10_{-1}+\bar{5}_3+1_{-4}$. Here the indices refer to charges under the $U(1)_X$ subgroup, which is orthogonal to $SU(5)$ and reflects the fact that $SO(10)$ has rank five. From the above, it is easy to see that $U(1)_X$ charges can be given as $2Y - 5(B-L)$. Intriguingly, all representations of $SO(10)$ are anomaly free in four dimensions (4d). Thus, the absence of anomalies in an $SU(5)$-GUT or a SM generation can be viewed as deriving from this feature.

We now describe in more detail how one family of quarks and leptons appears in the **16**. To understand this, recall that the $\Gamma$-matrices of the 10d Clifford algebra give rise to five independent, anticommuting 'creation-annihilation' operators $\Gamma^{a\pm} = (\Gamma^{2a-1} \pm i\Gamma^{2a})/2$ with $a = 1, ..., 5$. These correspond to five fermionic harmonic oscillators or "spin" $1/2$ systems. The 32-dimensional tensor product of those is reducible since the 10d rotation generators $M_{mn} = -i[\Gamma^m, \Gamma^n]/4$ ($m, n = 1, ..., 10$) always flip an even number of "spins". This gives rise to the **16** as displayed in Table 93.1. Next, one also recalls that the natural embedding of $SU(5)$ in $SO(10)$ relies on 'pairing up' the 10 real dimensions to produce 5 complex dimensions, $\mathbb{R}^{10} \equiv \mathbb{C}^5$, similarly to the paring up of $\Gamma^m$s used above. This makes it clear how to associate one $|\pm>$ system to each complex dimension of $SU(5)$, which explains the labeling of the "spin" columns in Table 93.1: The first three and last two "spins" correspond to $SU(3)_C$ and $SU(2)_L$, respectively. In fact, an $SU(3)_C$ rotation just raises one color index and lowers another, changing colors $\{r, g, b\}$, or changes relative phases between the three spin states. Similarly, an $SU(2)_L$ rotation raises one weak index and lowers another, thereby flipping the weak isospin from up to down or vice versa, or changes the relative phase between the two spin states. In this representation $U(1)_Y$ hypercharge is simply given by $Y = -2/3(\sum \text{color spins}) + (\sum \text{weak spins})$. $SU(5)$ rotations corresponding to $X$ bosons then raise (or lower) a color index, while at the same time lowering (or raising) a weak index. It is easy to see that such rotations can mix the states $\{Q, u^c, e^c\}$ and $\{d^c, L\}$ among themselves and $\nu^c$ is a singlet. Since $SO(10)$ has 45 generators, additional 21 gauge bosons are introduced including the $U(1)_X$ above. The 20 new $SO(10)$ rotations not in $SU(5)$ are then given by either raising any two spins or lowering them. With these rotations, **1** and **5** are connected with **10**. The last $SO(10)$ rotation changes phases of states with weight $2(\sum \text{color spins}) + 2(\sum \text{weak spins})$, which corresponds to $U(1)_X$.

$SO(10)$ has two inequivalent maximal subgroups and hence breaking patterns, $SO(10) \to SU(5) \times U(1)_X$ and $SO(10) \to SU(4)_C \times SU(2)_L \times SU(2)_R$. In the first case, one can carry on breaking to $G_{SM} \subset SU(5)$ precisely as in the minimal $SU(5)$ case above. Alternatively, one can identify $U(1)_Y$ as an appropriate linear combination of $U(1)_X$ and the $U(1)$ factor from $SU(5)$, leading to the so-called flipped $SU(5)$ [17] as an intermediate step in breaking $SO(10)$ to $G_{SM}$. In the second case, we have an intermediate Pati-Salam model thanks to the branching rule **16** = $(4, 2, 1) + (\bar{4}, 1, 2)$. Finally, $SO(10)$ can break directly to the SM at $M_G$. Gauge coupling unification remains intact in

<table><caption>**Table 93.1:** Quantum numbers of **16**-dimensional representation of $SO(10)$.</caption><thead><tr><th>state</th><th>Y</th><th>Color</th><th>Weak</th><th>$SU(5)$</th><th>$SO(10)$</th></tr></thead><tbody><tr><td>$\nu^c$</td><td>0</td><td>--</td><td>--</td><td>1</td><td></td></tr><tr><td>$e^c$</td><td>2</td><td>--</td><td>++</td><td></td><td></td></tr><tr><td>$\bar{u}_r$</td><td>1/3</td><td>+-</td><td>--</td><td rowspan="2">10</td><td></td></tr><tr><td>$\bar{d}_r$</td><td>1/3</td><td>+-</td><td>+-</td><td></td></tr><tr><td>$\bar{u}_g$</td><td>1/3</td><td>--</td><td>--</td><td></td><td></td></tr><tr><td>$\bar{d}_g$</td><td>1/3</td><td>--</td><td>+-</td><td></td><td></td></tr><tr><td>$\bar{u}_b$</td><td>1/3</td><td>--</td><td>+-</td><td></td><td rowspan="2">16</td></tr><tr><td>$\bar{d}_b$</td><td>1/3</td><td>--</td><td>+-</td><td></td></tr><tr><td>$\bar{u}_r^c$</td><td>-4/3</td><td>--</td><td>--</td><td></td><td></td></tr><tr><td>$\bar{u}_g^c$</td><td>-4/3</td><td>+-</td><td>--</td><td></td><td></td></tr><tr><td>$\bar{u}_b^c$</td><td>-4/3</td><td>++</td><td>--</td><td></td><td></td></tr><tr><td>$\bar{d}_r^c$</td><td>2/3</td><td>--</td><td>++</td><td></td><td></td></tr><tr><td>$\bar{d}_g^c$</td><td>2/3</td><td>+-</td><td>++</td><td rowspan="2">5</td><td></td></tr><tr><td>$\bar{d}_b^c$</td><td>2/3</td><td>++</td><td>++</td><td></td></tr><tr><td>$\nu$</td><td>-1</td><td>+++</td><td>--</td><td></td><td></td></tr><tr><td>$e$</td><td>-1</td><td>+++</td><td>--</td><td></td><td></td></tr></tbody></table>

the case of this 'direct' breaking and for the breaking pattern $SO(10) \to SU(5) \to G_{SM}$ (with $SU(5)$ broken at $M_G$). In the case of intermediate-scale Pati-Salam or flipped $SU(5)$ models, gauge coupling predictions are modified. The Higgs multiplets in the minimal $SO(10)$ come from the fundamental representation, $10_H = 5_H + \bar{5}_H$. Note, only in $SO(10)$ does the representation type distinguish SM matter from Higgs fields.

### 93.2.4 Beyond $SO(10)$

Finally, larger symmetry groups can be considered. For example, the exceptional group $E_6$ has maximal subgroup $SO(10) \times U(1)$ [18]. Its fundamental representation branches as **27** = **16**$_1$ + **10**$_{-2}$ + **1**$_4$. Another maximal subgroup is $SU(3)_C \times SU(3)_L \times SU(3)_R \subset E_6$ with branching rule **27** = $(3, 3, 1) + (\bar{3}, 1, \bar{3}) + (1, \bar{3}, 3)$. Independently of any underlying $E_6$, the group $[SU(3)]^3$ with additional permutation symmetry $Z_3$ interchanging the three factors can be considered. This is known as "trinification" [19]. The $E_6 \to [SU(3)]^3$ breaking pattern has been used in phenomenological analyses of the heterotic string [20]. However, in larger symmetry groups, such as $E_6$, $SU(6)$, etc., there are now many more states which have not been observed and must be removed from the effective low-energy theory.

Intriguingly, the logic by which $G_{SM}$ is a maximal subgroup of $SU(5)$, which together with $U(1)_X$ is a maximal subgroup of $SO(10)$, continues in a very elegant and systematic way up to the largest exceptional group. The resulting famous breaking chain $E_8 \to E_7 \to E_6 \to SO(10) \to SU(5) \to G_{SM}$ together with the special role played by $E_8$ in group and in string theory is a tantalizing hint at deeper structures. However, since all representations of $E_8$ and $E_7$ are real and can not lead to 4d chiral fermions, this is necessarily outside the 4d GUT framework.

## 93.3 GUT breaking and doublet-triplet splitting

In the standard, 4d field-theoretic approach to GUTs, the unified gauge group is broken spontaneously by an appropriate GUT Higgs sector. Scalar potentials (or superpotentials in SUSY GUTs) exist whose vacua spontaneously break $SU(5)$ or $SO(10)$. While these potentials are ad hoc (just like the Higgs potential in the SM), the most naive expectation is that all their dimensionful parameters are $O(M_G)$. In the simplest case of $SU(5)$, the **24** (adjoint) GUT Higgs develops a VEV along the $G_{SM}$-singlet direction as $\langle\Phi\rangle \propto \text{diag}(-2/3, -2/3, -2/3, 1, 1)$. In order for $SO(10)$ to break to $SU(5)$, the **16** or **126**, which have a $G_{SM}$-singlet with non-zero $U(1)_X$ charge, get a VEV.

The masses of doublet and triplet in the $5_H$ (and $\bar{5}_H$) generically split due to their coupling to the GUT Higgs. In addition, both the doublet and the triplet masses also get an equal contribution from an $SU(5)$-invariant GUT-scale mass term. Without any further structure, an extreme fine-tuning between two large effects is then necessary to keep the doublet mass at the electroweak scale. Supersymmetry plays an important role in for-

<footnote n="3">³ Useful references on group theory in the present context include [16] and refs. therein.</footnote>
bidding large radiative correction to the doublet mass due to the non-renormalization theorem [5]. However, even in this case we have to fine tune parameters at tree level. This is the doublet-triplet splitting problem which, in the SUSY context, is clearly related to the μ-term problem of the MSSM (the smallness of the coefficient of μH_u H_d).

Several mechanisms for natural doublet-triplet splitting have been suggested under the assumption of supersymmetry, such as the sliding singlet [21], missing partner [22], missing VEV [23], and pseudo-Nambu-Goldstone boson mechanisms [24]. Particular examples of the missing partner mechanism for SU(5) [25], the missing VEV mechanism for SO(10) [26, 27] and the pseudo-Nambu-Goldstone boson mechanism for SU(6) [28] have been shown to be consistent with gauge coupling unification and nucleon decay. From the GUT-scale perspective, one is satisfied if the triplets are naturally heavy and the doublets are massless. In other words, one wants to ensure that the fundamental superpotential term μH_u H_d is absent (μ ≃ 0) while the corresponding term for the Higgs triplets is large. We note however that implementing this by a global symmetry is problematic, see e.g. [29–31].

Once a model with vanishing fundamental μ term has been realized, an *effective* μ term of order the SUSY breaking scale can then be induced by, for example, higher-mass-dimension terms in the Kahler potential [32]. For a review of the μ problem and some suggested solutions in SUSY GUTs and string theory, see [30, 33–35] and references therein.

In general, GUT-breaking sectors successfully resolving the doublet-triplet splitting problem, dynamically stabilizing all GUT-scale VEVs and allowing for realistic neutrino masses and Yukawa couplings (including the GUT-symmetry violation in the latter) require a number of ingredients. However, for validity of the effective theory, introduction of higher or many representations is limited, otherwise a Landau pole may appear below the Planck scale. In addition, GUTs are only effective theories below the Planck scale in the 4d field-theoretic approach. Since $M_G$ is close to this scale, the effects of higher-dimension operators are not obviously negligible. In particular, operators including the GUT-breaking Higgs may affect low-energy predictions, such as quark and lepton masses.

Thus, especially in the context of GUT breaking and doublet-triplet splitting, models beyond 4d field theory appear attractive. While this is mainly the subject of the next section, some advantages can already be noted: In models with extra dimensions, in particular string constructions, GUT breaking may occur due to boundary conditions in the compactified dimensions [36–39]. No complicated GUT breaking sector is then required. Moreover, boundary conditions can give mass only to the triplet, leaving the doublet massless. This is similar to the ‘missing partner mechanism’ since the effective mass term does not ‘pair up’ the triplets from $\mathfrak{S}_H$ and $\bar{\mathfrak{S}}_H$ but rather each of them with further spaces which are automatically present in the higher-dimensional theory. This can eliminate dimension-five nucleon decay (cf. Sec. 93.6).

## 93.4 String-theoretic and higher-dimensional unified models

As noted earlier, the GUT scale is dangerously close to the scale of quantum gravity. It may hence be necessary to discuss unified models of particle physics in the latter, more ambitious context. Among the models of quantum gravity, superstring or M-theory stands out as the best-studied and technically most developed proposal, possessing in particular a high level of internal, mathematical consistency. For our purposes, it is sufficient to know that five 10d and one 11d low-energy effective supergravity theories arise in this setting (cf. [40] and refs. therein).

Grand unification is realized most naturally in the context of the two ‘heterotic’ theories with gauge groups $E_8 \times E_8$ and $SO(32)$, respectively [38, 41] (see [42] for some of the more recent results). Justified in part by the intriguing breaking path $E_8 \to \cdots \to G_{SM}$ mentioned above, the focus has historically largely been on $E_8 \times E_8$. To describe particle physics, solutions of the 10d theory with geometry $\mathbb{R}^{1,3} \times M_6$ are considered, where $M_6$ is a Calabi-Yau (CY) 3-fold (with 6 real dimensions) [38]. The background solution involves expectation values of higher-

dimensional components of the $E_8 \times E_8$ gauge fields. This includes both Wilson lines [36] and non-vanishing field-strength and leads, in general, to a reduced gauge symmetry and to chirality in the resulting 4d effective theory. The 4d fermions arise from 10d gauginos.

Given an appropriate embedding<fnref n="4" /> of $G_{SM}$ in $E_8 \times E_8$, gauge coupling unification is automatic at leading order. Corrections arise mainly through (string)-loop effects and are similar to the familiar field-theory thresholds of 4d GUTs<fnref n="5" /> [43]. Thus, one may say that coupling unification is a generic prediction in spite of the complete absence<fnref n="6" /> of a 4d GUT at any energy scale. This absence is both an advantage and a weakness. On the up side, GUT breaking and doublet-triplet splitting [45] are more naturally realized and dimension-five nucleon decay is relatively easy to avoid. On the down side, there is no reason to expect full GUT representations in the matter sector and flavor model building is much less tied to the GUT structure than in 4d.

Let us pause to explain the beautiful idea behind the advertised solution of the doublet-triplet splitting problem: One starts with a simply connected CY X and mods out the action of a discrete group G (say $\mathbb{Z}_2$). In the absence of fixed points, $X/G$ is smooth and has a non-contractible 1-cycle. Furthermore, let G also act on the gauge bundle, according to an embedding $G \to E_8$. Now the parallel transport around the 1-cycle is tied to a gauge rotation (one says a non-trivial Wilson-line is present). Moreover, this Wilson line can not be continuously turned off since, e.g. in the case of $\mathbb{Z}_2$, its square is the unit element of the group. The induced ‘Wilson-line breaking’, which comes on top of the breaking by non-zero field strengths, may remove certain sub-representations (e.g. the triplet of $SU(5) \to G_{SM}$) while keeping others exactly massless. A simpler and, due to fixed points, singular version of this will appear below in the context of orbifold GUTs.

One technical problem of heterotic constructions is the dependence on the numerous size and shape parameters of $M_6$ (the so-called moduli), the stabilization of which is poorly understood (see [46] for recent developments). Another is the sheer mathematical complexity of the analysis, involving in particular the study of (non-Abelian) gauge-bundles on CY spaces [47] (see however [48]).

An interesting aspect of heterotic string constructions is represented by orbifold models [37]. Here the internal space is given by a six-torus, modded out by a discrete symmetry group (e.g. $T^6/\mathbb{Z}_n$). More recent progress is reported in [49, 50], including in particular the systematic exploration of the phenomenological advantages of so-called ‘non-prime’ (referring to n) orbifolds. The symmetry breaking to $G_{SM}$ as well as the survival of Higgs doublets without triplet partners is ensured by the appropriate embedding of the discrete orbifold group in $E_8 \times E_8$. String theory on such spaces, which are locally flat but include singularities, is much more calculable than in the CY case. The orbifold geometries can be viewed as singular limits of CYs.

An even simpler approach to unified models, which includes many of the advantages of full-fledged string constructions, is provided by Orbifold GUTs [39]. These are (mostly) 5d or 6d SUSY field theories with unified gauge group (e.g. $SU(5)$ or $SO(10)$), broken in the process of compactifying to 4d. To give a particularly simple example, consider $SU(5)$ on $\mathbb{R}^{1,3} \times S^1/(\mathbb{Z}_2 \times \mathbb{Z}_2)$. Here the compact space is an interval of length $\pi R/2$ and the embedding of $\mathbb{Z}'_2$ in the hypercharge direction of $SU(5)$ realizes the breaking to $\tilde{G}_{SM}$. Concretely, 5d X bosons are given Dirichlet BCs at one endpoint of the interval and thus have no Kaluza-Klein (KK) zero mode. Their lightest modes have mass $\sim 1/R$, making the KK-scale the effective GUT scale. As an implication, the boundary theory has no $SU(5)$ invariance. Nevertheless, since the $SU(5)$-symmetric 5d bulk dominates 4d gauge couplings, unifica-

<footnote n="4">⁴ All embeddings of $G_{SM}$ in one $E_8$ factor which are consistent with a breaking-pattern $E_8 \to SU(5) \to G_{SM}$ are suitable (cf. for example the natural breaking chain from $E_8$ to $G_{SM}$ through maximal subgroups mentioned at the end of Sect. 93.3). Other embeddings can change the ratios between the three resulting $G_{SM}$ couplings at the GUT scale by group-theoretic factors. Crucially, due to the single 10d gauge coupling, no continuous tuning is possible.</footnote>

<footnote n="5">⁵ Field-theory thresholds of 4d GUTs are discussed in 93.5.</footnote>

<footnote n="6">⁶ See however [44].</footnote>
tion remains a prediction. Many other features but also problems of 4d GUTs can be circumvented, especially doublet-triplet splitting is easily realized.

With the advent of the string-theory 'flux landscape' [51], which is best understood in 10d type-IIB supergravity, the focus in string model building has shifted to this framework. While type II string theories have no gauge group in 10d, brane-stacks support gauge dynamics. A particularly appealing setting (see e.g. [52]) is provided by type IIB models with D7 branes (defining 8d submanifolds). However, in the SO(10) context the **16** is not available and, for SU(5), the top-Yukawa coupling vanishes at leading order [53]. As a crucial insight, this can be overcome on the non-perturbative branch of type IIB, also known as F-theory [54, 55]. This setting allows for more general branes, thus avoiding constraints of the Dp-brane framework. GUT breaking can be realized using hypercharge flux (the VEV of the U(1)$_Y$ field strength), an option not available in heterotic models. The whole framework combines the advantages of the heterotic or higher-dimensional unification approach with the more recent progress in understanding moduli stabilization. It thus represents at this moment the most active and promising branch of theory-driven GUT model building (see e.g. [56] and refs. therein).

As a result of the flux-breaking, a characteristic 'type IIB' or 'F-theoretic' tree-level correction to gauge unification arises [57]. The fact that this correction can be rather significant numerically is occasionally held against the framework of F-theory GUTs. However, at a parametric level, this correction nevertheless behaves like a 4d threshold, i.e., it provides $O(1)$ additive contributions to the inverse 4d gauge couplings $\alpha_i^{-1}(M_G)$.

A final important issue in string GUTs is the so-called string-scale/GUT-scale problem [58]. It arises since, in heterotic compactifications, the Planck scale and the high-scale value of the gauge coupling unambiguously fix the string-scale to about $10^{18}$ GeV. As the compactification radius $R$ is raised above the string length, the GUT scale (identified with $1/R$) goes down and the string coupling goes up. Within the domain of perturbative string theory, a gap of about a factor $\sim 20$ remains between the lowest GUT scale achievable in this way and the phenomenological goal of $2 \times 10^{16}$ GeV. The situation can be improved by venturing into the non-perturbative regime [58], by considering 'anisotropic' geometries with hierarchically different radii $R$ [58, 59] or by including GUT scale threshold corrections [60, 61].

In F-theory GUTs, the situation is dramatically improved since the gauge theory lives only in four out of the six compact dimensions. This allows for models with a 'decoupling limit', where the GUT scale is parametrically below the Planck scale [55]. However, moduli stabilization may not be without problems in such constructions, in part due to a tension between the required large volume and the desirable low SUSY breaking scale.

## 93.5 Gauge coupling unification

The quantitative unification of the three SM gauge couplings at the energy scale $M_G$ is one of the cornerstones of the GUT paradigm. It is obviously of direct phenomenological relevance. Gauge coupling unification is well understood in the framework of effective field theory (EFT) [62]. In the simplest case, the relevant EFT at energies $\mu \gg M_G$ has a unified gauge symmetry (say SU(5) for definiteness) and a single running gauge coupling $\alpha_G(\mu)$. At energies $\mu \ll M_G$, states with mass $\sim M_G$ (such as X bosons, GUT Higgs, color-triplet Higgs) have to be integrated out. The EFT now has three independent couplings and SM (or SUSY SM) matter content. One-loop renormalization group equations readily allow for an extrapolation to the weak scale,

$$ \alpha_i^{-1}(m_Z) = \alpha_G^{-1}(M_G) + \frac{b_i}{2\pi} \log \left( \frac{M_G}{m_Z} \right) + \delta_i, \quad (93.3) $$

($i=1,2,3$). Here we defined $\delta_i$ to absorb all sub-leading effects, such as threshold corrections at or near the weak scale (e.g. from superpartners and the additional Higgs bosons in the case of the MSSM) and at the GUT scale, and also higher-order corrections. We will discuss them momentarily.

It is apparent from Eq. (93.3) that the three low-scale couplings can be very different. This is due to the large energy range

$m_Z \ll \mu \ll M_G$ and the non-universal $\beta$-function coefficients ($b_i^{\text{SM}} = \{41/10, -19/6, -7\}$ or $b_i^{\text{MSSM}} = \{33/5, 1, -3\}$). Incomplete GUT multiplets, such as gauge and Higgs bosons in the SM and also their superpartners and the additional Higgs bosons in the MSSM, contribute to the differences between the $\beta$ functions. Inverting the argument, one expects that extrapolating the measured couplings to the high scale, we find quantitative unification at $\mu \sim M_G$. While this fails in the SM, it works intriguingly well in the MSSM (cf. Fig. 93.1).

The three equations contained in Eq. (93.3) can be used to determine the three 'unknowns' $\alpha_3(m_Z)$, $\alpha_G(M_G)$ and $M_G$, assuming that all other parameters entering the equations are given. Focusing on the SUSY case and using the $\overline{\text{MS}}$ coupling constants $\alpha_{\text{EM}}^{-1}(m_Z)$ and $\sin^2 \theta_W(m_Z)$ from [64],

$$ \alpha_{\text{EM}}^{-1}(m_Z) = 127.951 \pm 0.009, \quad (93.4) $$

$$ \sin^2 \theta_W(m_Z) = 0.23122 \pm 0.00004, \quad (93.5) $$

as input, one determines $\alpha_{1,2}^{-1}(m_Z)$, which then gives

$$ \alpha_G^{-1}(M_G) \approx 24.3 \quad \text{and} \quad M_G \approx 2 \times 10^{16} \text{ GeV}. \quad (93.6) $$

Here we have set $\delta_i = 0$ for simplicity. Crucially, one in addition obtains a prediction for the low-energy observable $\alpha_3$,

$$ \alpha_3^{-1}(m_Z) = -\frac{5}{7}\alpha_1^{-1}(m_Z) + \frac{12}{7}\alpha_2^{-1}(m_Z) + \Delta_3, \quad (93.7) $$

where

$$ \Delta_3 = \frac{5}{7}\delta_1 - \frac{12}{7}\delta_2 + \delta_3. \quad (93.8) $$

Here we followed the elegant formulation in Ref. [65] of the classical analyses of [7]. Of course, it is a matter of convention which of the three low-energy gauge coupling parameters one 'predicts' and indeed, early works on the subject discussed the prediction of $\sin^2 \theta_W$ in terms of $\alpha_{\text{EM}}$ and $\alpha_3$ [66, 67].

Remarkably, the leading order result (i.e. Eq. (93.7) with $\delta_i = 0$) is in excellent agreement with experiments [64]:

$$ \alpha_3^{\text{LO}}(m_Z) = 0.117 \quad \text{vs.} \quad \alpha_3^{\text{EXP}}(m_Z) = 0.1179 \pm 0.0009. \quad (93.9) $$

However, this near perfection is to some extent accidental. To see this, we now discuss the various contributions to the $\delta_i$ (and hence to $\Delta_3$).

The two-loop running correction from the gauge sector $\Delta_3^{(2)}$ and the low-scale threshold correction $\Delta_3^{(l)}$ from superpartners can be summarized as [65]

$$ \Delta_3^{(2)} \approx -0.82 \quad \text{and} \quad \Delta_3^{(l)} \approx \frac{19}{28\pi} \log \left( \frac{m_{\text{SUSY}}}{m_Z} \right). \quad (93.10) $$

The relevant scale $m_{\text{SUSY}}$ can be estimated as [68]

$$ m_{\text{SUSY}} \rightarrow m_H^{3/19} m_{\tilde{H}}^{12/19} m_{\tilde{W}}^{4/19} \times \left( \frac{m_{\tilde{W}}}{m_{\tilde{g}}} \right)^{28/19} \left( \frac{m_{\tilde{l}}}{m_{\tilde{q}}} \right)^{3/19}, \quad (93.11) $$

where $m_H$ stands for the masses of non-SM Higgs states and superpartner masses are given in self-evident notation. Detailed analyses including the above effects are best done using appropriate software packages, such as SOFTSUSY [63] (or alternatively SuSpect [69] or SPheno [70]). See also [63] for references to the underlying theoretical two-loop analyses.

To get a very rough feeling for these effects, let us assume that all superpartners are degenerate at $m_{\text{SUSY}} = 1$ TeV, except for heavier gluinos: $m_{\tilde{W}}/m_{\tilde{g}} \simeq 1/3$. This gives $\Delta_3^{(l)} \simeq -0.35 + 0.22 \ln(m_{\text{SUSY}}/m_Z) \simeq 0.18$. The resulting prediction of $\alpha_3(m_Z) \simeq 0.126$ significantly upsets the perfect one-loop agreement found earlier. Before discussing this issue further, it is useful to introduce yet another important type of correction, the high or GUT scale thresholds.
<figure><img src="image_392.png" /><figcaption>Figure 93.1: Running couplings in SM and MSSM using two-loop RG evolution. The SUSY threshold at 2 TeV is clearly visible on the MSSM side. (We thank Ben Allanach for providing the plots created using SOFTSUSY [63].)</figcaption></figure>

To discuss high scale thresholds, let us set all other corrections to zero for the moment and write down a version of Eq. (93.3) that captures the running near and above the GUT scale more correctly. The threshold correction at one-loop level can be evaluated accurately by the simple step-function approximation for the $\beta$ functions in the DR scheme<fnref n="7" /> [74],

$$ \alpha_i^{-1}(m_Z) = \alpha_G^{-1}(\mu) + \frac{1}{2\pi} \left[ b_i \ln \frac{\mu}{m_Z} + b_i^C \ln \frac{\mu}{M_C} + b_i^X \ln \frac{\mu}{M_X} + b_i^\Phi \ln \frac{\mu}{M_\Phi} \right]. \quad (93.12) $$

Here we started the running at some scale $\mu \gg M_G$, including the contribution of the minimal set of states relevant for the transition from the high-scale $SU(5)$ model to the MSSM. These are the color-triplet Higgs multiplets with mass $M_C$, massive vector multiplets of X-bosons with mass $M_X$ (including GUT Higgs degrees of freedom), and the remaining GUT-Higgs fields and superpartners with mass $M_\Phi$. The coefficients $b_i^{C,X,\Phi}$ can be found in Ref. [75]. Crucially, the $b_i$ in Eq. (93.12) conspire to make the running GUT-universal at high scales, such that the resulting prediction for $\alpha_3$ does not depend on the value of $\mu$.

To relate this to our previous discussion, we can, for example, define $M_G \equiv M_X$ and then choose $\mu = M_G$ in Eq. (93.12). This gives the high-scale threshold corrections

$$ \delta_i^{(h)} = \frac{1}{2\pi} \left[ b_i^C \ln \frac{M_G}{M_C} + b_i^\Phi \ln \frac{M_G}{M_\Phi} \right], \quad (93.13) $$

and a corresponding correction $\Delta_3^{(h)}$. To get some intuition for the magnitude, one can furthermore assume $M_\Phi = M_G$, finding (with $b_i^C = \{2/5, 0, 1\}$)

$$ \Delta_3^{(h)} = \frac{9}{14\pi} \ln \left( \frac{M_G}{M_C} \right). \quad (93.14) $$

To obtain the desired effect of $-\Delta_3^{(2)} - \Delta_3^{(l)} \simeq +0.64$, the triplet Higgs would have to be by about a factor 20 lighter than the GUT scale. While this is ruled out by nucleon decay in the minimal model [76] as will be discussed Sec. 93.6, it is also clear that

threshold corrections of this order of magnitude can, in general, be realized with a certain amount of GUT-scale model building, e.g. in specific $SU(5)$ [25] or $SO(10)$ [26,27] constructions. Corrections can also be much larger or of different sign if, as is required in many fully realistic 4d GUT models, many additional (and in particular higher) representations are introduced. Thus, there is considerable model building freedom. Nevertheless, a significant constraint from getting the right GUT threshold corrections while keeping the triplet Higgs heavy remains.

The above analysis implicitly assumes universal soft SUSY breaking masses at the GUT scale, which directly affect the spectrum of SUSY particles at the weak scale. In the simplest case we have a universal gaugino mass $M_{1/2}$, a universal mass for squarks and sleptons $m_{16}$ and a universal Higgs mass $m_{10}$, as motivated by $SO(10)$. In some cases, threshold corrections to gauge coupling unification can be exchanged for threshold corrections to soft SUSY parameters (see [77] and refs. therein). For example, if gaugino masses were not unified at $M_G$ and, in particular, gluinos were lighter than winos at the weak scale (cf. Eq. (93.11)), then it is possible that, due to weak scale threshold corrections, a much smaller or even slightly negative threshold correction at the GUT scale would be consistent with gauge coupling unification [78].

It is also noteworthy that perfect unification can be realized without significant GUT-scale corrections, simply by slightly raising the (universal) SUSY breaking scale. In this case the dark matter abundance produced by thermal processes in the early Universe (if the lightest neutralino is the dark matter particle) is too high. However, even if the gaugino mass in the MSSM is about 1 TeV to explain the dark matter abundance, if the Higgsino and the non-SM Higgs boson masses are about 10-100 TeV, the effective SUSY scale can be raised [79]. This setup is realized in split SUSY [80] or the pure gravity mediation model [81] based on anomaly mediation [82]. Since the squarks and sleptons are much heavier than the gaugino masses in those setups, a gauge hierarchy problem is reintroduced. The facts that no superpartners have so far been seen at the LHC and that the observed Higgs mass favors heavier stop masses than about 1 TeV force one to accept a certain amount of fine-tuning anyway.

For non-SUSY GUTs or GUTs with a very high SUSY breaking scale to fit the data, new light states in incomplete GUT multiplets or multiple GUT breaking scales are required. For example, non-SUSY models $SO(10) \to SU(4)_C \times SU(2)_L \times SU(2)_R \to SM$, with the second breaking scale of order an intermediate scale, determined by light neutrino masses using the see-saw mechanism, can fit the low-energy data for gauge couplings [83] and at the

<footnote n="7">⁷The DR scheme is frequently used in a supersymmetric regularization [71]. The renormalization transformation of the gauge coupling constants from MS to DR scheme is given in Ref. [72]. For an alternative treatment using holomorphic gauge couplings and NSVZ $\beta$-functions see e.g. [73].</footnote>
same time survive nucleon decay bounds [84]. Alternatively, one
can appeal to string-theoretic corrections discussed in Sec. 93.4
to compensate for a high SUSY breaking scale. This has, for ex-
ample, been concretely analyzed in the context of F-theory GUTs
in [85]. Similarly, one may even wonder whether particularly large
GUT threshold corrections could be sufficient to ensure non-SUSY
precision unification. Notice here that the gauge coupling unifica-
tion predicts just one parameter. When introducing new states,
typically one can fit the data by choice of their masses. This is
not the case in SUSY GUTs with low-scale SUSY breaking scale
where the masses are constrained by fine-tuning.

In 5d or 6d orbifold GUTs, certain “GUT scale” threshold cor-
rections come from the Kaluza-Klein modes between the compact-
ification scale, M_c ~ 1/R, and the effective cutoff scale M*. In
string theory, this cutoff scale is the string scale. Gauge coupling
unification at two loops then constrains the values of M_c and
M*.<fnref n="8" /> Often, one finds M_c to be lower than the 4d GUT scale.
Since the X-bosons, responsible for nucleon decay, get mass at
the compactification scale, this has significant consequences for
nucleon decay.

Finally, it has been shown that non-supersymmetric GUTs in
warped 5d orbifolds can be consistent with gauge coupling unifi-
cation. This assumes (in 4d language) that the r.h. top quark and
the Higgs doublets are composite-like objects with a composite-
ness scale in the TeV range [87].

93.6 Nucleon decay

Quarks and leptons are indistinguishable in any 4d GUT, and
both the baryon (B) and lepton number (L) are not conserved.
This leads to baryon-number-violating nucleon decay. In addi-
tion to baryon-number violation, lepton-number violation is also
required for nucleon decay since, in the SM, leptons are the
only free fermions which are lighter than nucleons. The lowest-
dimension operators relevant for nucleon decay are (B+L) violat-
ing dimension-six four-fermion-terms in the SM, and all baryon-
violating operators with dimension less than seven preserve (B-L)
[1,88].

In SU(5) GUTs, the dimension-six operators are induced by X
boson exchange. These operators are suppressed by $(1/M_X^2)$ ($M_X$ is the X boson mass), and the nucleon lifetime is given by $\tau_N \propto M_X^4/(\alpha_G^2 m_p^5)$ ($m_p$ is proton mass). The dominant decay mode of the proton (and the baryon-violating decay mode of the neutron), via X boson exchange, is $p \to e^+ \pi^0$ ($n \to e^+ \pi^-$). In any simple gauge symmetry, with one universal GUT coupling $\alpha_G$ and scale $M_X$, the nucleon lifetime from gauge boson exchange is calculable. Hence, the GUT scale may be directly observed via the extremely rare decay of the nucleon. The present experimental bounds on the modes come from Super-Kamiokande. With 450 kton-years of data they find $\tau_p/\text{Br}(p \to e^+\pi^0) > 2.4 \times 10^{34}$ years at 90% CL [89]. In addition, Hyper-Kamiokande [90] is planned to reach to $\tau_p/\text{Br}(p \to e^+\pi^0) \sim 10^{35}$ years. The reader may consult [91] for other planned experiments and some historical context. The hadronic matrix elements for baryon-number-violating operators are evaluated with lattice QCD simulations [92]. In SUSY SU(5) GUTs, the lower bound on the X boson mass from null results in nucleon decay searches is approaching $10^{16}$ GeV [93], which is close to the GUT scale suggested by gauge coupling unification. On the other hand, the prediction for nucleon decay in non-SUSY GUTs is hard to quantify. The reason is that gauge couplings do not unify with just the SM particle content. Once extra states or large thresholds are included to ensure precision unification, a certain range of unification scales is allowed.

In SUSY GUTs there are additional sources for baryon and/or
lepton-number violation – dimension-four and five operators [14].
These arise since, in the SUSY SM, quarks and leptons have scalar
partners (squarks and sleptons). Although our notation does
not change, when discussing SUSY models our fields are chiral
superfields and both fermionic and bosonic matter is implicitly
represented by those. In this language, baryon- and/or lepton-
number-violating dimension-four and five operators are given as

so-called F terms of products of chiral superfields, which con-
tain two fermionic components and the rest scalars or products of
scalsars. Within the context of SU(5) the dimension-four and five
operators have the form

$$
\begin{align*}
(10 \bar{5} \bar{5}) \supset & (u^c d^c d^c) + (Q L d^c) + (e^c L L), \\
(10 \ 10 \ 10 \bar{5}) \supset & (Q Q Q L) + (u^c u^c d^c e^c) \\
& + B^- \text{ and } L^- \text{-conserving terms,}
\end{align*}
$$

respectively.

The dimension-four operators in (10 5 5) violate either baryon number or lepton number. The nucleon lifetime is extremely short if both types of dimension-four operators are present in the SUSY SM since squark or slepton exchange induces the dangerous dimension-six SM operators. Even in the case that they violate baryon number or lepton number only but not both, they are constrained by various phenomena [94]. For example, the primordial baryon number in the Universe is washed out unless the dimensionless coupling constants are less than 10⁻⁷. Both types of operators can be eliminated by requiring R parity, which distinguishes Higgs from ordinary matter multiplets. R parity [95] or its cousin, matter parity [6, 96], act as F → −F, H → H with F = {10, 5}, H = {5H, 5H} in SU(5).<fnref n="9" /> In SU(5), the Higgs multiplet 5H and the matter multiplets 5 have identical gauge quantum numbers. In E₆, Higgs and matter multiplets could be unified within the fundamental 27 representation. Only in SO(10) are Higgs and matter multiplets distinguished by their gauge quantum numbers. The Z₄ center of SO(10) distinguishes 10s from 16s and can be associated with R parity [97].

The baryon-number violating dimension-five operators have a dimensionful coupling. They are generated by integrating out the color-triplet Higgs with GUT-scale mass in SUSY GUTs such that the coefficient is suppressed by $1/M_G$. Note that both triplet Higgsinos (due to their fermionic nature) and Higgs scalars (due to their mass-enhanced trilinear coupling with matter) contribute to the operators. The dimension-five operators include squarks and/or sleptons. To allow for nucleon decay, these must be converted to light quarks or leptons by exchange of a gaugino or Higgsino in the SUSY SM. The nucleon lifetime is proportional to $M_G^2 m_{\text{SUSY}}^2/m_p^5$, where $m_{\text{SUSY}}$ is the SUSY breaking scale. Thus, dimension-five operators may predict a shorter nucleon lifetime than dimension-six operators. Unless accidental cancellations are present, the dominant decay modes from dimension-five operators include a K meson, such as $p \to K^+ \bar{\nu}$ ($n \to K^0 \bar{\nu}$). This is due to a simple symmetry argument: The operators are given as $(Q_i Q_j Q_k L_l)$ and $(u_i^c u_j^c d_k^c e_l^c)$, where $i, j, k, l (= 1, 2, 3)$ are family indices and color and weak indices are implicit. They must be invariant under SU(3)C and SU(2)L so that their color and weak doublet indices must be anti-symmetrized. Since these operators are given by bosonic superfields, they must be totally symmetric under interchange of all indices. Thus, the first operator vanishes for $i = j = k$ and the second vanishes for $i = j$. Hence a second or third generation member exists in the dominant modes of nucleon decay unless these modes are accidentally suppressed [96].

The Super-Kamiokande bounds on the proton lifetime severely constrain the dimension-five operators. With 306 kton-years of data they find $\tau_p/\text{Br}(p \to K^+\bar{\nu}) > 6.61 \times 10^{33}$ years at 90% CL [98]. In the minimal SUSY SU(5), $\tau_p/\text{Br}(p \to K^+\bar{\nu})$ is smaller than about $10^{31}$ years if the triplet Higgs mass is $10^{16}$ GeV and $m_{\text{SUSY}} = 1$ TeV [99]. The triplet Higgs mass bound from nucleon decay is then in conflict with gauge coupling unification so that this model is considered to be ruled out [76].

Since nucleon decay induced by the triplet Higgs is a severe
problem in SUSY GUTs, various proposals for its suppression
have been made. First, some accidental symmetry or accidental
structure in non-minimal Higgs sectors in SU(5) or SO(10) theo-
ries may suppress the dimension-five operators [22, 26, 27, 100].

<footnote n="8">⁸ It is interesting to note that a ratio $M_*/M_c \sim 100$, needed for gauge coupling unification to work in orbifold GUTs, is typically the maximum value for this ratio consistent with perturbativity [86].</footnote>

<footnote n="9">⁹ This forbids the dimension-four operator (10 5 5), but allows the Yukawa couplings for quark and lepton masses of the form (10 5 5H) and (10 10 5H). It also forbids the dimension-three, lepton-number-violating operator (5 5H) ⊃ (L H_u) as well as the dimension-five, baryon-number-violating operator (10 10 10 5H) ⊃ (Q Q Q H_d) + ...</footnote>
Symmetries to suppress the dimension-five operators are typically broken by the VEVs responsible for the color-triplet Higgs masses. Consequently, the dimension-five operators are generically generated via the triplet Higgs exchange in SUSY SU(5) GUTs, as mentioned above. In other words, the nucleon decay is suppressed if the Higgs triplets in $\bar{5}_H$ and $5_H$ do not have a common mass term but, instead, their mass terms involve partners from other SU(5) multiplets. Second, the SUSY breaking scale may be around $\mathcal{O}(10-100)$ TeV in order to explain the observed Higgs boson mass at the LHC. In this case, nucleon decay is automatically suppressed [80, 101, 102]. Third, accidental cancellations among diagrams due to a fine-tuned structure of squark and slepton flavor mixing might suppress nucleon decay [103]. Last, we have also implicitly assumed a hierarchical structure for Yukawa matrices in the analysis. It is however possible to fine-tune a hierarchical structure for quarks and leptons which baffles the family structure so that the nucleon decay is suppressed [104]. The upper bound on the proton lifetime from some of these theories is approximately a factor of 10 above the experimental bounds. Future experiments with larger neutrino detectors, such as JUNO [105], Hyper-Kamiokande [90], DUNE [106] and Theia [107], are planned and will have higher sensitivities to nucleon decay.

Are there ways to avoid the stringent predictions for proton decay discussed above? Orbifold GUTs and string theories, see Sec. 93.4, contain grand unified symmetries realized in higher dimensions. In the process of compactification and GUT symmetry breaking, the triplet Higgs states may be removed (projected out of the massless sector of the theory). In such models, the nucleon decay due to dimension-five operators can be severely suppressed or eliminated completely. However, nucleon decay due to dimension-six operators may be enhanced, since the gauge-bosons mediating proton decay obtain mass at the compactification scale, $M_c$, which is typically less than the 4d GUT scale (cf. Sec. 93.5). Alternatively, the same projections which eliminate the triplet Higgs may rearrange the quark and lepton states such that the massless states of one family come from different higher-dimensional GUT multiplets. This can suppress or completely eliminate even dimension-six proton decay. Thus, enhancement or suppression of dimension-six proton decay is model-dependent. In some complete 5d orbifold GUT models [65, 108] the lifetime for the decay $\tau_p/\text{Br}(p \to e^+\pi^0)$ can be near the bound of $1 \times 10^{34}$ years with, however, large model-dependence and/or theoretical uncertainties. In other cases, the modes $p \to K^+\bar{\nu}$ and $p \to K^0\mu^+$ may be dominant [65]. Thus, interestingly, the observation of nucleon decay may distinguish string or higher-dimensional GUTs from 4d ones.

In orbifold GUTs or string theory, new discrete symmetries consistent with SUSY GUTs can forbid all dimension-three and four baryon- and lepton-number-violating operators. Even the $\mu$ term and dimension-five baryon- and lepton-number-violating operators can be forbidden to all orders in perturbation theory [35]. The $\mu$ term and dimension-five baryon- and lepton-number-violating operators may then be generated, albeit sufficiently suppressed, via non-perturbative effects. The simplest example of this is a $Z_4^R$ symmetry which is the unique discrete R symmetry consistent with SO(10) [35]. Even though it forbids the dimension-five proton decay operator to the desired level, it allows the required dimension-five neutrino mass term. In this case, proton decay is dominated by dimension-six operators, leading to decays such as $p \to e^+\pi^0$.

## 93.7 Yukawa coupling unification

In the SM, masses and mixings for quarks and leptons come from the Yukawa couplings with the Higgs doublet, but the values of these couplings remain a mystery. GUTs provide at least a partial understanding since each generation is embedded in unified multiplet(s). Specifically, since quarks and leptons are two sides of the same coin, the GUT symmetry relates the Yukawa couplings (and hence the masses) of quarks and leptons.

In SU(5), there are two types of independent renormalizable Yukawa interactions given by $\lambda_{ij}$ ($10_i, 10_j, \bar{5}_H$) + $\lambda'_{ij}$ ($10_i, \bar{5}_j, \bar{5}_H$). These contain the SM interactions $\lambda_{ij}$ ($Q_i u_j^c H_u$) + $\lambda'_{ij}$ ($Q_i d_j^c H_d$ + $e_i^c L_j H_d$). Here i, j (=1, 2, 3)

are, as before, family indices. Hence, at the GUT scale we have tree-level relations between Yukawa coupling constants for charged lepton and down quark masses, such as $\lambda_b = \lambda_\tau$ in which $\lambda_{b/\tau}$ are the bottom quark $l/\tau$ lepton Yukawa coupling constants [109, 110]. In SO(10), there is only one type of independent renormalizable Yukawa interaction given by $\lambda_{ij}$ ($16_i, 16_j, 10_H$), leading to relations among all Yukawa coupling constants and quark and lepton masses within one generation [111, 112] (such as $\lambda_t = \lambda_b = \lambda_\tau$, with $\lambda_t$ the top quark Yukawa coupling constant).

In addition to gauge coupling unification, the ratio of bottom quark and $\tau$ lepton mass has been a central target in the study of GUTs since it was found that this ratio was almost consistent with observations after including the QCD correction [109]. Today the quark masses and the gauge coupling constants are known precisely such that, as discussed below, the ratio of bottom quark and $\tau$ lepton mass has become a target of precision analyses in the GUT context.

### 93.7.1 The third generation, $b-\tau$ or $l-b-\tau$ unification

Third generation Yukawa couplings are larger than those of the first two generations. Hence, the fermion mass relations predicted from renormalizable GUT interactions which we introduced above are expected to be more reliable. In order to compare them with data, we have to include the radiative correction to these relations from the RG evolution between GUT and fermion mass scale, from integrating out heavy particles at the GUT scale, and from weak scale thresholds.

Since testing Yukawa coupling unification is only possible in models with successful gauge coupling unification, we here focus on SUSY GUTs. In the MSSM, top and bottom quark and $\tau$ lepton masses are related to the Yukawa coupling constants at the scale $m_Z$ as

$$ \begin{aligned} m_t(m_Z) &= \lambda_t(m_Z) v_u(1 + \delta m_t / m_t), \\ m_{b/\tau}(m_Z) &= \lambda_{b/\tau}(m_Z) v_d(1 + \delta m_{b/\tau} / m_{b/\tau}), \end{aligned} $$

where $\langle H_u^0 \rangle \equiv v_u = \sin\beta v/\sqrt{2}$, $\langle H_d^0 \rangle \equiv v_d = \cos\beta v/\sqrt{2}$, $v_u/v_d \equiv \tan\beta$ and $v \sim 246$ GeV is fixed by the Fermi constant, $G_\mu$. Here, $\delta m_f / m_f$ ($f = t, b, \tau$) represents the threshold correction due to integrating out SUSY partners. For the bottom quark mass, it is found [113] that the dominant corrections come from the gluino-sbottom and from the Higgsino-stop loops,

$$ \begin{aligned} \left(\frac{\delta m_b}{m_b}\right)_{g_3} &\sim \frac{g_3^2}{6\pi^2} \frac{m_{\tilde{g}}^{\sim \mu}}{m_{\text{SUSY}}^2} \tan \beta \quad \text{and} \\ \left(\frac{\delta m_b}{m_b}\right)_{\lambda_t} &\sim \frac{\lambda_t^2}{16\pi^2} \frac{A_t \mu}{m_{\text{SUSY}}^2} \tan \beta, \end{aligned} \quad (93.16) $$

where $m_{\tilde{g}}$, $\mu$, and $A_t$ stand for gluino and Higgsino masses and trilinear stop coupling, respectively. Note that Eq. (93.16) only illustrates the structure of the corrections – non-trivial functional dependences on several soft parameters $\sim m_{\text{SUSY}}$ have been suppressed. For the full one-loop correction to the bottom quark mass see, for example, Ref. [114].

Note also that the corrections do not go to zero as SUSY particles become much heavier than $m_Z$. They may change the bottom quark mass at the $\mathcal{O}(10)\%$ level for $\tan\beta = \mathcal{O}(10)$. The total effect is sensitive to the relative phase between gluino and Higgsino masses since $A_t \sim -m_{\tilde{g}}$ due to the infrared fixed point nature of the RG equation for $A_t$ [115] in settings where SUSY breaking terms come from Planck scale dynamics, such as gravity mediation. The $\tau$ lepton mass also receives a similar correction, though only at the few % level. The top quark mass correction, not being proportional to $\tan\beta$, is at most 10% [116].

Including one loop threshold corrections at $m_Z$ and additional RG running, one finds the top, bottom and $\tau$ pole masses. In SUSY GUTs, $b-\tau$ unification has two possible solutions with $\tan\beta \sim 1$ or $\mathcal{O}(10)$. The small $\tan\beta$ solution may be realized in the MSSM if superpartner masses are $\mathcal{O}(10)$ TeV, as suggested by the observed Higgs mass [101]. The large $\tan\beta$ limit such as $\tan\beta \sim 40-50$ overlaps the SO(10) symmetry relation [116, 117]. When $\tan\beta$ is large, there are significant threshold corrections to