84. Pentaquarks

Revised March 2024 by M. Karliner (Tel Aviv U.) and T. Skwar-
nicki (Syracuse U.).

Experimental searches for pentaquark hadrons comprised of
light flavors have a long and vivid history. No undisputed can-
didates had been found in 50 years. The first wave of claimed
observations of pentaquark candidates containing a strange anti-
quark occurred in the early seventies, see e.g. a review in the 1976
edition of Particle Data Group listings for Z₀(1780), Z₀(1865) and
Z₁(1900) [1]. The last mention of these candidates can be found
in the 1992 edition [2] with the perhaps prophetic comment “the
results permit no definite conclusion - the same story for 20 years.
[...] The skepticism about baryons not made of three quarks, and
lack of any experimental activity in this area, make it likely that
another 20 years will pass before the issue is decided.” A decade
later, a second wave of observations occurred, possibly motivated
by specific theoretical predictions for their existence [3–5]. The ev-
idence for pentaquarks was based on observations of peaks in the
invariant mass distributions of their decay products. More data
and subsequent more sensitive experiments did not confirm these
claims [6]. In the last mention of the best known candidate from
that period, Θ(1540)⁺, the 2006 Particle Data Group listing [7] in-
cluded a statement: “The conclusion that pentaquarks in general,
and that Θ⁺, in particular, do not exist, appears compelling.”
which well reflected the prevailing mood in the particle physics
community until a study of Λ₀ᵇ → J/ψpK⁻ (J/ψ → µ⁺µ⁻) de-
cays by LHCb [8] (charge conjugate modes are implied). From
an analysis of 3 fb⁻¹ Run 1 data at 7 and 8 TeV at the LHC,
the LHCb collaboration reported a significant J/ψp structure in
Λ₀ᵇ → J/ψpK⁻ decays [8]. The exotic character of this structure,
with the minimal quark content of uudc<span style="text-decoration: overline;">c</span>c, was demonstrated in a
nearly model-independent way in Ref. [9], where it was shown that
the J/ψp mass (m<sub>J/ψp</sub>) peak near 4450 MeV was too narrow to be
accounted for by Λ* → pK⁻ reflections (Λ* denotes a generic Λ ex-
citation), reinforcing the results from the earlier model-dependent
six-dimensional amplitude analysis of invariant masses and decay
angles describing the Λ₀ᵇ decay in the same data [8]. The LHCb 6
fb⁻¹ Run 2 LHC data at 13 TeV, together with the improvements
in the data selection for both runs, resulted in a nine-fold increase
in the number of reconstructed Λ₀ᵇ → J/ψpK⁻ decays (246,000
events) [10] and observation of new narrow J/ψp structures which
were too faint to had been significant in the Run 1 data analysis.
A second horizontal band is observed in the Dalitz plot (Fig. 84.1)
near 4312 MeV in the J/ψp mass.

The 4450 MeV structure also appears to consist of two nar-
rower peaks at 4440 and 4457 MeV. Performing a rigorous six-

<figure><img src="image_354.png" /><figcaption>**Figure 84.1:** Dalitz plot distributions for Λ�0_b → J/ψpK⁻ decays as observed by LHCb.</figcaption></figure>

dimensional amplitude analysis of these faint J/ψp structures is
challenging and has not been accomplished yet. Fortunately, the
newly observed peaks are so narrow that it is not necessary to con-
struct an amplitude model to prove that these states are not arti-

facts of interfering Λ* resonances, as was previously demonstrated
in Ref. [9]. Their masses and widths have been characterized by
the LHCb (see Table 84.1) from one-dimensional fits to J/ψp mass
distributions, with different levels of suppression of the Λ* contri-
butions, which peak at the lower pK⁻ masses (Fig. 84.1). Such
analysis is not sensitive to any broad J/ψp contributions. The
histograms analyzed by the LHCb are available in tabular form
at https://www.hepdata.net/record/89271.

The fit chosen by the LHCb for the central mass and width
values is displayed in Fig. 84.2. The P<sub>c<span style="text-decoration: overline;">c</span></sub>(4312)<sup>+</sup> state (formerly
known as P<sub>c</sub>(4312)<sup>+</sup>) peaks right below the Σ<sub>c</sub><sup>+</sup>D<sup>0</sup> threshold and
has statistical significance over 7.6σ. The P<sub>c<span style="text-decoration: overline;">c</span></sub>(4457)<sup>+</sup> state peaks
right below the Σ<sub>c</sub><sup>+</sup>D<sup>*0</sup> threshold, while the P<sub>c<span style="text-decoration: overline;">c</span></sub>(4440)<sup>+</sup> state
peaks about 20 MeV below it. The significance of the two-peak
versus one-peak hypothesis for the 4450 MeV structure is over
5.4σ, rendering the single peak interpretation of this region obso-
lete. The six-dimensional amplitude analysis reported in Ref. [8],
which in addition to the structure near 4450 MeV, provided also
evidence for the broad P<sub>c<span style="text-decoration: overline;">c</span></sub>(4380)<sup>+</sup> state, is obsolete since it used
the single P<sub>c<span style="text-decoration: overline;">c</span></sub>(4450)<sup>+</sup> state and it lacked the P<sub>c<span style="text-decoration: overline;">c</span></sub>(4312)<sup>+</sup> state.
Furthermore, it used the helicity formalism in which the half-
integer spin of the proton was not aligned properly between the
different Λ<sub>b</sub><sup>0</sup> decay chains [11, 12]. The newer one-dimensional
analysis by LHCb [10] was not sensitive to wide P<sub>c<span style="text-decoration: overline;">c</span></sub><sup>+</sup> states. The
LHCb result from the six-dimensional amplitude analysis of the
Cabibbo suppressed channel Λ<sub>b</sub><sup>0</sup> → J/ψpπ<sup>-</sup> [13], which contains a
statistically marginal evidence for the sum of the P<sub>c<span style="text-decoration: overline;">c</span></sub><sup>+</sup> and the T<sub>c<span style="text-decoration: overline;">c</span>1</sub>
(formerly known as Z<sub>c</sub>(4200)<sup>-</sup>) contributions, took extensive in-
put from Ref. [8] and, like the P<sub>c<span style="text-decoration: overline;">c</span></sub>(4380)<sup>+</sup> state, should be treated
with caution until the both amplitude analyses are completed on
the enlarged data sets with the modified helicity formalism.

<figure><img src="image_355.png" /><figcaption>**Figure 84.2:** Fit to the J/ψp mass distribution, in which events were weighted to suppress Λ* → pK⁻ backgrounds, of three Breit-Wigner functions and a sixth-order polynomial background. This fit was used to determine the central values of the masses and widths of the P<sub>c<span style="text-decoration: overline;">c</span></sub> states (formerly known as P<sub>c</sub><sup>+</sup>) reported by LHCb. The mass thresholds for the Σ<sub>c</sub><sup>+</sup>D<sup>0</sup> and Σ<sub>c</sub><sup>+</sup>D<sup>*0</sup> final states are superimposed.</figcaption></figure>

While Σ_c D*(*) states had been predicted [14–17] before the first
LHCb results [8], after these results became known, many theo-
retical groups interpreted the P_{c\bar{c}}(4450)⁺ and P_{c\bar{c}}(4380)⁺ states
in terms of diquarks and triquarks as building blocks of a compact
pentaquark [18–24]. In a different strategy, a tentative attempt
has been made to treat the full 5-body dynamics, leading to states
below the lowest threshold for spontaneous dissociation [25]. In
the first implementation of the former approach [18], the pen-
<table><caption>**Table 84.1:** Summary of the narrow $P_{c\bar{c}}^+$ (formerly known as $P_c^+$) properties, interpreted as Breit-Wigner resonances. The central values are based on the fit displayed in Fig. 84.2.</caption><thead><tr><th>State</th><th><i>M</i> [MeV]</th><th><i>Γ</i> [MeV] (95% CL)</th><th><i>R</i> [%]</th></tr></thead><tbody><tr><td><i>P</i><sub>c<span style="text-decoration: overline">c</span></sub>(4312)<sup>+</sup></td><td>4311.9 ± 0.7<sup>+6.8</sup><sub>-0.6</sub></td><td>9.8 ± 2.7<sup>+3.7</sup><sub>-4.5</sub> (< 27)</td><td>0.30 ± 0.07<sup>+0.34</sup><sub>-0.09</sub></td></tr><tr><td><i>P</i><sub>c<span style="text-decoration: overline">c</span></sub>(4440)<sup>+</sup></td><td>4440.3 ± 1.3<sup>+4.1</sup><sub>-4.7</sub></td><td>20.6 ± 4.9<sup>+8.7</sup><sub>-10.1</sub> (< 49)</td><td>1.11 ± 0.33<sup>+0.22</sup><sub>-0.10</sub></td></tr><tr><td><i>P</i><sub>c<span style="text-decoration: overline">c</span></sub>(4457)<sup>+</sup></td><td>4457.3 ± 0.6<sup>+4.1</sup><sub>-1.7</sub></td><td>6.4 ± 2.0<sup>+5.7</sup><sub>-1.9</sub> (< 20)</td><td>0.53 ± 0.16<sup>+0.15</sup><sub>-0.13</sub></td></tr></tbody></table>

taquark mass splitting was generated mostly by the change of angular momentum between the sub-components ($L$) from zero to one, which would also make the heavier state narrower and of opposite parity. Explicit modeling of multiquark systems [26] questions if centrifugal barrier factor provides enough width suppression via spatial separation of $c$ and $\bar{c}$ quarks at these masses, as the phase space for $J/\psi p$ decay is very large (more than 400 MeV). Also, the observed mass splitting was too small to be only due to the mechanism proposed in Ref. [18] and required fine-tuning of such models. A variation of this model, in which the heavy (cu) diquark couples with heavy $\bar{c}$ to form colored triquark attracting the light diquark (ud), has been re-implemented for the narrow $P_{c\bar{c}}^+$ states [27]. In this model, the $P_{c\bar{c}}(4440)^+$ and $P_{c\bar{c}}(4457)^+$ states are accommodated via spin-orbit interactions for the $L=1$ states, while the $P_{c\bar{c}}(4312)^+$ is one of the $L=0$ states. However, the mass prediction for the latter is off by (-72 ± 29) MeV [27]. This work was later extended to $SU(3)_F$ [28]. The width dilemma becomes more severe in view of the narrow widths of the newly observed states (Table 84.1), especially for the $L=0$ $P_{c\bar{c}}(4312)^+$ state, and requires a different origin of potential barrier between $c$ and $\bar{c}$ than angular momentum [27, 29], which remains a subject of theoretical controversy. Measurement of spin-parity of $P_{c\bar{c}}(4312)^+$, $P_{c\bar{c}}(4440)^+$ and $P_{c\bar{c}}(4457)^+$ will be crucial for testing these and alternative theoretical ideas discussed in the following.

More effective width suppression mechanism is offered by a loosely bound charmed baryon-anticharmed meson molecular model, in which $c$ and $\bar{c}$ can be separated by much larger distances, resulting in a smaller probability of them getting close enough to each other in order to make a $J/\psi$. Since molecular binding energy cannot be large, such molecules are in S-wave, so it has been realized early on that their masses must be near the sum of the baryon and meson masses and their spin and parity are inherited from their constituent hadrons, see e.g. discussion in [17]. The mass coincidence of the $P_{c\bar{c}}(4312)^+$ and of $P_{c\bar{c}}(4457)^+$ states, with the two related thresholds, $\Sigma_c^+\bar{D}^0$ and $\Sigma_c^+\bar{D}^*$, provides very strong experimental evidence in favor of this interpretation. Given how close $P_{c\bar{c}}(4312)^+$ is to the $\Sigma_c^+\bar{D}^0$ threshold, it might be a virtual rather than a bound state [30]. Since the spins of $\Sigma_c^+$ and of $\bar{D}^*$ can be combined in two different ways, the narrow $P_{c\bar{c}}(4440)^+$ peak also finds natural explanation in this physical picture. It cannot be a virtual state since it is sufficiently below the $\Sigma_c^+\bar{D}^*$ threshold.

It is worth stressing that other baryon-meson combinations, $\Lambda_c^{(*)+}\bar{D}^{(*)0}$ and $\chi_{cJp}$ are not expected to bind [14, 31]. Before the first pentaquark observation [8] heavy quark symmetry was used to show that, in addition to the three $\Sigma_c\bar{D}^{(*)0}$ states, one expects four $\Sigma_c^+\bar{D}^{(*)0}$ states, for a total of seven [32]. Indeed, additional states at, or below, the $\Sigma_c^{*+}\bar{D}$ and $\Sigma_c^{*+}\bar{D}^*$ thresholds, are expected [33–36]. Since $\Sigma_c^{*+}$ width is likely around 15 MeV [37], more than the width of either $P_{c\bar{c}}(4312)^+$ or $P_{c\bar{c}}(4457)^+$, it is important to keep in mind that a molecule is typically as broad as the constituents<fnref n="1" /> [38–40]. There is no significant evidence for $\Sigma_c^*\bar{D}^{(*)}$ states in the present LHCb data set [10, 41]. Larger data sets are expected from the Upgraded LHCb experiment. For a review on hadronic molecules, and a recent survey in a particular approach, see respectively Refs. [42, 43].

It is useful to consider the $P_{c\bar{c}}(4312)^+$, $P_{c\bar{c}}(4440)^+$ and $P_{c\bar{c}}(4457)^+$ narrow pentaquarks together with several analogous exotic states with hidden charm and bottom in the meson sector. This provides additional significant motivation for the molecular model. At least five exotic mesons are close to thresholds of

two heavy-light mesons: $\chi_{c1}(3872)$ (formerly known as $X(3872)$) [44–47], $T_{b\bar{b}1}(10610)$ and $T_{b\bar{b}1}(10650)$ (formerly $Z_b(10610)$ and $Z_b(10650)$) in the bottomonium sector [48–52] and $T_{c\bar{c}1}(3900)$ (formerly $Z_c(3900)$) [53–57] and $T_{c\bar{c}1}(4020)$ (formerly $Z_c(4020)$) [58–60] in the charmonium sector (see Table II in Ref. [61]; for reviews of experimental information see Ref. [62, 63], as well as Spectroscopy of Mesons Containing Two Heavy Quarks and Heavy Non-q\bar{q} Mesons in the current Review of Particle Properties. These states share several important features: a) their masses are near thresholds and their spin and parity correspond to S-wave combination of the two mesons; b) they are very narrow, despite very large phase space for decay into quarkonium + pion(s); c) the branching fractions for “fall apart” mode into two mesons are much larger than branching fractions for decay into quarkonium and pion(s). So far, there is no experimental evidence for states at two pseudoscalar thresholds ($DD$ and $BB$), suggesting that pseudoscalar exchange is essential for binding in meson-meson systems.

The above provide a strong hint that these states are deuteron-like loosely bound states of two heavy mesons [64–72]. It is then natural to conjecture that similar bound states might exist of two heavy baryons [73, 74], or a meson and a baryon or a baryon and an antibaryon, leading to a rather accurate prediction of the $P_{c\bar{c}}(4457)^+$ mass as $3/2^- \Sigma_c\bar{D}^*$ molecule (the mass threshold is 4460 MeV for $\Sigma_c^+\bar{D}^*$ and 4464 MeV for $\Sigma_c^{++}\bar{D}^{*-}$) [17, 61], following similar predictions obtained in a wider framework of doubly heavy baryon-meson hadronic molecules, which might include mixtures of various two-hadron states [14–16, 31]. However, single pion exchange is not possible in $\Sigma_c^+\bar{D}^0$ system. Thus the existence of $P_{c\bar{c}}(4312)^+$ suggests a more general binding mechanism in baryon-meson molecules, e.g. vector or two-pion exchanges, and/or coupled-channel interactions. Two-pion exchange in $D\bar{D}$ system is highly suppressed, because the intermediate state is $D^*\bar{D}^*$, which is 282 MeV heavier than $D\bar{D}$. On the other hand, in the $\Sigma_c\bar{D}$ system the intermediate $\Lambda_c^+\bar{D}^*$ state is only 25 MeV lighter than $\Sigma_c\bar{D}$, and therefore does not have to suffer significant suppression, providing a possible binding mechanism [75]. In a generic hadronic molecule it is essential that the two hadrons are heavy, in order to minimize the repulsive kinetic energy [73, 74, 76].

Following the initial LHCb discovery [8], several groups carried out a detailed analysis of the $P_{c\bar{c}}^+$ states as hadronic molecules [77–86] followed by further analyses [35, 87–111] after the updated LHCb results [10]. Partial widths of all the allowed decay channels for the $P_{c\bar{c}}$ states have been estimated within a specific model in the molecular picture [112]. The most striking suggestion is that $P_{c\bar{c}}(4312)^+$ decays are totally dominated by the $\Lambda_c^+\bar{D}^*$ channel. This channel is also suggested to be very prominent in decays of $P_{c\bar{c}}(4440)^+$ and $P_{c\bar{c}}(4457)^+$. But other scenarios are possible, as shown by a recent theoretical analysis, allowing for the most general scattering potential of the baryon-meson system consistent with QCD symmetries at leading order in a momentum counting [113]. They found that both large and vanishing transitions to $\Lambda_c^+\bar{D}^{(*)0}$ channels are allowed. Clearly, experimental determination of the branching fraction of these channels is of high priority.

The $P_{c\bar{c}}$ states have also been interpreted as so called hadrocharmonium [114], a bound state of relatively compact charmonium states with light hadronic matter. It was proposed that $P_{c\bar{c}}(4440)^+$ and $P_{c\bar{c}}(4457)^+$ are spin-split $\psi(2S)p$ bound states with $J^P = \frac{1}{2}^-$ and $\frac{3}{2}^-$, while $P_{c\bar{c}}(4312)^+$ is a $\chi_{c0p}$ bound state with $J^P = \frac{1}{2}^+$ [115]. While very interesting from the theoretical point of view, it is not at all clear why the binding energies

<footnote n="1">¹This feature gets changed if systems are more deeply bound.</footnote>
between charmonia and the nucleon should conspire to produce states so close to the $Σ_c \bar{D}$ and $Σ_c \bar{D}^*$ thresholds. Moreover, the predicted widths of $P_{c\bar{c}}(4440)^+$ and $P_{c\bar{c}}(4457)^+$ are too big by a factor ~ 2-3. One should also keep in mind that the molecular and hadro-charmonium pictures provide opposite predictions for the parity of $P_{c\bar{c}}(4312)^+$. In principle LHCb can check the spin and parity through partial wave analysis, but at present it is not known if systematic uncertainties can be sufficiently reduced to make such an analysis conclusive.

Shortly after the initial experimental discovery it was conjectured that the $P_{c\bar{c}}(4450)^+$ reflects the presence of a triangle singularity near the $\chi_{c1}p$ threshold [116–119]. These explanations are no longer viable, since the $P_{c\bar{c}}(4440)^+$ mass is not at any threshold and the $P_{c\bar{c}}(4312)^+$ and $P_{c\bar{c}}(4457)^+$ peak slightly below the $Σ_c^+ \bar{D}^0$ and $Σ_c^+ \bar{D}^{*0}$ thresholds. The $P_{c\bar{c}}(4457)^+$ mass is exactly at $\Lambda_c^+ \bar{D}^0$ threshold, but LHCb has demonstrated that the observed peaking is narrower in the data than expected from the triangle-diagram when a realistic width of the excited $D_s^-$ state exchanged in the triangle is used (Supplemental Material in Ref. [10]).

More extensive pre-2019 reviews of some of the theoretical issues can be found in Refs. [120,121]. Two recent relevant reviews are Refs. [122,123].

So far the $P_{c\bar{c}}^+$ states have been observed by only one experiment in only one channel. It is essential to explore other possible experimental channels, such as $P_{c\bar{c}}^+ \to \Lambda_c^+ \bar{D}^{(*)0}$, $\eta_c p$ [113]. These channels are however much more experimentally challenging than $P_{c\bar{c}}^+ \to J/\psi p$. Proposals have also been made to search for heavy pentaquarks in photo-production [124–131]. Ref. [132] discusses photoproduction within the string-junction physical picture of the pentaquarks. Photoproduction is also related to recent work on $J/\psi(\eta_c)N$ scattering on the lattice [133] and on computation of $J/\psi(\eta_c)N$ and $\Upsilon(\eta_b)N$ cross sections [134]. In addition, pentaquark production has been discussed in the context of antiproton-deuterium collisions [135], of heavy ion collisions at LHC [136], in pA collisions [137] and in pion-induced processes [138–140]. The GlueX Collaboration reported negative search results for the $P_{c\bar{c}}^+$ states in photo-production at JLAB [141, 142], $\gamma p \to (J/\psi \to e^+e^-)p$. Within the large experimental errors and considerable theoretical model dependence these results do not contradict the molecular interpretations of the narrow $P_{c\bar{c}}^+$ states. In particular, vector-meson dominance often assumed in theoretical modelling may not hold for near-threshold $J/\psi$ production, and $\Lambda_c \bar{D}^{(*)0}$ intermediate states may significantly contribute to the $J/\psi p$ production complicating probing for the $P_{c\bar{c}}^+$ states [143, 144].

As noted in e.g. [17], bottom analogues of the $P_{c\bar{c}}^+$ might well exist, but experimental search for such states involves very significant challenges. It is therefore hardly surprising that they have not been observed so far. A detailed discussion is beyond the scope of the current review.

The LHCb collaboration obtained a 3.1σ evidence for a $P_{c\bar{c}}^+ \to J/\psi p$ state in the four-dimensional amplitude analysis of about $800 B_s^0 \to J/\psi p \bar{p} \ (J/\psi \to \mu^+ \mu^-)$ decays [145]. The mass of the state, $4337_{-4}^{+7} \pm 2$ MeV, is not compatible with the $P_{c\bar{c}}(4312)^+$ state at 3.1 standard deviations. The width is relatively small, $29_{-12}^{+26} \pm 14$ MeV. The present data do not provide sufficient discrimination between various $J^P$ assignments. Its mass is about 19 MeV higher than the $Σ_c^+ \bar{D}^0$ threshold and about 16 MeV lower than the $\chi_{c0}p$ threshold. At present, no specific explanation for this structure has been proposed, but some intriguing ideas have been raised in Ref. [146]. Since the statistical significance of this evidence is marginal, more data are required before this state is considered experimentally established.

Similarly inconclusive 3.1σ evidence for a $P_{c\bar{c} s}^0 \to J/\psi \Lambda$ structure was observed by the LHCb collaboration in the six-dimensional amplitude analysis of about 1750 $\Xi_b^-$ to $J/\psi \Lambda K^-$ ($J/\psi \to \mu^+ \mu^-$, $\Lambda \to p\pi^-$ decays [12]. The strange counterparts of $P_{c\bar{c}}^+$ states have been predicted in both molecular and compact pentaquark models [15, 99, 101, 147–149]. When interpreted as a single peak, its mass, $4458.8 \pm 2.9_{-1.1}^{+4.7}$ MeV, is 15 MeV above the $\Xi_c^0 \bar{D}^0$ threshold and 18 MeV below the $\Xi_c^0 \bar{D}^{*0}$ threshold. Its width, $17.3 \pm 6.5_{-5.7}^{+8.0}$ MeV, is relatively narrow, thus plausibly

could be interpreted as a loosely bound $\Xi_c^0 \bar{D}^{*0}$ state. However, in the latter model two states are expected with $J^P$ equal to $1/2^-$ and $3/2^-$ . In fact, the LHCb data are consistent with such hypothesis, under which $4454.9 \pm 2.7$ MeV ($\Gamma = 7.5 \pm 9.7$ MeV) and $4467.8 \pm 3.7$ MeV ($\Gamma = 5.2 \pm 5.3$ MeV) mass (width) estimates are obtained (statistical errors only). It is worth noting, that the $SU(3)$ flavor structure of $\Xi_c^0 \bar{D}^{*0}$ states is different from that of $Σ_c^+ \bar{D}^{*0}$ states, since $\Xi_c^0$ belongs to the charmed baryon triplet, together with $\Xi_c^+$ and $\Lambda_c^+$, where the two light quarks form a spin-zero diquark, unlike $Σ_c^+$, which together with $\Xi_c^{0}$ and $\Omega_c^0$ belongs to the charmed baryon sextet, where the two light quarks form a spin-1 diquark. More data are required to experimentally establish this structure, clarify its composition, and determine the related quantum numbers.

If the two states at 4455 MeV and 4468 MeV indeed correspond to a $\Xi_c^0 \bar{D}^{*0}$ molecule, there perhaps exist two other states, corresponding to $\Xi_c' \bar{D}^*$ [15, 99, 101, 147–149], with a mass shifted upwards by approximately $\Xi_c' - \Xi_c$ mass difference [150,151], i.e. about 108 MeV [37].

More recently, the LHCb experiment observed highly significant (> 14σ) pentaquark state $P_{c\bar{c}s}(4338)^0 \to J/\psi \Lambda$ (formerly known as $P_{c\bar{c}}(4338)$) in the six-dimensional amplitude analysis of $4620 \pm 70 B^- \to J/\psi \Lambda \bar{p}$ decays [152]. Its mass $4338.2 \pm 0.7 \pm 0.4$ MeV is right at the $\Xi_c \bar{D}$ threshold (Fig. 84.3), its width is narrow $7.0 \pm 1.2 \pm 1.3$ MeV, and its spin is 1/2, with a preference for negative parity, which all fits the molecular model very well. This makes the $P_{c\bar{c}s}(4338)^0$ state a likely $\Xi_c^0 \bar{D}^0$ analog of the $P_{c\bar{c}}(4312)^+$ state in $Σ_c^0 \bar{D}^0$ system. In addition, as in the $P_{c\bar{c}s}(4459)^0$ case, a $\Xi_c' \bar{D}$ state is expected, for a total of three additional strange pentaquarks [151]. Experimental determination whether yet additional $\Xi_c^{(*)'} \bar{D}^{(*)0}$ states actually exist will provide a useful testing ground for the various theoretical approaches which have been used to predict them.

<figure><img src="image_356.png" /><figcaption>**Figure 84.3:** Distribution of invariant $J/\psi \Lambda$ mass, with the projection of the amplitude fit model superimposed. The model without the $P_{c\bar{c}s}(4338)^0$ state (labeled as $P_{\psi s}^{\Lambda 0}$) is also shown in grey. The $\Xi_c^+ \bar{D}^-$ baryon-meson threshold is indicated with a vertical dashed line.</figcaption></figure>

In hidden-charm exotics discussed above, the charmed and the anticharmed quarks can form a charmonium and thereby decouple from the light quarks. The molecular model provides an efficient mechanism to suppress such decays, thus making these states narrow. Compact direct color couplings between quarks are missing an obvious width suppression mechanism, thus could lead to broader states. In fact, there is plenty of evidence for them in form of broader tetraquark mass structures in $J/\psi \phi$ [153–159], $J/\psi J/\psi$ [160–162], $T_{c\bar{c}1}(4430)^+ \to \psi(2S)\pi^+$ (formerly
$Z_c(4430)^+$ [163–165] and others, although alternative explanations of these structures have been also proposed. Analogous configurations may also exist for pentaquarks, but there is no firm evidence for them yet.

The hidden-charm exotic states, should not be confused with exotic hadrons containing two heavy quarks, rather than a heavy quark and a heavy antiquark. Decoupling from the light quarks is impossible in exotics which contain two heavy quarks. This has far-reaching consequences. The first tetraquark of the latter type, i.e. containing two heavy quarks, has been reported by LHCb [166, 167]. A comprehensive discussion of tetraquark candidates can be found in the “Heavy non-$Q\bar{Q}$ Mesons” review.

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Hypothetical Particles and Concepts

<table><tr><td>85. Extra dimensions (rev.)</td><td>1021</td></tr><tr><td>86. W'-boson searches (rev.)</td><td>1029</td></tr><tr><td>87. Z'-boson searches (rev.)</td><td>1033</td></tr><tr><td>88. Supersymmetry: theory (rev.)</td><td>1038</td></tr><tr><td>89. Supersymmetry: experiment (rev.)</td><td>1058</td></tr><tr><td>90. Axions and other similar particles (rev.)</td><td>1079</td></tr><tr><td>91. Quark and lepton compositeness, searches for (rev.)</td><td>1098</td></tr><tr><td>92. Dynamical electroweak symmetry breaking: implications of the H<sup>0</sup> (rev.)</td><td>1104</td></tr><tr><td>93. Grand unified theories (rev.)</td><td>1119</td></tr><tr><td>94. Leptoquarks (rev.)</td><td>1134</td></tr><tr><td>95. Magnetic monopoles (rev.)</td><td>1138</td></tr></table>
<figure><img src="image_357.png" /></figure>
# 85. Extra Dimensions

Revised August 2023 by Z. Demiragli (Boston U.) and A. Pomarol (U. Autònoma de Barcelona; IFAE).

## 85.1 Introduction

Proposals for a spacetime with more than three spatial dimensions date back to the 1920s, mainly through the work of Kaluza and Klein, in an attempt to unify the forces of nature [1]. Although their initial idea failed, the formalism that they and others developed is still useful nowadays. Around 1980, string theory proposed again to enlarge the number of space dimensions, this time as a requirement for describing a consistent theory of quantum gravity. The extra dimensions were supposed to be compactified at a scale close to the Planck scale, and thus not testable experimentally in the near future.

A different approach was given by Arkani-Hamed, Dimopoulos, and Dvali (ADD) in their seminal paper in 1998 [2], where they showed that the weakness of gravity could be explained by postulating two or more flat extra dimensions in which only gravity could propagate. The size of these extra dimensions should range between roughly a millimeter and ~1/TeV, leading to possible observable consequences in current and future experiments. A year later, Randall and Sundrum (RS) [3] found a new possibility using a warped geometry, postulating a five-dimensional Anti-de Sitter (AdS) spacetime with a compactification scale of order 1/TeV. The origin of the smallness of the electroweak scale versus the Planck scale was explained by the gravitational redshift factor present in the warped AdS metric. As in the ADD model, originally only gravity was assumed to propagate in the extra dimensions, although it was soon clear that this was not necessary in warped extra dimensions and also the SM gauge fields [4, 5] and SM fermions [6, 7] could propagate in the five-dimensional spacetime.

The physics of warped extra-dimensional models has an alternative interpretation by means of the AdS/CFT correspondence [8–10]. Models with warped extra dimensions are related to four-dimensional strongly-interacting theories, allowing an understanding of the properties of five-dimensional fields as those of four-dimensional composite states [11]. This approach has opened new directions for tackling outstanding questions in particle physics, such as the flavor problem, grand unification, and the origin of electroweak symmetry breaking or supersymmetry breaking.

### 85.1.1 Experimental Constraints

Constraints on extra-dimensional models arise from astrophysical and cosmological considerations, tabletop experiments exploring gravity at sub-mm distances, and collider experiments. Collider limits on extra-dimensional models are dominated by LHC results, which can be found on the public WWW pages of ATLAS [12] and CMS [13]. This review includes the most recent limits, most of which are published results based on 140 fb⁻¹LHC data collected in 2015-18 at a center-of-mass energy of 13 TeV and legacy results from 20 fb⁻¹of 8 TeV data collected in Run 1. For most of the models, Run 2 results surpass the sensitivity of Run 1.

### 85.1.2 Kaluza-Klein Theories

Field theories with compact extra dimensions can be written as theories in ordinary four dimensions (4D) by performing a Kaluza-Klein (KK) reduction. As an illustration, consider a simple example, namely a field theory of a complex scalar in flat five-dimensional (5D) spacetime. The action will be given by <fnref n="1" />

$$S_5 = - \int d^4 x dy M_5 \left[ |\partial_\mu \phi|^2 + |\partial_y \phi|^2 + \lambda_5 |\phi|^4 \right], \quad (85.1)$$

where y refers to the extra (fifth) dimension. A universal scale $M_5$ has been extracted in front of the action in order to keep the 5D field with the same mass-dimension as in 4D. This theory is perturbative for energies $E \lesssim \ell_5 M_5 / \lambda_5$ where $\ell_5 = 24\pi^3$ [14].

Let us now consider that the fifth dimension is compact with the topology of a circle $S^1$ of radius $R$, which corresponds to the identification of $y$ with $y + 2\pi R$. In such a case, the 5D complex scalar field can be expanded in a Fourier series:

$$\phi(x, y) = \frac{1}{\sqrt{2\pi R M_5}} \sum_{n=-\infty}^{\infty} e^{iny/R} \phi^{(n)}(x), \quad (85.2)$$

that, inserted in Eq. (85.1) and integrating over $y$, gives

$$S_5 = S_4^{(0)} + S_4^{(n)}, \quad (85.3)$$

where

$$S_4^{(0)} = - \int d^4 x \left[ |\partial_\mu \phi^{(0)}|^2 + \lambda_4 |\phi^{(0)}|^4 \right], \quad (85.4)$$

and,

$$S_4^{(n)} = - \int d^4 x \sum_{n \neq 0} \left[ |\partial_\mu \phi^{(n)}|^2 + \left(\frac{n}{R}\right)^2 |\phi^{(n)}|^2 \right. \\ \left. + \text{quartic interactions.} \right] \quad (85.5)$$

The $n=0$ mode self-coupling is given by

$$\lambda_4 = \frac{\lambda_5}{2\pi R M_5}. \quad (85.6)$$

The above action corresponds to a 4D theory with a massless scalar $\phi^{(0)}$, referred to as the zero mode, and an infinite tower of massive modes $\phi^{(n)}$ with $n > 0$, known as KK modes. The KK reduction thus allows a treatment of 5D theories as 4D field theories with an infinite number of fields. At energies smaller than $1/R$, the KK modes can be neglected, leaving the zero-mode action of Eq. (85.4). The strength of the interaction of the zero-mode, given by Eq. (85.6), decreases as $R$ increases. Thus, for a large extra dimension $R \gg 1/M_5$, the massless scalar is very weakly coupled.

## 85.2 Large Extra Dimensions for Gravity

### 85.2.1 The ADD Scenario

The ADD scenario [2, 15] (for a review see, for example, [16]) assumes a $D = 4 + \delta$ dimensional spacetime, with $\delta$ compactified spatial dimensions. The apparent weakness of gravity arises since it propagates in the higher-dimensional space. The SM is assumed to be localized in a 4D subspace, a 3-brane, as can be found in certain string theory constructions [17, 18]. Gravity is described by the Einstein-Hilbert action in $D = 4 + \delta$ spacetime dimensions

$$S_D = -\frac{\bar{M}_D^{2+\delta}}{2} \int d^4 x d^\delta y \sqrt{-g} \mathcal{R} + \int d^4 x \sqrt{-g_{\text{ind}}} \mathcal{L}_{\text{SM}}, \quad (85.7)$$

where $x$ labels the ordinary four coordinates, $y$ the $\delta$ extra coordinates, $g$ refers to the determinant of the $D$-dimensional metric whose Ricci scalar is defined by $\mathcal{R}$, and $\bar{M}_D$ is called the reduced Planck scale of the $D$-dimensional theory. In the second term of Eq. (85.7), which gives the gravitational interactions of SM fields, the $D$-dimensional metric reduces to the induced metric on the 3-brane where the SM fields propagate. The extra dimensions are assumed to be flat and compactified in a volume $V_\delta$. As an example, consider a toroidal compactification of equal radii $R$ and volume $V_\delta = (2\pi R)^\delta$. After a KK reduction, one finds that the fields that couple to the SM are the spin-2 gravitational field $G_{\mu\nu}(x, y)$ and a tower of spin-0 KK graviscals [19]. The graviscals, however, only couple to SM fields through the trace of the energy-momentum tensor, resulting in weaker couplings to the SM fields. The Fourier expansion of the spin-2 field is given by

$$G_{\mu\nu}(x, y) = G_{\mu\nu}^{(0)}(x) + \frac{1}{\sqrt{V_\delta}} \sum_{\vec{n} \neq 0} e^{i \vec{n} \cdot \vec{y} / R} G_{\mu\nu}^{(\vec{n})}(x), \quad (85.8)$$

where $\vec{y} = (y_1, y_2, \dots, y_\delta)$ are the extra-dimensional coordinates and $\vec{n} = (n_1, n_2, \dots, n_\delta)$. Eq. (85.8) contains a massless state,

<footnote n="1">¹Our convention for the metric is $\eta_{MN} = \text{Diag}(-1, 1, 1, 1, 1)$.</footnote>
the 4D graviton $G_{\mu\nu}^{(0)}$, and its KK tower $G_{\mu\nu}^{(\bar{n})}$ with masses $m_{\bar{n}}^2 = |\bar{n}|^2/R^2$. At energies below $1/R$ the action is that of the zero mode

$$S_4^{(0)} = -\frac{\bar{M}_D^{2+\delta}}{2} \int d^4x V_\delta \sqrt{-g^{(0)}} \mathcal{R}^{(0)} + \int d^4x \sqrt{-g_{\text{ind}}^{(0)}} \mathcal{L}_{\text{SM}}, \quad (85.9)$$

where we can identify the 4D reduced Planck mass, $M_P \equiv 1/\sqrt{8\pi G_N} \approx 2.4 \times 10^{18}$ GeV, as a function of the D-dimensional parameters:

$$M_P^2 = V^\delta \bar{M}_D^{2+\delta} \equiv R^\delta M_D^{2+\delta}. \quad (85.10)$$

Fixing $M_D$ at around the electroweak scale $M_D \sim \text{TeV}$ to avoid introducing a new mass scale in the model, Eq. (85.10) gives a prediction for $R$:

$$\delta = 1, 2, \dots, 6 \rightarrow R \sim 10^9 \text{ km}, 0.5 \text{ mm}, \dots, 0.1 \text{ MeV}^{-1}. \quad (85.11)$$

The option $\delta = 1$ is clearly ruled out, as it leads to modifications of Newton's law at solar system distances. However this is not the case for $\delta \ge 2$, and possible observable consequences can be sought in present and future experiments.

Consistency of the model requires a stabilization mechanism for the radii of the extra dimensions, to the values shown in Eq. (85.11). The fact that we need $R \gg 1/M_D$ leads to a new hierarchy problem, the solution of which might require imposing supersymmetry in the extra-dimensional bulk (for the case of two extra dimensions see for example [20]).

### 85.2.2 Tests of the Gravitational Force Law at Sub-mm Distances

The KK modes of the graviton give rise to deviations from Newton's law of gravitation for distances $\lesssim R$. Such deviations are usually parameterized by a modified Newtonian potential of the form

$$V(r) = -G_N \frac{m_1 m_2}{r} \left[ 1 + \alpha e^{-r/\lambda} \right]. \quad (85.12)$$

For a 2-torus compactification, $\alpha = 16/3$ and $\lambda = R$. Searches for deviations from Newton's law of gravitation have been performed in several experiments [21–24]. From Ref. [23] we have the constraint $R < 30 \mu\text{m}$ at 95% CL for $\delta = 2$, corresponding to $M_D > 4.0$ TeV. We see then that bounds from Newton's law deviations are already pushing the scale $M_D$ beyond the TeV for two extra dimensions.

### 85.2.3 Astrophysical and Cosmological Constraints

The light KK gravitons could be copiously produced in stars, carrying away energy. Ensuring that the graviton luminosity is low enough to preserve the agreement of stellar models with observations provides powerful bounds on the scale $M_D$. The most stringent bound arises from supernova SN1987A, giving $M_D > 27$ (2.4) TeV for $\delta = 2$ (3) [25]. After a supernova explosion, most of the KK gravitons stay gravitationally trapped in the remnant neutron star. The requirement that neutron stars are not excessively heated by KK decays into photons leads to $M_D > 1700$ (76) TeV for $\delta = 2$ (3) [26].

Cosmological constraints are also quite stringent [27]. To avoid overclosure of the Universe by relic gravitons one needs $M_D > 7$ TeV for $\delta = 2$. Relic KK gravitons decaying into photons contribute to the cosmic diffuse gamma radiation, from which one can derive the bound $M_D > 100$ TeV for $\delta = 2$.

We must mention however that bounds coming from the decays of KK gravitons into photons can be reduced if we assume that KK gravitons decay mainly into other non-SM states. This could happen, for example, if there were other 3-branes with hidden sectors residing on them [15].

### 85.2.4 Collider Signals

#### 85.2.4.1 Graviton and Other Particle Production

Although each KK graviton has a purely gravitational coupling, suppressed by $1/M_P$, inclusive processes in which one sums over the almost continuous spectrum of available gravitons have cross sections suppressed only by powers of $M_D$. Processes involving gravitons are therefore detectable in collider experiments if $M_D \sim \text{TeV}$. A number of experimental searches for evidence of

large extra dimensions have been performed at colliders, and interpreted in the context of the ADD model.

One signature arises from direct graviton emission. By making a derivative expansion of Einstein gravity, one can construct an effective theory, valid for energies much lower than $M_D$, and use it to make predictions for graviton-emission processes at colliders [19, 28, 29]. Gravitons produced in the final state would escape detection, giving rise to missing transverse momentum ($p_T^{\text{miss}}$). The results quoted below are 95% CL lower limits on $M_D$ for a range of values of $\delta$ between 2 and 6, with more stringent limits corresponding to lower $\delta$ values.

At hadron colliders, experimentally sensitive channels include the jet (j) + $p_T^{\text{miss}}$ and $\gamma + p_T^{\text{miss}}$ final states. ATLAS (CMS) $j + p_T^{\text{miss}}$ results with 139 (137) fb$^{-1}$ of 13 TeV data provide limits on $M_D > 5.9 - 11.2$ TeV [30] ($M_D > 5.5 - 10.7$ TeV [31]). For these analyses, both experiments are assuming leading order (LO) cross sections. Since the effective theory is only valid for energies much less than $M_D$, the results are quoted for the full space, and include the information that suppressing the graviton cross section by a factor $M_D^4/\hat{s}^2$ for $\sqrt{\hat{s}} > M_D$, where $\sqrt{\hat{s}}$ is the parton-level center-of-mass energy of the hard collision, weakens the limits on $M_D$ by a negligible amount for $\delta = 2$ (~3% for $\delta = 6$). Less stringent limits are obtained by CMS [32] from analysis of 36 fb$^{-1}$ of 13 TeV data in the $\gamma + p_T^{\text{miss}}$ final state ($M_D > 2.85 - 2.90$ TeV for $\delta = 3 - 6$). The analogous ATLAS search [33] uses full Run 2 statistics but does not quote ADD interpretation of the results.

In models in which the ADD scenario is embedded in a string theory at the TeV scale [18], we expect the string scale $M_s$ to be smaller than $M_D$, and therefore expect production of string resonances at the LHC [34]. A result from CMS analyzing the dijet invariant mass distribution for 137 fb$^{-1}$ of 13 TeV data excludes string resonances that decay predominantly to $q + g$ with masses below 7.9 TeV [35]. ATLAS dijet analysis [36] provides their results in the context of model-independent limits on the cross section times acceptance for generic resonances of a variety of possible widths, from which one can deduce similar lower mass limits ~ 8 TeV for string resonances decaying to $q + g$.

#### 85.2.4.2 Virtual graviton effects

One can also search for virtual graviton effects, the calculation of which however depends on the ultraviolet cut-off of the theory and is therefore very model dependent. In the literature, several different formulations exist [19, 29, 37] for the dimension-eight operator for gravity exchange at tree level:

$$\mathcal{L}_8 = \pm \frac{4}{M_{TT}^4} \left( T_{\mu\nu} T^{\mu\nu} - \frac{1}{\delta + 2} T^\mu_\mu T^\nu_\nu \right), \quad (85.13)$$

where $T_{\mu\nu}$ is the energy-momentum tensor and $M_{TT}$ is related to $M_D$ by some model-dependent coefficient [38]. The relations with the parametrization of Refs. [37] and [19] are, respectively, $M_{TT} = M_S$ and $M_{TT} = (2/\pi)^{1/4} \Lambda_T$. The experimental results below are given as 95% CL lower limits on $M_{TT}$, including in some cases the possibility of both constructive or destructive interference, depending on the sign chosen in Eq. (9).

The most stringent limits arise from LHC analyses of the dijet angular distribution. Using 35.9 fb$^{-1}$ of 13 TeV data, CMS [39] obtains results that correspond to an approximate limit of $M_{TT} \gtrsim 9$ TeV.

The next most restrictive results come from the analyses of diphoton ($M_{TT} > 6.1$ TeV from ATLAS [40] and $M_{TT} > 7.0$ TeV from CMS [41]) and dilepton mass spectra ($M_{TT} > 6.5$ TeV from CMS [42]). The complete Run 2 (139 fb$^{-1}$) analysis of ATLAS di-lepton data [43] does not quote the limits on ADD.

At the one-loop level, gravitons can also generate dimension-six operators with coefficients that are also model dependent. Experimental bounds on these operators can also give stringent constraints on $M_D$ [38].

#### 85.2.4.3 Black Hole Production

The physics at energies $\sqrt{s} \sim M_D$ is sensitive to the details of the unknown quantum theory of gravity. Nevertheless, in the transplanckian regime, $\sqrt{s} \gg M_D$, one can rely on a semiclassical description of gravity to obtain predictions. An interesting feature
of transplanckian physics is the creation of black holes [44,45] (for a review see, for example, [46]). A black hole is expected to be formed in a collision in which the impact parameter is smaller than the Schwarzschild radius [47]:

$$R_S = \frac{1}{M_D} \left[ \frac{2^\delta \pi^{(\delta-3)/2}}{\delta + 2} \Gamma \left( \frac{\delta + 3}{2} \right) \frac{M_{BH}}{M_D} \right]^{1/(\delta+1)}, \quad (85.14)$$

where $M_{BH}$ is the mass of the black hole, which would roughly correspond to the total energy in the collision. The cross section for black hole production can be estimated to be of the same order as the geometric area $\sigma \sim \pi R_S^2$. For $M_D \sim$ TeV, this gives a production of $\sim 10^7$ black holes at the $\sqrt{s} = 14$ TeV LHC with an integrated luminosity of 30 fb$^{-1}$ [44,45]. A black hole would provide a striking experimental signature since it is expected to thermally radiate with a Hawking temperature $T_H = (\delta+1)/(4\pi R_S)$, and therefore would evaporate democratically into all SM states. Nevertheless, given the present constraints on $M_D$, the LHC will not be able to reach energies much above $M_D$. This implies that predictions based on the semiclassical approximation could receive sizable modifications from model-dependent quantum-gravity effects.

The most stringent limits on microscopic black holes arise from LHC searches which observed no excesses above the SM background in high-multiplicity final states. The results are usually quoted as model-independent limits on the cross section for new physics in the final state and kinematic region analyzed. These results can then be used to provide constraints of models of low-scale gravity and weakly-coupled string theory. In addition, limits are sometimes quoted on particular implementations of models, which are used as benchmarks to illustrate the sensitivity.

A CMS analysis [48] of multi-object final states using 36 fb$^{-1}$ of 13 TeV data, excludes semiclassical black holes below masses of up to 10.1 TeV for $M_D = 2$ TeV and $\delta = 6$. Analogous Run 2 ATLAS analysis [49], using 3.0 fb$^{-1}$ of 13 TeV data, excludes black hole masses up to 9.0–9.7 TeV, depending on $M_D$, for $\delta = 6$. Another ATLAS search [50] for an excess of events with multiple high transverse momentum objects, including charged leptons and jets, using 3.2 fb$^{-1}$ of 13 TeV data, excludes semiclassical black holes below masses of $\sim 8.7$ TeV for $M_D = 2$ TeV and $\delta = 6$.

A complementary approach is to look for jet extinction at high transverse momenta, as we expect hard short distance scattering processes to be highly suppressed at energies above $M_D$ [51]. The CMS analysis [52] of inclusive jet $p_T$ spectrum in 10.7 fb$^{-1}$ of 8 TeV data set a lower limit of 3.3 TeV on the extinction mass scale.

For black hole masses near $M_D$, the semi-classical approximation is not valid, and one could instead expect quantum black holes (QBH) that decay primarily into two-body final states [53]. In the context of both ADD model with $\delta = 6$ the QBHs have been searched in few final state analysis (based on 2.3 fb$^{-1}$ [54]), dijet [36,39,55], (same and different flavor) dilepton [56–60], photon+jet [61,62] and lepton+jet [63] channels by both ATLAS and CMS experiments. The Run 2 results at 13 TeV provide lower limits on QBH masses of up-to 9.4 TeV in an ADD model with $\delta = 6$. The strongest constraints are from dijet searches.

In weakly-coupled string models the semiclassical description of gravity fails in the energy range between $M_s$ and $M_s/g_5^2$ where stringy effects are important. In this regime one expects, instead of black holes, the formation of string balls, made of highly excited long strings, that could be copiously produced at the LHC for $M_s \sim$ TeV [64], and would evaporate thermally at the Hagedorn temperature giving rise to high-multiplicity events. The same analyses used to search for black holes can be interpreted in the context of string balls. For example, for the case of $\delta = 6$ with $M_s = M_D/1.26 = 3$ TeV, the ATLAS multiple high transverse momentum object analysis [49] excludes string balls with masses below 6.5 to 9.0 TeV for values of $0.2 < g_s < 0.8$. The CMS multi-object analysis [48] studies string ball production in two scenarios, both assuming $\delta = 6$. For the constant $g_s = 0.2$ and $1 < M_s < 3.5$ TeV the string ball masses below 7.2 to 9.4 TeV are excluded, while at constant $M_s = 3.6$ TeV and $0.2 < g_s < 0.4$ masses below 7.2 to 8.1 TeV are excluded.

# 85.3 TeV-Scale Extra Dimensions

## 85.3.1 Warped Extra Dimensions

The RS model [3] is the most attractive setup of warped extra dimensions at the TeV scale, since it provides an alternative solution to the hierarchy problem. The RS model is based on a 5D theory with the extra dimension compactified in an orbifold, $S^1/Z_2$, a circle $S^1$ with the extra identification of $y$ with $-y$. This corresponds to the segment $y \in [0, \pi R]$, a manifold with boundaries at $y = 0$ and $y = \pi R$. Let us now assume that this 5D theory has a cosmological constant in the bulk $\Lambda$, and on the two boundaries $\Lambda_0$ and $\Lambda_{\pi R}$. The action is given by

$$S_5 = - \int d^4 x dy \left\{ \sqrt{-g} \left[ \frac{1}{2} M_5^3 R + \Lambda \right] + \sqrt{-g_0} \delta(y) \Lambda_0 \right. \\ \left. + \sqrt{-g_{\pi R}} \delta(y - \pi R) \Lambda_{\pi R} \right\}, \quad (85.15)$$

where $g_0$ and $g_{\pi R}$ are the values of the determinant of the induced metric on the two respective boundaries. Einstein's equations can be solved, giving in this case the metric

$$ds^2 = a(y)^2 dx^\mu dx^\nu \eta_{\mu\nu} + dy^2, \quad a(y) = e^{-ky}, \quad (85.16)$$

where $k = \sqrt{-\Lambda/6M_5^3}$. Consistency of the solution requires $\Lambda_0 = -\Lambda_{\pi R} = -\Lambda/k$. The metric in Eq. (85.16) corresponds to a 5D AdS space. The factor $a(y)$ is called the “warp” factor and determines how 4D scales change as a function of the position in the extra dimension. In particular, this implies that energy scales for 4D fields localized at the boundary at $y = \pi R$ are redshifted by a factor $e^{-k\pi R}$ with respect to those localized at $y = 0$. For this reason, the boundaries at $y = 0$ and $y = \pi R$ are usually referred to as the ultraviolet (UV) and infrared (IR) boundaries, respectively.

As in the ADD case, we can perform a KK reduction and obtain the low-energy effective theory of the 4D massless graviton. In this case we obtain

$$M_P^2 = \int_0^{\pi R} dy e^{-2ky} M_5^3 = \frac{M_5^3}{2k} (1 - e^{-2k\pi R}). \quad (85.17)$$

Taking $M_5 \sim k \sim M_P$, we can generate an IR-boundary scale of order $ke^{-k\pi R} \sim$ TeV for an extra dimension of radius $R \simeq 11/k$. Mechanisms to stabilize $R$ to this value have been proposed [65,66] that, contrary to the ADD case, do not require introducing any new small or large parameter. Therefore a natural solution to the hierarchy problem can be achieved in this framework if the Higgs field, whose vacuum expectation value (VEV) is responsible for electroweak symmetry breaking, is localized at the IR-boundary where the effective mass scales are of order TeV. The radion field is generically heavy in models with a stabilized $R$. Nevertheless, it has been recently discussed that under some conditions a naturally light radion can arise [67–70]. In these cases the radion is identified with the dilaton, the Nambu-Goldstone boson associated to the spontaneous breaking of scale invariance, and its mass can be naturally below $ke^{-k\pi R} \sim$ TeV.

In the RS model [3], all the SM fields were assumed to be localized on the IR-boundary. Nevertheless, for the hierarchy problem, only the Higgs field has to be localized there. SM gauge bosons and fermions can propagate in the 5D bulk [4–7] (for a review see, for example, [71,72]). By performing a KK reduction from the 5D action of a gauge boson, we find [4,5]

$$\frac{1}{g_4^2} = \int_0^{\pi R} dy \frac{1}{g_5^2} = \frac{\pi R}{g_5^2}, \quad (85.18)$$

where $g_D$ ($D = 4,5$) is the gauge coupling in D-dimensions. Therefore the 4D gauge couplings can be of order one, as is the case of the SM, if one demands $g_5^2 \sim \pi R$. Using $kR \sim 10$ and $g_4 \sim 0.5$, one obtains the 5D gauge coupling

$$g_5 \sim 4/\sqrt{k}. \quad (85.19)$$

Boundary kinetic terms for the gauge bosons can modify this relation, allowing for larger values of $g_5/\sqrt{k}$.
Fermions propagating in a warped extra dimension have 4D massless zero-modes with wavefunctions which vary as $f_0 \sim \exp[(1/2-c_f)ky]$, where $c_f k$ is their 5D mass [7,73]. Depending on the free parameter $c_f k$, fermions can be localized either towards the UV-boundary ($c_f > 1/2$) or IR-boundary ($c_f < 1/2$). Since the Higgs boson is localized on the IR-boundary, one can generate exponentially suppressed Yukawa couplings by having the fermion zero-modes localized towards the UV-boundary, generating naturally the light SM fermion spectrum [7]. A large overlap with the wavefunction of the Higgs is needed for the top quark, in order to generate its large mass, thus requiring it to be localized towards the IR-boundary. In conclusion, the large mass hierarchies present in the SM fermion spectrum can be easily obtained in warped models via suitable choices of the order-one parameters $c_f$ [74]. In these scenarios, deviations in flavor physics from the SM predictions are expected to arise from flavor-changing KK gluon couplings [75], putting certain constraints on the parameters of the models and predicting new physics effects to be observed in B-physics processes (see, for example, [76,77]).

The masses of the KK states can also be calculated. One finds [7]

$$m_n \approx \left(n + \frac{\alpha}{2} - \frac{1}{4}\right) \pi k e^{-\pi k R}, \qquad (85.20)$$

where $n = 1, 2, ...$ and $\alpha = \{|c_f - 1/2|, 0, 1\}$ for KK fermions, KK gauge bosons and KK gravitons, respectively. Their masses are of order $ke^{-\pi k R} \sim \text{TeV}$ (for this reason we refer to these scenarios as TeV-scale extra dimensions). The first KK state of the gauge bosons would be the lightest, while gravitons are expected to be the heaviest.

### 85.3.1.1 Models of Electroweak Symmetry Breaking

Theories in warped extra dimensions can be used to implement symmetry breaking at low energies by boundary conditions (for a review see, for example, [78]). For example, for a $U(1)$ gauge symmetry in the 5D bulk, this can be easily achieved by imposing a Dirichlet boundary condition on the IR-boundary for the gauge-boson field, $A_\mu|_{y=\pi R} = 0$. This makes the zero-mode gauge boson get a mass, given by $m_A = g_4 \sqrt{2k/g_5^2} e^{-\pi k R}$. A very different situation occurs if the Dirichlet boundary condition is imposed on the UV-boundary, $A_\mu|_{y=0} = 0$. In this case the zero-mode gauge boson disappears from the spectrum. Finally, if a Dirichlet boundary condition is imposed on the two boundaries, one obtains a massless 4D scalar corresponding to the fifth component of the 5D gauge boson, $A_5$. Thus, different scenarios can be implemented by appropriately choosing the 5D bulk gauge symmetry, $G_5$, and the symmetries to which it reduces on the UV and IR-boundary, $H_{UV}$ and $H_{IR}$, respectively. In all cases the KK spectrum comes in representations of the group $G_5$.

Among the most interesting scenarios are those called gauge-Higgs unified models, where the Higgs boson appears as the fifth component of a 5D gauge boson, $A_5$. The Higgs mass is protected by the 5D gauge invariance and can only get a nonzero value from non-local one-loop effects [79]. To guarantee the relation $M_W^2 \simeq M_Z^2 \cos^2 \theta_W$, a custodial $SU(2)_V$ symmetry is needed in the bulk and IR-boundary [80]. The simplest realization [81,82] has

$$G_5 = SU(3)_c \times SO(5) \times U(1)_X, \\ H_{IR} = SU(3)_c \times SO(4) \times U(1)_X, \\ H_{UV} = G_{SM}.$$

The Higgs boson gets a potential at the one-loop level that triggers a VEV, breaking the electroweak symmetry. In these models there is a light Higgs boson whose mass can be around 125 GeV, as required by the discovered Higgs boson [83]. This state, as will be explained in Sec. 85.3.2, behaves as a composite pseudo-Nambu-Goldstone boson with couplings that deviate from the SM Higgs [84]. The present experimental determination of the Higgs couplings at the LHC, that agrees with the SM predictions, put important constraints on these scenarios [83]. The lightest KK modes of the model are color fermions with charges $Q = -1/3, 2/3$ and $5/3$ [85].

### 85.3.1.2 Constraints from Electroweak Precision Tests

Models in which the SM gauge bosons propagate in 1/TeV-sized extra dimensions give generically large corrections to electroweak observables. When the SM fermions are confined on a boundary these corrections are universal and can be parameterized by four quantities: $\hat{S}$, $\hat{T}$, W and Y, as defined in Ref. [86]. For warped models, where the 5D gauge coupling of Eq. (85.19) is large, the most relevant parameter is $\hat{T}$, which gives the bound $m_{KK} \gtrsim 10$ TeV [71]. When a custodial symmetry is imposed [80], the main constraint comes from the $\hat{S}$ parameter, requiring $m_{KK} \gtrsim 3$ TeV, independent of the value of $g_5$. Corrections to the $Zb_L b_L$ coupling can also be important [71], especially in warped models for electroweak symmetry breaking as the ones described above.

### 85.3.1.3 Kaluza-Klein Searches

The main prediction of 1/TeV-sized extra dimensions is the presence of a discretized KK spectrum, with masses around the TeV scale, associated with the SM fields that propagate in the extra dimension.

In the RS model [3], only gravity propagates in the 5D bulk. Experimental searches have been performed for the lightest KK graviton through its decay to a variety of SM particle-antiparticle pairs. The results are usually interpreted in the plane of the dimensionless coupling $k/M_P$ versus $m_1$, where $M_P$ is the reduced Planck mass defined previously and $m_1$ is the mass of the lightest KK excitation of the graviton. Since the AdS curvature $\sim k$ cannot exceed the cut-off scale of the model, which is estimated to be $\ell_5^{1/3} M_5$ [38], one must demand $k \ll \sqrt{2\ell_5} M_P \simeq 40 M_P$. The most stringent limits currently arise from LHC searches for resonances in the dilepton and diphoton final states, using 13 TeV collisions. Searches with the $\gamma\gamma$ final state are an especially powerful approach, given that these final states have a branching fraction twice that of any individual lepton flavor. The CMS analysis [41] of 36 fb$^{-1}$ of 13 TeV data excludes KK gravitons below 2.3 to 4.6 TeV, depending on the value of the coupling $k/M_P$, which is varied between 0.01 and 0.2, while ATLAS [87] uses the full 139 fb$^{-1}$ and provides a lower limit on the KK graviton mass of 4.5 TeV for the coupling parameter 0.1. The CMS [42] dilepton analyses, combining results from the $ee$ and $\mu\mu$ channels, exclude KK gravitons with masses 2.47-4.78 TeV for $k/M_P$ values of 0.01-0.1. The ATLAS [88] analysis of 139 fb$^{-1}$ of Run 2 data does not include a RS KK graviton interpretation of the results. Less stringent limits on the KK graviton mass can be derived from analyses of the dijet [35, 36, 89, 90], $HH$ [91-99], and $VV$ [100-104] final states, where V can represent either a W or Z boson.

In addition, both ATLAS and CMS experiments directly search for heavy radii, with masses above 1 TeV where the dominant decay mode is to pairs of bosons. The main production mechanism is gluon fusion. ATLAS [102] (CMS [105]) excludes radion masses below 3.2 TeV (3.1 TeV) for a radion decaying into $WW, ZZ$. Radions are also searched in final states with pairs of Higgs bosons $HH$ [91, 94, 96, 97, 99, 106], and dijet final states [107] with additional gluons. Bounds for a light radion (1 keV-10 GeV mass range) can be found in [108].

In warped extra-dimensional models in which the SM fields propagate in the 5D bulk, the couplings of the KK graviton to $ee/\mu\mu/\gamma\gamma$ are suppressed [109], and the above bounds do not apply. Furthermore, the KK graviton is the heaviest KK state (see Eq. (85.20)), and therefore experimental searches for KK gauge bosons and fermions are more appropriate discovery channels in these scenarios. For the scenarios discussed above in which only the Higgs boson and the top quark are localized close to the IR-boundary, the KK gauge bosons mainly decay into top quarks, longitudinal $W/Z$ bosons, and Higgs bosons. Couplings to light SM fermions are suppressed by a factor $g/\sqrt{g_5^2}k \sim 0.2$ [7] for the value of Eq. (85.19) that is considered from now on. Searches have been made for evidence of the lightest KK excitation of the gluon, through its decay to $t\bar{t}$ pairs. The searches take into account the natural KK gluon width, which is typically $\sim 15\%$ of its mass. The decay of a heavy particle to $t\bar{t}$ would tend to produce highly boosted top (anti-)quarks in the final state. Products of the subsequent top decays would therefore tend to be close to each other in the detector. In the case of $t \to Wb \to jjb$ decays, the three
jets could overlap with one another and not be individually reconstructed with the standard jet algorithms, while $t \to Wb \to \ell v b$ decays could result in the lepton failing standard isolation requirements due to its proximity to the b-jet; in both cases, the efficiency for properly reconstructing the final state would fall as the mass of the original particle increases. To avoid the loss in sensitivity which would result, a number of techniques, known generally as top quark tagging [110, 111], have been developed to reconstruct and identify highly boosted top quarks, for example by using a single wide jet to contain all the decay products of a hadronic top decay. The large backgrounds from QCD jets can then be reduced by requiring the jet mass be consistent with that of a top quark, and also by examining the substructure of the wide jet for indication that it resulted from the hadronic decay of a top quark. These techniques are key to extending to very high masses the range of accessible resonances decaying to $t\bar{t}$ pairs. While the ATLAS search in 139 fb⁻¹ of Run 2 data [112] does not provide a KK interpretation, dedicated analysis from CMS [113] of 36 fb⁻¹ of 13 TeV data combines di-lepton, lepton-plus-jet, and all-hadronic $t\bar{t}$ decays and excludes KK gluons with masses below 4.55 TeV. ATLAS uses all-hadronic [114] and lepton-plus-jet [115] final states to exclude KK gluons up to 3.4 and 3.8 TeV respectively with 36fb⁻¹ of 13 TeV data. The results are not directly comparable between the two LHC experiments, since they employ in their respective analyses different implementations of the theoretical model. For masses between 3 and 5 TeV, the cross-section limits are around 20 fb for CMS analysis of 36 fb⁻¹ and 30 fb (4 fb) for ATLAS analyses of 36 (139) fb⁻¹.

A gauge boson KK excitation could be also sought through its decay to longitudinal W/Z bosons. Recent analyses from ATLAS [116] (and CMS [117]) with 139 (137) fb⁻¹ of 13 TeV data searching for heavy vector resonances decaying to a W or Z boson and a Higgs in the $q\bar{q}b\bar{b}$ final state have set a lower limit on the mass of these KK of 3.2 (3.7) TeV (warped models are equivalent to the Model B considered in the analyses with $g_V \sim g_5V/\sqrt{k}$). The decay to a pair of intermediate vector bosons has also been exploited to search for KK gravitons in models in which the SM fields propagate in the 5D bulk. The analyses typically reconstruct hadronic W/Z decays using variants of the boosted techniques mentioned previously. An ATLAS analysis [118] combines leptonic and hadronic final states from the KK graviton decay $G^* \to VV$, where V can represent either a W or Z boson, exclude KK gravitons with masses below 2.3 TeV, for a value of $k/M_P = 1$. CMS VV analyses [105, 119, 120] using 137 fb⁻¹ of 13 TeV data also exclude KK gravitons with masses below 1.8 TeV in the context of bulk gravitons for a maximum value of $k/M_P = 0.5$.

The lightest KK states are, in certain models, the partners of the top quark. For example, in 5D composite Higgs models these are colored states with charges $Q = -1/3, 2/3$ and $5/3$ (arising from $SU(2)_L$ doublets with $Y = 7/6, 1/6$), and masses expected to be below the TeV [85]. They can be either singly or pair-produced, and mainly decay into a combination of W/Z with top/bottom quarks [121–124]. An exhaustive review of these searches can be found in Ref. [125]. Of particular note, the $Q = 5/3$ state decays mainly into $W^+t \to W^+W^+b$, giving a very clean signature of a pair of same-sign leptons in the final state. An analysis by ATLAS [126] searching in the lepton-plus-jets final state for evidence of pair production of the $Q = 5/3$ state provides a lower mass limit of 1.25 TeV. A CMS analysis [127] searching for pair production of the $Q = 5/3$ state using both lepton-plus-jets and same sign lepton final states excludes masses below 1.3 TeV. Similarly, searches for single production of the $Q = 5/3$ state [128], also excludes masses up to 0.9 to 1.5 TeV depending on the model parameters. Both LHC experiments have searched for pair production of vector-like quarks T and B of charges $Q = 2/3$ and $-1/3$ respectively, assuming the allowable decays are $T \to Wb/Zt/Ht$ and $B \to Wt/Zb/Hb$. In each case, it is assumed the branching fractions of the three decay modes sum to unity, but the individual branching fractions, which are model-dependent, are allowed to vary within this constraint. Both ATLAS [129–131] and CMS [132, 133] obtain lower limits on the mass of the T and B vector-like quarks up to 1.5 TeV and 1.6 TeV respectively.

Analyses from ATLAS [134–138] and CMS [128, 139–142] also

search for a single top partner and single bottom partner production, the cross section for which is model-dependent [143] but does not carry the kinematic penalty for producing two heavy objects.

### 85.3.2 Connection with Strongly Coupled Models via the AdS/CFT Correspondence

The AdS/CFT correspondence [8] provides a connection between warped extra-dimensional models and strongly-coupled theories in ordinary 4D. Although the exact connection is only known for certain cases, the AdS/CFT techniques have been very useful to obtain, at the qualitative level, a 4D holographic description of the various phenomena in warped extra-dimensional models [11, 72].

The connection goes as follows. The physics of the bulk AdS₅ models can be interpreted as that of a 4D conformal field theory (CFT) which is strongly coupled. The extra-dimensional coordinate $y$ plays the role of the renormalization scale $\mu$ of the CFT by means of the identification $\mu \equiv ke^{-ky}$. Therefore the UV-boundary corresponds in the CFT to a UV cut-off scale at $\Lambda_{UV} = k \sim M_P$, breaking explicitly conformal invariance, while the IR-boundary can be interpreted as a spontaneous breaking of the conformal symmetry at energies $ke^{-k\pi R} \sim$ TeV. Fields localized on the UV-boundary are elementary fields external to the CFT, while fields localized on the IR-boundary and KK states correspond to composite resonances of the CFT. Furthermore, local gauge symmetries in the 5D models, $G_5$, correspond to global symmetries of the CFT, while the UV-boundary symmetry can be interpreted as a gauging of the subgroup $H_{UV}$ of $G_5$ in the CFT. Breaking gauge symmetries by IR-boundary conditions corresponds to the spontaneous breaking $G_5 \to H_{IR}$ in the CFT at energies $\sim ke^{-k\pi R}$. Using this correspondence one can easily derive the 4D massless spectrum of the compactified AdS₅ models. One also has the identification $k^3/M_5^3 \approx 16\pi^2/N^2$ and $g_5^2k \approx 16\pi^2/N^r$ ($r = 1$ or 2 for CFT fields in the fundamental or adjoint representation of the gauge group), where $N$ plays the role of the number of colors of the CFT. Therefore the weak-coupling limit in AdS₅ corresponds to a large-$N$ expansion in the CFT.

Following the above AdS/CFT dictionary one can understand the RS solution to the hierarchy problem from a 4D viewpoint. The equivalent 4D model is a CFT with a TeV mass gap and a Higgs boson emerging as a composite state. In the particular case where the Higgs is the fifth-component of the gauge-boson, $A_5$ [144], this corresponds to models, similar to those proposed in Ref. [145–148], where the Higgs is a composite pseudo-Nambu-Goldstone boson arising from the spontaneous breaking $G_5 \to H_{IR}$ in the CFT. The AdS/CFT dictionary tells us that KK states must behave as composite resonances. For example, if the SM gauge bosons propagate in the 5D bulk, the lowest KK $SU(2)_L$-gauge boson must have properties similar to those of the Techni rho $\rho_T$ [125] with a coupling to longitudinal W/Z bosons given by $g_5\sqrt{k} \approx g_{\rho_T}$, while the coupling to elementary fermions is $g^2/\sqrt{g_5^2k} \approx g^2F_{\rho_T}/M_{\rho_T}$.

Fermions in compactified AdS₅ also have a simple 4D holographic interpretation. The 4D massless mode described in Sec. 85.3.1 corresponds to an external fermion $\psi_i$ linearly coupled to a fermionic CFT operator $O_i: \mathcal{L}_{\text{int}} = \lambda_i \psi_i O_i + \text{h.c.}$. The dimension of the operator $O_i$ is related to the 5D fermion mass according to $\text{Dim}[O_i] = |c_f + 1/2| - 1$. Therefore, by varying $c_f$ one varies $\text{Dim}[O_i]$, making the coupling $\lambda_i$ irrelevant ($c_f > 1/2$), marginal ($c_f = 1/2$) or relevant ($c_f < 1/2$). When irrelevant, the coupling is exponentially suppressed at low energies, and then the coupling of $\psi_i$ to the CFT (and eventually to the composite Higgs) is very small. When relevant, the coupling grows in the IR and become as large as $g_5$ (in units of $k$), meaning that the fermion is as strongly coupled as the CFT states [81]. In this latter case $\psi_i$ behaves as a composite fermion.

### 85.3.3 Linear dilaton geometry

The warp factor $a(y)$ in Eq. (85.16) can be different from the exponential considered above, giving rise to a different KK mass spectrum. A particularly interesting case is the linear dilaton geometry [149] where the KK states become very narrowly-spaced. These scenarios can also be understood in 4D via the clockwork mechanism [150], and for this reason are usually referred to as
Clockwork/Linear Dilaton (CW/LD) models. The collider phenomenology becomes different in these models since the decay of the narrowly-spaced KK spectrum appears as a periodic signal in the invariant mass of the decaying products. An excellent detector resolution is needed in order to resolve these KK modes which can be done by looking at 2 electrons or 2 photons in the final state. A search for this type of KK gravitons decaying into $ee$ [151] and $\gamma\gamma$ [151, 152] put bounds for $M_5$ in the range of 10 TeV to 1 TeV for values of the lightest KK graviton mass (referred as $k$ in [151, 152]) in the range of few TeVs.

### 85.3.4 Flat Extra Dimensions

Models with quantum gravity at the TeV scale, as in the ADD scenario, can have extra (flat) dimensions of $1/\text{TeV}$ size, as happens in string scenarios (see, for example, [153]). All SM fields may propagate in these extra dimensions, leading to the possibility of observing their corresponding KK states.

A simple example is to assume that the SM gauge bosons propagate in a flat five-dimensional orbifold $S^1/Z_2$ of radius $R$, with the fermions localized on a 4D boundary. The KK gauge bosons behave as sequential SM gauge bosons with a coupling to fermions enhanced by a factor $\sqrt{2}$ [153]. The experimental limits on such sequential gauge bosons could therefore be recast as limits on KK gauge bosons. Such an interpretation of the ATLAS 7 TeV dilepton analysis [154] yielded the bound $1/R > 4.16$ TeV, while a CMS 8 TeV search with a lepton and missing transverse energy in the final state [155] give $1/R > 3.4$ TeV. Indirect bounds from LEP2 require however $1/R \gtrsim 6$ TeV [86, 156], a bound that can considerably improve in the future by high-energy measurements of the dilepton invariant mass spectrum from Drell-Yan processes at the LHC [157]. More recent LHC limits on leptonically decaying gauge bosons [88, 158–162] are not interpreted as bounds on $1/R$ by the collaborations, but the published results allow for independent derivation of such bounds.

An alternative scenario, known as Universal Extra Dimensions (UED) [163] (for a review see, for example, [164]), assumes that all SM fields propagate universally in a flat orbifold $S^1/Z_2$ with an extra $Z_2$ parity, called KK-parity, that interchanges the two boundaries. In this case, the lowest KK state is stable and is a Dark Matter candidate. At colliders, the KK particles would have to be created in pairs, and would then cascade decay to the lightest KK particle, which would be stable and escape detection. The UED mass-spectrum depends not only on the extra-dimensional radius $R$, but also on the cut-off of the 5D theory $\Lambda$, since quantum corrections sensitive to $\Lambda R$ induce mass-splittings between the KK states. Experimental signatures, such as jets or leptons and $p_T^{\text{miss}}$, would be similar to those of typical R-parity conserving SUSY searches. An interpretation of the recent LHC experimental SUSY searches for UED models has been presented in Refs. [165, 166]. A lower bound $1/R > 1.4–1.5$ TeV was derived for $\Lambda R$ in the range $5–35$ [165]. A recent analysis is given in Ref. [167] where it is shown that the minimal UED model is ruled out when LHC data is combined with Dark Matter relic density data. Extensions to the minimal UED model where boundary terms are included can however be compatible with experiments [167].

Finally, realistic models of electroweak symmetry breaking can also be constructed with flat extra spatial dimensions, similarly to those in the warped case, requiring, however, the presence of sizeable boundary kinetic terms [168]. There is also the possibility of breaking supersymmetry by boundary conditions [169]. Models of this type could explain naturally the presence of a Higgs boson lighter than $M_D \sim$ TeV (see, for example, [170–172]).

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86. W'-Boson Searches

Revised September 10, 2023 by B.A. Dobrescu (FNAL) and S. Willocq (U. Massachusetts).

The W' boson is a massive hypothetical particle of spin 1 and electric charge $\pm 1$, which is a color singlet and is predicted in various extensions of the Standard Model (SM).

## 86.1 W' couplings to quarks and leptons

The Lagrangian terms describing the couplings of a $W'^{+}$ boson to fermions are given by

$$ \frac{W'_{\mu}^{+}}{\sqrt{2}} \left[ \bar{u}_i (C_{q_{ij}}^R P_R + C_{q_{ij}}^L P_L) \gamma^{\mu} d_j + \bar{v}_i (C_{\ell_{ij}}^R P_R + C_{\ell_{ij}}^L P_L) \gamma^{\mu} e_j \right] . \quad (86.1) $$

Here, $u, d, \nu$, and $e$ are the SM fermions in the mass eigenstate basis, $i, j = 1, 2, 3$ label the fermion generation, and $P_{R,L} = (1 \pm \gamma_5)/2$. The coefficients $C_{q_{ij}}^L, C_{q_{ij}}^R, C_{\ell_{ij}}^L$, and $C_{\ell_{ij}}^R$ are complex dimensionless parameters. If $C_{\ell_{ij}}^R \neq 0$, then the $i$th generation includes a right-handed neutrino. Using this notation, the SM W couplings are $C_q^L = gV_{CKM}, C_\ell^L = g \approx 0.63$ and $C_q^R = C_\ell^R = 0$.

Unitarity considerations imply that the W' boson is associated with a spontaneously-broken gauge symmetry. This is true even when it is a composite particle (e.g. $\rho^{\pm}$-like bound states [1]) if its mass is much smaller than the compositeness scale, or a Kaluza-Klein mode in theories where the W boson propagates in extra dimensions [2]. The simplest extension of the electroweak gauge group that includes a W' boson is $SU(2)_1 \times SU(2)_2 \times U(1)$, but larger groups are encountered in some theories. A generic property of these gauge theories is that they also include a Z' boson [3]; the W'-to-Z' mass ratio is often a free parameter.

A tree-level mass mixing may be induced between the electrically-charged gauge bosons. Upon diagonalization of their mass matrix, the W-to-Z mass ratio and the couplings of the observed W boson are shifted from the SM values. Their measurements imply that the mixing angle, $\theta_+$, between the gauge eigenstates must be smaller than about $10^{-2}$ [4]. In certain theories the mixing is negligible (e.g., due to a new parity [5]), even when the W' mass is near the electroweak scale. Note that $SU(2)$ gauge invariance suppresses the kinetic mixing between the W and W' bosons (in contrast to the case of a Z' boson [3]).

The W' coupling to WZ is fixed by Lorentz and gauge invariances, and to leading order in $\theta_+$ is given by [6]

$$ \frac{g \theta_+ i}{\cos \theta_W} \left[ W'_{\mu}^{+} (W_{\nu}^{-} Z^{\nu \mu} + Z_{\nu} W^{- \mu \nu}) + Z^{\nu} W^{- \mu} W_{\nu \mu}^{+} \right] + \text{H.c.}, \quad (86.2) $$

where $W^{\mu\nu} \equiv \partial^\mu W^\nu - \partial^\nu W^\mu$, etc. The $\theta_W$ dependence shown here corrects the one given in Ref. [7], which has been referred to as the Extended Gauge Model by the experimental collaborations. The W' coupling to $W h^0$, where $h^0$ is the SM Higgs boson, is

$$ - \xi_h g_{W'} M_W W'_{\mu}^{+} W^{\mu-} h^0 + \text{H.c.}, \quad (86.3) $$

where $g_{W'}$ is the gauge coupling of the W' boson, and the coefficient $\xi_h$ satisfies $\xi_h \le 1$ in simple Higgs sectors [6].

In models based on the “left-right symmetric” gauge group [8], $SU(2)_L \times SU(2)_R \times U(1)_{B-L}$, the SM fermions that couple to the W boson transform as doublets under $SU(2)_L$ while the other fermions transform as doublets under $SU(2)_R$. Consequently, the W' boson couples primarily to right-handed fermions; its coupling to left-handed fermions arises due to the $\theta_+$ mixing, so that $C_q^L$ is proportional to the CKM matrix and its elements are much smaller than the diagonal elements of $C_q^R$. Generically, $C_q^R$ does not need to be proportional to $V_{CKM}$.

There are many other models based on the $SU(2)_1 \times SU(2)_2 \times U(1)$ gauge symmetry. In the “alternate left-right” model [9], all the couplings shown in Eq. (86.1) vanish, but there are some new fermions such that the W' boson couples to pairs involving a SM fermion and a new fermion. In the “ununified SM” [10], the left-handed quarks are doublets under one $SU(2)$, and the left-handed leptons are doublets under a different $SU(2)$, leading to a mostly leptophobic W' boson: $C_{\ell_{ij}}^L \ll C_{q_{ij}}^L$ and $C_{\ell_{ij}}^R = C_{q_{ij}}^R = 0$. Fermions of different generations may also transform

as doublets under different $SU(2)$ gauge groups [11]. In particular, the couplings to third generation quarks may be enhanced [12].

It is also possible that the W' couplings to SM fermions are highly suppressed. For example, if the quarks and leptons are singlets under one $SU(2)$ [13], then the couplings are proportional to the tiny mixing angle $\theta_+$. Similar suppressions may arise if some vectorlike fermions mix with the SM fermions [14].

Gauge groups that embed the electroweak symmetry, such as $SU(3)_W \times U(1)$ or $SU(4)_W \times U(1)$, also include one or more W' bosons [15].

## 86.2 Collider searches

At hadron colliders, W' bosons can be detected through resonant pair production of fermions ($f$ and $f'$) or electroweak bosons with a net electric charge equal to $\pm 1$. When $W'$ has a width much smaller than its mass ($\Gamma_{W'}/M_{W'} \lesssim 7\%$), the contribution of the s-channel W' exchange to the total rate for $pp \to f f' X$, where $X$ is any final state, may be approximated by the branching fraction $B(W' \to f f')$ times the production cross section [16], which may be written as

$$ \sigma(pp \to W'X) \approx \frac{\pi}{6s} \sum_{i,j} \left[ (C_{q_{ij}}^L)^2 + (C_{q_{ij}}^R)^2 \right] w_{ij} \left( M_{W'}^2/s, M_{W'} \right). \quad (86.4) $$

The functions $w_{ij}$ include the information about proton structure, and are given to leading order in $\alpha_s$ by

$$ w_{ij}(z, \mu) = \int_z^1 \frac{dx}{x} \left[ u_i(x, \mu) \bar{d}_j \left( \frac{z}{x}, \mu \right) + \bar{u}_i(x, \mu) d_j \left( \frac{z}{x}, \mu \right) \right], \quad (86.5) $$

where $u_i(x, \mu)$ and $d_i(x, \mu)$ are the parton distributions inside the proton at the factorization scale $\mu$ and parton momentum fraction $x$ for the up- and down-type quarks of the $i$th generation, respectively. QCD corrections to W' production are sizable (they also include quark-gluon initial states), but preserve the above factorization of couplings at next-to-leading order [17].

The most commonly studied W' signal consists of a high-momentum electron or muon and large missing transverse momentum. The signal transverse mass distribution forms a Jacobian peak with its endpoint at $M_{W'}$ (see Fig. 1 (top) of Ref. [18]). Given that the branching fractions for $W' \to e\nu$ and $W' \to \mu\nu$ could be very different, the results in these channels should be presented separately. Searches in these channels often implicitly assume that the left-handed couplings vanish (no interference between W and W'), and that the right-handed neutrino is light compared to the W' boson and escapes the detector. An example of parameter values that satisfy these assumptions is $C_q^R = gV_{CKM}, C_\ell^R = g, C_q^L = C_\ell^L = 0$, which define a model that preserves lepton universality and predicts the same total cross section as the Sequential SM used in many W' searches. However, if a W' boson were discovered and the final state fermions have left-handed helicity, then the effects of W-W' interference could be observed [22], providing information about the W' couplings. The effects of the W' width on interference are discussed in Ref. [23].

In the $e\nu$ channel, the ATLAS and CMS collaborations set limits on the W' production cross section times branching fraction $\sigma \times B$ (and thus indirectly on the W' couplings). These limits are set for $M_{W'}$ in the $0.15-7$ TeV range and are based on approximately $140 \text{ fb}^{-1}$ at $\sqrt{s} = 13$ TeV [18, 19], with the most stringent limits reproduced in Fig. 86.1. ATLAS sets the strongest mass limit $M_{W'} > 6.0$ TeV in the Sequential SM (all limits in this mini-review are at the 95% CL). The coupling limits are much weaker for $M_{W'} < 150$ GeV, a range last explored with the Tevatron at $\sqrt{s} = 1.8$ TeV [24].

In the $\mu\nu$ channel, ATLAS and CMS set rate limits for $M_{W'}$ in the $0.15-7$ TeV range [18, 19], with the strongest mass lower limit of $5.6$ TeV in the Sequential SM set by CMS [19] using $138 \text{ fb}^{-1}$ of $\sqrt{s} = 13$ TeV data, as shown in Fig. 86.1. When combined with the $e\nu$ channel assuming lepton universality, the upper limit on the $\sqrt{s} = 13$ TeV cross section times branching fraction to $\ell\nu$ varies between 0.05 and 2.1 fb for $M_{W'}$ values between 1 and 6 TeV [18]. Only weak limits on $W' \to \mu\nu$ exist for $M_{W'} < 150$ GeV [25]. Note that masses of the order of the electroweak
<figure><img src="image_358.png" /><figcaption>**Figure 86.1:** Upper limit on $\sigma(pp \to W'X) \times B(W' \to \ell\nu)$ in the $\ell\nu$ channel from ATLAS [18] (left) and the $\mu\nu$ channel from CMS [19] (right). The red (black) line shows the theoretical prediction in the Sequential SM in the $\ell\nu$ ($\mu\nu$) channel.</figcaption></figure>

<figure><img src="image_359.png" /><figcaption>**Figure 86.2:** Upper limits on $W'$ production cross section times branching fraction into a $W$ and a SM Higgs boson decaying into heavy-flavor quarks, from ATLAS [20] (left) and CMS [21] (right).</figcaption></figure>

scale are interesting from a theoretical point of view, while lepton universality does not necessarily apply to a $W'$ boson.

Searches for $W' \to \tau\nu$ have been performed at 13 TeV by CMS with $138 \text{ fb}^{-1}$ [26], and by ATLAS with $139 \text{ fb}^{-1}$ [27]. Limits are set on $\sigma \times B$ for $M_{W'}$ between 0.5 and 6 TeV. A mass lower limit of 5.0 TeV is set in the Sequential SM, and the upper limit on the cross section times branching fraction to $\tau\nu$ at 13 TeV varies between 0.4 and 9 fb for $M_{W'}$ values between 1 and 5 TeV [27].

The $W'$ decay into a charged lepton and a right-handed neutrino, $\nu_R$, may also be followed by the $\nu_R$ decay through a virtual $W'$ boson into a charged lepton and two quark jets. The CMS [28] and ATLAS [29] searches in the $eejj$ and $\mu\mu jj$ channels have set limits at $\sqrt{s} = 13$ TeV on the cross section times branching fractions as a function of the $\nu_R$ mass and of $M_{W'}$. No requirement is placed on the charge of the lepton pair. A related $W'$ search in the $\tau\tau jj$ channel with hadronic $\tau$ decays was also performed by CMS [30].

The $t\bar{b}$ channel is particularly important because a $W'$ boson that couples only to right-handed fermions cannot decay to leptons when the right-handed neutrinos are heavier than $M_{W'}$. Additional motivations are provided by a $W'$ boson with enhanced couplings to the third generation [12], and by a leptophobic $W'$ boson. The usual signature for $W' \to t\bar{b}$ consists of a leptonically-decaying $W$ boson and two b-jets. Recent studies have also incorporated the fully hadronic decay channel for $M_{W'} \gg m_t$ with the use of jet substructure techniques to tag highly boosted top-jets.

For a detailed discussion of this channel, see Ref. [31].

Searches for dijet resonances may be used to set limits on $W' \to q\bar{q}'$. ATLAS [32] and CMS [33] provide similar coverage in the $\sim 1.5-8$ TeV mass range with $139$ and $137 \text{ fb}^{-1}$ of data, respectively, collected at $\sqrt{s} = 13$ TeV. Interpretation in terms of $W'$ decays with $139 \text{ fb}^{-1}$ of 13 TeV data yields a $W'$ mass lower limit of 4.0 TeV in the Sequential SM [32]. For masses in the range $\sim 0.5-1.5$ TeV, analyses based on jets reconstructed online provide the best sensitivity because they circumvent trigger bandwidth limitations [34, 35]. For $W'$ masses below $\sim 0.5$ TeV, the best limits are set in novel analyses exploiting boosted technologies and initial state radiation [36–39]. Cross-section limits for $W'$ masses below $\sim 1.5$ TeV can be derived from the dijet limits on $Z'$ bosons summarized in Ref. [3].

In some theories [5] the $W'$ couplings to SM fermions are suppressed by discrete symmetries. $W'$ production then occurs in pairs, through a photon or $Z$ boson. The decay modes are model-dependent and often involve other new particles. The ensuing collider signals arise from cascade decays and often include missing transverse momentum or boosted multi-jet topologies. Even when single production of $W'$ is large enough, it is possible that its main decay mode is into a new particle and a SM one. An example is a search performed by CMS [40] for $W'$ decays into a vector-like quark and a top or a bottom quark. The final state studied in this analysis involves a boosted SM Higgs or $Z$ boson, as well as a $t\bar{t}$ or $b\bar{b}$ pair, with all heavy particles decaying into
jets. Another example is a search performed by ATLAS [41] for $W'$ decays into a $W$ boson and a $Z'$ boson, the latter decaying into a jet pair.

Searches for $WZ$ resonances at the LHC have focused on the process $pp \to W' \to WZ$ with the production mainly from $u\bar{d} \to W'$, assuming SM-like couplings to quarks. ATLAS and CMS have set upper limits on the $W'WZ$ coupling for $M_{W'}$ in the $0.2-5.0$ TeV range with a combination of fully leptonic, semi-leptonic and fully hadronic channels with $\sim 36-139$ fb$^{-1}$ at 13 TeV [42–44] (see also Ref. [31]). Recent constraints with 138–139 fb$^{-1}$ at 13 TeV were also set in individual channels by ATLAS [45–47] and CMS [21, 48–50]. The strongest lower limit on the $W'$ mass is set by ATLAS [46] in the semi-leptonic channel with a lower limit on $M_{W'}$ of 3.9 TeV [46] in the context of the Heavy Vector Triplet (HVT) weakly-coupled scenario A [51]. A similar result is obtained by CMS in the fully-hadronic channel [21]. A fermiophobic $W'$ boson that couples to $WZ$ may be produced at hadron colliders in association with a $Z$ boson, or via $WZ$ fusion. This would give rise to $(WZ)Z$ and $(WZ)jj$ final states [52]. Results of the search for the latter are reported in Refs. [21, 42, 46, 48, 50].

$W'$ bosons have also been searched for in final states with a $W$ boson and a SM Higgs boson in the channels $W \to \ell\nu$ or $W \to q\bar{q}'$ and $h^0 \to b\bar{b}$ by ATLAS [20, 53] and CMS [21, 50] with 138–139 fb$^{-1}$ at $\sqrt{s} = 13$ TeV. CMS also searched for $W'$ bosons in the $q\bar{q}'\tau\tau$ final state [54]. Cross-section limits are set for $W'$ masses between 0.4 and 6.0 TeV. The most stringent upper limit on the cross section is set by the ATLAS analysis with $W \to \ell\nu$ at low $M_{W'}$ and the CMS analysis with $W \to q\bar{q}'$ at high $M_{W'}$, as shown in Fig. 86.2.

At lepton colliders, $W'$ bosons may be produced in pairs via their photon coupling, which is model independent. At LEP-II, although dedicated searches for $W'$ bosons have not been performed, the large cross section for $e^+e^- \to \gamma^* \to W'^+W'^-$ and small backgrounds suggest that any $W'$ is ruled out to the kinematic limit, $M_{W'} < \sqrt{s}/2 \approx 105$ GeV, for most decay modes. Sensitivity to $M_{W'}$ above $\sqrt{s}$ could be achieved [55] using the $e^+e^- \to \gamma\nu\bar{\nu}$ process via a t-channel $W'$ exchange, if the $W'$ coupling to $e\nu$ is large enough.

## 86.3 Low-energy constraints

The properties of $W'$ bosons are also constrained by measurements of processes at energies much below $M_{W'}$. The bounds on $W-W'$ mixing [4] are mostly due to the change in $W$ properties compared to the SM. Limits on deviations in the $ZWW$ couplings provide a leading constraint for fermiophobic $W'$ bosons [14].

Constraints arising from low-energy effects of $W'$ exchange are strongly model-dependent. If the $W'$ couplings to quarks are not suppressed, then box diagrams involving a $W$ and a $W'$ boson contribute to neutral meson-mixing. In the case of $W'$ couplings to right-handed quarks as in the left-right symmetric model, the limit from $K_L - K_S$ mixing is severe: $M_{W'} > 2.9$ TeV for $C_R^q = gV_{CKM}$ [56]. However, if no correlation between the $W'$ and $W$ couplings is assumed, then the limit on $M_{W'}$ may be significantly relaxed [57].

$W'$ exchange also contributes at tree level to various low-energy processes. In particular, it would impact the measurement of the Fermi constant $G_F$ in muon decay, which in turn would change the predictions of many other electroweak processes. A recent test of parity violation in polarized muon decay [58] has set limits of about 600 GeV on $M_{W'}$, assuming $W'$ couplings to right-handed leptons as in left-right symmetric models and a light $\nu_R$. There are also $W'$ contributions to the neutron electric dipole moment, $\beta$ decays, and other processes [4].

If right-handed neutrinos have Majorana masses, then there are tree-level contributions to neutrinoless double-beta decay, and a limit on $M_{W'}$ versus the $\nu_R$ mass may be derived [59]. For $\nu_R$ masses below a few GeV, the $W'$ boson contributes to leptonic and semileptonic $B$ meson decays, so that limits may be placed on various combinations of $W'$ parameters [57]. For $\nu_R$ masses below $\sim 30$ MeV, the most stringent constraints on $M_{W'}$ are due to the limits on $\nu_R$ emission from supernovae [60].

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87. Z'-Boson Searches

Revised September 2023 by B.A. Dobrescu (FNAL) and S. Willocq (U. Massachusetts).

The Z' boson is a massive, electrically-neutral and color-singlet hypothetical particle of spin 1. This particle is predicted in many extensions of the Standard Model (SM) and has been the object of extensive phenomenological studies [1].

87.1 Z' boson couplings

The couplings of a Z' boson to the first-generation fermions are given by

$$
Z'_{\mu} \left( g_u^L \bar{u}_L \gamma^{\mu} u_L + g_d^L \bar{d}_L \gamma^{\mu} d_L + g_u^R \bar{u}_R \gamma^{\mu} u_R + g_d^R \bar{d}_R \gamma^{\mu} d_R \right. \\
\left. + g_{\nu}^L \bar{v}_L \gamma^{\mu} v_L + g_e^L \bar{e}_L \gamma^{\mu} e_L + g_e^R \bar{e}_R \gamma^{\mu} e_R \right), \qquad (87.1)
$$

where u, d, ν, e are the quark and lepton fields in the mass eigen-
state basis, and the coefficients g^L_u, g^L_d, g^R_u, g^R_d, g^L_ν, g^R_ν, g^L_e, g^R_e are
real dimensionless parameters. If the Z' couplings to quarks and
leptons are generation-independent, then these seven parameters
describe the couplings of the Z' boson to all SM fermions. More
generally, however, the Z' couplings to fermions are generation-
dependent, in which case Eq. (87.1) may be written with genera-
tion indices i, j = 1, 2, 3 labeling the quark and lepton fields, and
with the seven coefficients promoted to 3 × 3 Hermitian matrices
(e.g., g^L_{ij}^e \bar{e}_L \gamma^\mu e^c_L, where e^2_L is the left-handed muon, etc.).

The parameters describing the Z' boson interactions with quarks and leptons are subject to some theoretical constraints. Quantum field theories that include a heavy spin-1 particle are well behaved at high energies only if that particle is a gauge boson associated with a spontaneously broken gauge symmetry. Quantum effects preserve the gauge symmetry only if the couplings of the gauge boson to fermions satisfy the anomaly equations [2]. Furthermore, the fermion charges under the new gauge symmetry are constrained by the requirement that the quarks and leptons get masses from gauge-invariant interactions with the Higgs fields.

The relation between the couplings displayed in Eq. (87.1) and the gauge charges $z_{fi}^L$ and $z_{fi}^R$ of the fermions $f = u, d, \nu, e$ involves the unitary $3 \times 3$ matrices $V_f^L$ and $V_f^R$ that transform the gauge eigenstate fermions $f_L^i$ and $f_R^i$, respectively, into the mass eigenstates. The $Z'$ couplings also depend on the mixings of the new gauge boson in the gauge eigenstate basis ($\tilde{Z}'_\mu$). The main mixings are a kinetic mixing ($-\chi/2)B^{\mu\nu}\tilde{Z}'_{\mu\nu}$ with the hypercharge gauge boson $B^\mu$ ($\chi$ is a dimensionless parameter), and a mass mixing $\delta M^2 \tilde{Z}' \tilde{Z}'_\mu$ with the linear combination ($\tilde{Z}'_\mu$) of neutral bosons that couples as the SM Z boson [3]. Since both the kinetic and mass mixings shift the mass and couplings of the Z boson, electroweak measurements impose upper limits on $\chi$ and $\delta M^2/(M_{Z'}^2 - M_Z^2)$ of the order of $10^{-3}$ [4]. Keeping only linear terms in these two small quantities, the couplings of the mass-eigenstate $Z'$ boson are given by

$$
g_{f \, ij}^{L} = g_{z} V_{f \, i'j}^{L} z_{f \, i'}^{L} (V_{f}^{L})_{i'j}^{\dagger} \\
\qquad + \frac{e}{c_{W}} \left( \frac{s_{W} \chi M_{Z'}^{2} + \delta M^{2}}{2 s_{W} (M_{Z'}^{2} - M_{Z}^{2})} \sigma_{f}^{3} - \epsilon Q_{f} \right), \qquad (87.2)
$$

$$
g_f^R_{ij} = g_z V_{f_{i'}}^R z_{f'i}^R (V_f^R)_{i'j}^\dagger - \frac{e}{c_W} \epsilon Q_f, \qquad (87.3)
$$

where g_z is the new gauge coupling, Q_f is the electric charge of f, e is the electromagnetic gauge coupling, s_W and c_W are the sine and cosine of the weak mixing angle, σ_f^3 = +1 for f = u, ν and σ_f^3 = -1 for f = d, e, and

$$
\epsilon = \frac{\chi (M_{Z'}^2 - c_W^2 M_Z^2) + s_W \delta M^2}{M_{Z'}^2 - M_Z^2}. \qquad (87.4)
$$

The interaction of the Z' boson with a pair of W bosons has

the form

$$
\left[ i (W_{\mu}^{-} Z'_{\nu} - W_{\nu}^{-} Z'_{\mu}) \partial^{\mu} W^{+\nu} + \text{H.c.} \right] \\
+ i (W_{\mu}^{+} W_{\nu}^{-} - W_{\nu}^{+} W_{\mu}^{-}) \partial^{\mu} Z'^{\nu} \qquad (87.5)
$$

with a coefficient of order $M_W^2/M_Z^2$, [5]. The $Z'$ also couples to one SM Higgs boson and one Z boson, $Z'_\mu Z^\mu h^0$, with a coefficient of order $M_Z$.

87.2 Z' models

A simple origin of a Z' boson is a new U(1)' gauge symme-
try. In that case, the matricial equalities z_L^u = z_L^d and z_L^ν = z_L^e
are required by the SM SU(2)_W gauge symmetry. Given that
the U(1)' interaction is not asymptotically free, the theory may
be well-behaved at high energies (e.g., by embedding U(1)' in a
non-Abelian gauge group) only if the charges are commensurate
numbers, i.e. any ratio of charges is a rational number. Satis-
fying the anomaly equations [6] with rational numbers is highly
nontrivial, and typically new fermions charged under U(1)' are
necessary.

If the couplings are generation-independent ($V_f^{L,R}$ are then unit matrices in Eq. (87.2)) and the mixings of $\tilde{Z}'$ are negligible, then there are five commensurate couplings: $g_u^R, g_d^R, g_e^R, g_q^L$ ($q = u$ or $d$), $g_l^L$ ($l = \nu$ or $e$). Four sets of charges are displayed in Table 87.1, each of them spanned by a free parameter $x$ [6]. The first set, labeled $B - xL$, has charges proportional to the baryon number minus $x$ times the lepton number. These charges allow all SM Yukawa couplings to the Higgs doublet which is neutral under $U(1)_{B-xL}$, so that there is no tree-level $\tilde{Z} - \tilde{Z}'$ mixing. For $x = 1$ one recovers the $U(1)_{B-L}$ group, which is non-anomalous in the presence of one “right-handed neutrino” (a chiral fermion that is a singlet under the SM gauge group) per generation. For $x \neq 1$, it is necessary to include some fermions that are vectorlike (i.e. their mass terms are gauge invariant) with respect to the electroweak gauge group and chiral with respect to $U(1)_{B-xL}$. In the particular cases $x = 0$ or $x \gg 1$, the $Z'$ is leptophobic or quark-phobic, respectively.

The second set, U(1)$_{10+x5}$, has charges that commute with the representations of the SU(5) grand unified group. Here x is related to the mixing angle between the two U(1) bosons encountered in the E$_6 \to$ SU(5)×U(1)×U(1) symmetry breaking patterns of grand unified theories [1,7]. With these charges, two Higgs doublets are typically required to generate masses for both up- and down- type fermions. This set leads to $\tilde{Z} - \tilde{Z}'$ mass mixing at tree level, such that for a $Z'$ mass close to the electroweak scale, the measurements at the Z-pole require some fine tuning between the charges and VEVs of the two Higgs doublets. Vectorlike quarks charged under the electroweak gauge group are required (except for $x = -3$) to make this set anomaly free. The particular cases $x = -3, 1, -1/2$ are usually labeled U(1)$_\chi$, U(1)$_\psi$, and U(1)$_\eta$, respectively. Under the third set, U(1)$_{d-xu}$, the weak-doublet quarks are neutral, and the ratio of $u_R$ and $d_R$ charges is $-x$. For $x = 1$, this is the “right-handed” group U(1)$_R$. For $x = 0$, the charges are those of the E$_6$-inspired U(1)$_I$ group, which requires new quarks and leptons. Other generation-independent sets of U(1)' charges are given in Ref. [8].

In the absence of new fermions charged under the SM group, the most general generation-independent charge assignment is U(1)$_{q+xu}$, which is a linear combination of hypercharge and B-L. Many other anomaly-free solutions exist if generation-dependent charges are allowed. An example is B - xL$_e$ - yL$_\mu$ + (y - 3)L$_\tau$, with x, y free parameters. This allows all fermion masses to be generated by Yukawa couplings to a single Higgs doublet, without inducing tree-level flavor-changing neutral current (FCNC) processes. There are also lepton-flavor dependent charges that allow neutrino masses to arise only from operators of high dimensionality [9].

If the SU(2)_W-doublet quarks have generation-dependent U(1)'
charges, then the mass eigenstate quarks have flavor off-diagonal
couplings to the Z' boson (see Eq. (87.1), and note that V_u^L
(V_d^L)^\dagger
<table><caption>**Table 87.1:** Examples of generation-independent $U(1)'$ charges for quarks and leptons. The parameter $x$ is an arbitrary rational number. Gauge anomaly cancellation requires certain new fermions [6].</caption><thead><tr><th>fermion</th><th>$U(1)_{B-xL}$</th><th>$U(1)_{10+x5}$</th><th>$U(1)_{d-xu}$</th><th>$U(1)_{q+ xu}$</th></tr></thead><tbody><tr><td>$(u_L, d_L)$</td><td>1/3</td><td>1/3</td><td>0</td><td>1/3</td></tr><tr><td>$u_R$</td><td>1/3</td><td>-1/3</td><td>-x/3</td><td>x/3</td></tr><tr><td>$d_R$</td><td>1/3</td><td>-x/3</td><td>1/3</td><td>(2-x)/3</td></tr><tr><td>$(\nu_L, e_L)$</td><td>-x</td><td>x/3</td><td>(-1+x)/3</td><td>-1</td></tr><tr><td>$e_R$</td><td>-x</td><td>-1/3</td><td>x/3</td><td>-(2+x)/3</td></tr></tbody></table>

is the CKM matrix). These are severely constrained by measurements of FCNC processes, which in this case are mediated at tree-level by $Z'$ boson exchange [10]. The constraints are relaxed if the first and second generation charges are the same, although they are increasingly tightened by the measurements of B meson properties [11]. If only the $SU(2)_W$-singlet quarks have generation-dependent $U(1)'$ charges, there is more freedom in adjusting the flavor off-diagonal couplings because the $V_{u,d}^R$ matrices are not observable in the SM.

The anomaly equations for $U(1)'$ could be circumvented only if there is an axion with certain dimension-5 couplings to the gauge bosons. However, such a scenario violates unitarity unless the quantum field theory description breaks down at a scale near $M_{Z'}$ [12]. It is possible, though, that the SM fermions are not charged under the $U(1)'$, but have mass mixing with some vectorlike fermions which are charged under $U(1)'$, implying that the physical fermions have couplings to the $Z'$ [13]. For example, if the only SM quarks that have large mixings with some vectorlike quarks are the b quarks, then the only currently known particles that couple to the $Z'$ would be the physical b quarks.

$Z'$ bosons may also arise from larger gauge groups. These may extend the electroweak group, as in $SU(2) \times SU(2) \times U(1)$, or may embed the electroweak group, as in $SU(3)_W \times U(1)$ [14]. If the larger group is spontaneously broken down to $SU(2)_W \times U(1)_Y \times U(1)'$ at a scale $v_\star \gg M_{Z'}/g_Z$, then the above discussion applies up to corrections of order $M_{Z'}/(g_Z v_\star)^2$. For $v_\star \sim M_{Z'}/g_Z$, additional gauge bosons have masses comparable to $M_{Z'}$, including at least a $W'$ boson [14]. If the larger gauge group breaks together with the electroweak symmetry directly to the electromagnetic $U(1)_{em}$, then the left-handed fermion charges are no longer correlated ($z_u^L \neq z_d^L$, $z_\nu^L \neq z_e^L$) and a $Z'W^+W^-$ coupling is induced.

If the electroweak gauge bosons propagate in extra dimensions, then their Kaluza-Klein (KK) excitations include a series of $Z'$ boson pairs. Each of these pairs can be associated with a different $SU(2) \times U(1)$ gauge group in four dimensions. The properties of the KK particles depend strongly on the extra dimensional theory [15]. For example, in universal extra dimensions there is a parity that forces all couplings of Eq. (87.1) to vanish in the case of the lightest KK bosons, while allowing couplings to pairs of fermions involving a SM and a heavy vectorlike fermion. There are also 4-dimensional gauge theories (e.g. little Higgs with $T$ parity) with $Z'$ bosons exhibiting similar properties. By contrast, in a warped extra dimension, the couplings of Eq. (87.1) may be sizable even when SM fields propagate along the extra dimension.

$Z'$ bosons may also be composite particles. For example, in confining gauge theories [16], the $\rho$-like bound state is a spin-1 boson that may be interpreted as arising from a spontaneously broken gauge symmetry [17].

## 87.3 Non-resonant $Z'$ signatures at colliders

In the presence of the couplings shown in Eq. (87.1), the $Z'$ boson may be produced in the s-channel at colliders, and would decay to pairs of fermions. The decay width into a pair of electrons is given by

$$ \Gamma(Z' \to e^+e^-) \approx [(g_e^L)^2 + (g_e^R)^2] \frac{M_{Z'}}{24\pi}, \quad (87.6) $$

where small corrections from electroweak loops are not included. The decay width into $q\bar{q}$ is similar, except for an additional color factor of 3, QCD radiative corrections, and fermion mass corrections. Thus, one may compute the $Z'$ branching fractions in terms of the couplings of Eq. (87.1). However, other decay channels, such as WW or a pair of new particles, could have large widths and need to be added to the total decay width.

As mentioned above, there are theories in which the $Z'$ couplings are controlled by a discrete symmetry that forbids decays into a pair of SM particles. Typically, such theories involve several new particles, which may be produced only in pairs and undergo cascade decays through $Z'$ bosons, leading to signals involving missing transverse momentum. Given that the cascade decays depend on the properties of new particles other than the $Z'$ boson (see, e.g. Ref. [18]), this case is not discussed further here.

The $Z'$ contribution to the cross sections for $e^+e^- \to \bar{f}f$ proceeds through an s-channel $Z'$ exchange (when $f = e$, there are also t- and u-channel exchanges). For $M_{Z'} < \sqrt{s}$, the $Z'$ appears as an $\bar{f}f$ resonance in the radiative return process where photon emission tunes the effective center-of-mass energy to $M_{Z'}$. The agreement between the LEP-II measurements and the SM predictions implies that either the $Z'$ couplings are smaller than or of order $10^{-2}$, or else $M_{Z'}$ is above 209 GeV, the maximum energy of LEP-II. In the latter case, the $Z'$ exchange may be approximated up to corrections of order $s/M_{Z'}^2$, by the contact interactions

$$ \frac{g_z^2}{M_{Z'}^2 - s} \left[ \bar{e}\gamma\mu \left( z_e^L P_L + z_e^R P_R \right) e \right] \left[ \bar{f}\gamma^\mu \left( z_f^L P_L + z_f^R P_R \right) f \right], \quad (87.7) $$

where $P_{L,R}$ are chirality projection operators, and the relation between $Z'$ couplings and charges (see Eq. (87.2) in the limit where the mass and kinetic mixings are neglected) is used, assuming generation-independent charges. The four LEP collaborations have set limits on the coefficients of such operators for all possible chiral structures and for various combinations of fermions [19]. Thus, one may derive bounds on $(M_{Z'}/g_Z)|z_e^L z_f^L|^{-1/2}$ and the analogous combinations of LR, RL and RR charges, which are typically on the order of a few TeV. LEP-II limits were derived [6] on the four sets of charges shown in Table 87.1.

Somewhat stronger bounds can be set on $M_{Z'}/g_Z$ for specific sets of $Z'$ couplings if the effects of several operators (87.7) are combined. Dedicated analyses by the LEP collaborations have set limits on $Z'$ bosons for particular values of the gauge coupling (see section 3.5 of Ref. [19]). For example, $M_{Z'_{SSM}} > 1.76$ TeV for a “sequential” $Z'$ of same couplings as the SM $Z$ boson, while $M_{Z'} > 0.785$ TeV for the $Z'$ associated with $U(1)_\chi$ assuming a unification condition for the gauge coupling.

## 87.4 Searches at hadron colliders

$Z'$ bosons with couplings to quarks (see Eq. (87.1)) may be produced at hadron colliders in the s-channel and would show up as resonances in the invariant mass distribution of the decay products. The cross section for producing a $Z'$ boson at the LHC, which then decays to some $\bar{f}f$ final state, takes the form [22]

$$ \sigma (pp \to Z'X \to f\bar{f}X) \approx \frac{\pi}{6s} \sum_q c_q^f w_q (s, M_{Z'}^2) \quad (87.8) $$

for flavor-diagonal couplings to quarks. Here, we have neglected the interference with the SM contribution to $\bar{f}f$ production, which is a good approximation for a narrow $Z'$ resonance (deviations from the narrow width approximation are discussed in Ref. [23]). The coefficients

$$ c_q^f = [(g_q^L)^2 + (g_q^R)^2] B(Z' \to f\bar{f}) \quad (87.9) $$

contain all the dependence on the $Z'$ couplings, while the functions $w_q$ include all the information about parton distributions and QCD corrections [6, 8]. This factorization holds exactly to NNLO and the deviations from it induced at NNLO are very small. Note that the $w_u$ and $w_d$ functions are substantially larger than the $w_q$ functions for the other quarks. Eq. (87.8) also applies to
<figure><img src="image_360.png" /></figure>

<figure><img src="image_361.png" /><figcaption>**Figure 87.1:** Upper limits on the cross section for $Z'$ production times the branching fraction for $Z' \to e^+e^-$ (left panel, set by ATLAS [20]) or $Z' \to \mu^+\mu^-$ (right panel, set by CMS [21]) as a function of $M_{Z'}$. The lines labeled by $Z'_\psi$ and $Z'_\chi$ are theoretical predictions for the $U(1)_{10+x\bar{5}}$ models in Table 87.1 with $x = -3$ and $x = +1$, respectively, for $g_z$ fixed by an $E_6$ unification condition. The $Z'_{SSM}$ line corresponds to $Z'$ couplings equal to those of the Z boson.</figcaption></figure>

the Tevatron, except for changing the $pp$ initial state to $p\bar{p}$, which implies that the $w_q(s, M_{Z'}^2)$ functions are replaced by some other functions $\bar{w}_q((1.96 \text{ TeV})^2, M_{Z'}^2)$.

It is common to present results of $Z'$ searches as limits on the cross section versus $M_{Z'}$ (see for example Fig. 87.1). An alternative is to plot exclusion curves for fixed $M_{Z'}$ values in the $c_u^\ell - c_d^\ell$ planes, allowing a simple derivation of the mass limit within any $Z'$ model. CMS upper limits in the $c_u^\ell - c_d^\ell$ plane ($\ell = e$ or $\mu$) for different $M_{Z'}$ are shown in Ref. [21] (for Tevatron limits, see Refs. [8, 24]).

The discovery of a dilepton resonance at the LHC would determine the $Z'$ mass and width. A measurement of the total cross section would define a band in the $c_u^\ell - c_d^\ell$ plane. Angular distributions can be used to measure several combinations of $Z'$ parameters (angular distributions were used in Ref. [25] to improve the Tevatron sensitivity). Even though the original quark direction in a $pp$ collider is unknown, the leptonic forward-backward asymmetry $A_{\text{FB}}^\ell$ can be extracted from the kinematics of the dilepton system, and is sensitive to parity-violating couplings. A fit to the $Z'$ rapidity distribution can distinguish between the couplings to up and down quarks. These measurements, combined with off-peak observables, have the potential to differentiate among various $Z'$ models [26]. In some cases, $A_{\text{FB}}^\ell$ may provide discovery sensitivity that is competitive with the mass distribution [27]. The spin of the $Z'$ boson may be determined from angular distributions [28].

Searches for $Z'$ decays into $e^+e^-$ and $\mu^+\mu^-$ by the ATLAS and CMS collaborations [20, 21] have set 95% C.L. upper cross-section limits as low as 0.02 fb (see Fig. 87.1), with the mass lower limits in specific models as high as 4.9 TeV in a single channel. Cross section limits in the dimuon channel for low mass regions, below 200 GeV but not near the $Z$ mass, have been set at the LHC by CMS [29, 30] and LHCb [31].

The $Z' \to \tau^+\tau^-$ decay has been searched for by ATLAS with 36 fb$^{-1}$ of 13 TeV data [32], and a limit $M_{Z'} > 2.4$ TeV has been set in the case of a sequential $Z'$. Limits on heavy scalars in the $\tau^+\tau^-$ final state, set by ATLAS [33] and CMS [34] with the full Run 2 datasets, can be recast into limits on $Z'$. Limits in the flavor-violating leptonic final states have also been reported by ATLAS [35] and CMS [36] at 13 TeV, for resonances in the $e^\pm\mu^\mp$, $e^\pm\tau^\mp$ and $\mu^\pm\tau^\mp$ channels.

Final states with higher background, $jj, b\bar{b}$ and $t\bar{t}$, are also important as they probe various combinations of $Z'$ couplings to quarks. Besides the improved sensitivity at masses of several TeV, the LHC searches in the dijet channel have been also extended to masses as low as 10 GeV, through the use of new techniques involving boosted topologies and initial state radiation [37]. Limits from such $Z'$ searches in hadronic final states are summarized in Fig. 87.2.

Searches for the $Z' \to b\bar{b}$ decay have been performed for two production mechanisms. ATLAS [39] and CMS [40] searched for $b\bar{b}$ resonances using the full Run 2 datasets, setting limits under the assumption that the $Z'$ boson couples to light quarks and thus is produced in the s-channel. The CMS upper limit on the cross-section times the $Z' \to b\bar{b}$ branching fraction decreases from 40 fb at $M_{Z'} = 1.8$ TeV to 0.4 fb at $M_{Z'} = 8$ TeV. The ATLAS search covered the 1.3 TeV – 5 TeV mass range, setting a lower mass limit of 2.7 TeV for the $Z'_{SSM}$. Independent of any other couplings, if the $Z'$ couples to the b quark, then it can be produced in association with a $b\bar{b}$ pair, and it decays to a second $b\bar{b}$ pair, which forms a resonance. The cross-section limit for that process, set by ATLAS [41] with 103 fb$^{-1}$ of 13 TeV data, is between 0.1 and 0.2 pb for $M_{Z'}$ in the 1.3 – 3 TeV range. For a vector coupling $g_b$ of $Z'$ to b quarks, i.e., $g_{d\bar{3}3}^L = g_{d\bar{3}3}^R = g_b$ using the notation of Eq. (87.1), the limit is $M_{Z'} > 1.45$ TeV when $g_b = 1$.

Another search that depends on the $Z'$ couplings to the b quark for production (more precisely, for a $b\bar{b}, s\bar{s}$ or $b\bar{s}$ initial state) was performed by CMS [42], under the assumption that the $Z'$ boson decays into a muon pair. Searches for the $Z' \to t\bar{t}$ decay are discussed in Ref. [16].

Heavy resonances decaying into $Zh^0$ with $Z \to \ell^+\ell^-$, $\nu\bar{\nu}$ or $q\bar{q}$ and $h^0 \to b\bar{b}$ have been studied by ATLAS [43, 44] and CMS [45, 46] using the full 13 TeV dataset. CMS also searched for $Z'$ bosons in the $q\bar{q}\tau\tau$ final state [47]. The most stringent constraint on $Z'$ production in the context of the Heavy Vector Triplet (HVT) model weakly-coupled scenario A [48] is set in the semi-leptonic channel, with a mass lower limit of 3.5 TeV [45].

Searches for a $Z'$ boson lighter than the SM $Z$ and which couples to leptons have been performed in the 4-lepton final state. ATLAS [49] and CMS [50] focused on the $Z$ decays into a muon pair followed by the radiation of a narrow $Z'$ boson which decays itself into a muon pair. A related CMS search [51] for $Z$ decays into four leptons, but without requiring that a pair of leptons form a narrow resonance, was used to set limits on $Z'$ couplings to electrons or muons as a function of mass, which are relevant for large widths. ATLAS [52] considered the $h^0 \to ZZ'$ and $h^0 \to Z'Z'$ processes followed by the leptonic decays of both spin-1 bosons.

The $pp \to Z'X \to W^+W^-X$ process has also been searched for at the LHC. The channel where the $Z'$ boson is produced through its couplings to quarks, and the $W$ bosons decay hadronically, has been explored using boosted techniques to analyze the 13 TeV data [46, 53–55] with a mass lower limit of 3.5 TeV in the HVT model A [54]. A search in the $e\nu\mu\nu$ final state sets cross-section limits at lower $Z'$ mass [56]. The $Z'$ boson may also be produced through its couplings to $W$ bosons [57], which has been explored with the use of forward jets consistent with a vector boson fusion event topology [46, 54, 55]. The latter process provides a test of
<figure><img src="image_362.png" /><figcaption>**Figure 87.2:** Upper limits on the $Z'$ coupling to quarks as a function of $M_{Z'}$ based on various searches performed by the ATLAS, CMS, CDF, and UA2 experiments [38].</figcaption></figure>

fermiophobic $Z'$ models.

At the Tevatron, the CDF and DØ collaborations have searched for $Z'$ bosons in the $e^{+}e^{-}$ [58], $\mu^{+}\mu^{-}$ [59], $e^{\pm}\mu^{\mp}$ [60], $\tau^{+}\tau^{-}$ [61], $t\bar{t}$ [62], $jj$ [63] and $W^{+}W^{-}$ [64] final states. These limits have been mostly superseded by the LHC results.

## 87.5 Low-energy constraints

$Z'$ boson properties are also constrained by a variety of low-energy experiments [65]. Polarized electron-nucleon scattering and atomic parity violation are sensitive to electron-quark contact interactions, which get contributions from $Z'$ exchange that can be expressed in terms of the couplings introduced in Eq. (87.1) and $M'_{Z'}$. Further corrections to the electron-quark contact interactions are induced in the presence of $\tilde{Z}-\tilde{Z}'$ mixing because of the shifts in the $Z$ couplings to quarks and leptons [3]. Deep-inelastic neutrino-nucleon scattering is similarly affected by $Z'$ bosons. Other low-energy observables are discussed in [4]. Viable models with $Z'$ bosons much lighter than the $Z$ boson have been constructed, despite many additional experimental constraints [66]. Limits on the $Z'$ coupling to leptons in the mass range 0.02–10.2 GeV have been set in $e^{+}e^{-}$ collisions by BaBar [67], assuming a dark photon, i.e., a $Z'$ boson whose couplings arise only from the kinetic mixing with the hypercharge gauge boson.

In some models, the lower limits on $M_{Z'}$ set by low-energy data are above 1 TeV. For example, $M_{Z'} > 1.1$ TeV and $M_{Z'_{\eta}} > 0.43$ TeV assuming that the Higgs sectors consist of electroweak doublets and singlets only [4], while the gauge coupling is fixed by an $SO(10)$ unification condition for $U(1)_{\chi}$ and $U(1)_{\eta}$. For more general models, see Refs. [1, 6, 68]. The mass bounds from direct searches at the LHC [20, 21] exceed the electroweak constraints by a factor of three or more for the models mentioned here. This conclusion could change if the collider bounds are weakened by exotic decay channels [69].

Although the LHC data are most constraining for many $Z'$ models, one should be careful in assessing the relative reach of various experiments given the freedom in $Z'$ couplings. For example, a $Z'$ coupled to $B-yL_{\mu}+(y-3)L_{\tau}$ has implications for the muon $g-2$, neutrino oscillations or $\tau$ decays, and would be hard to see in processes involving first-generation fermions. Moreover, the combination of LHC searches and low-energy measurements could allow a precise determination of the $Z'$ parameters [70].

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# 88. Supersymmetry, Part I (Theory)

Revised August 2023 by B.C. Allanach (DAMTP, Cambridge U.) and H.E. Haber (UC Santa Cruz).

<table><tbody><tr><td>88.1</td><td>Introduction</td><td>1038</td></tr><tr><td>88.2</td><td>Structure of the MSSM</td><td>1038</td></tr><tr><td>88.2.1</td><td>R-parity and the lightest supersymmetric particle</td><td>1039</td></tr><tr><td>88.2.2</td><td>The goldstino and gravitino</td><td>1039</td></tr><tr><td>88.2.3</td><td>Hidden sectors and the structure of SUSY breaking</td><td>1040</td></tr><tr><td>88.2.4</td><td>SUSY and extra dimensions</td><td>1040</td></tr><tr><td>88.2.5</td><td>Split-SUSY</td><td>1040</td></tr><tr><td>88.3</td><td>Parameters of the MSSM</td><td>1041</td></tr><tr><td>88.3.1</td><td>The SUSY-conserving parameters</td><td>1041</td></tr><tr><td>88.3.2</td><td>The SUSY-breaking parameters</td><td>1041</td></tr><tr><td>88.3.3</td><td>MSSM-124</td><td>1041</td></tr><tr><td>88.4</td><td>The supersymmetric-particle spectrum</td><td>1042</td></tr><tr><td>88.4.1</td><td>The charginos and neutralinos</td><td>1042</td></tr><tr><td>88.4.2</td><td>The squarks and sleptons</td><td>1043</td></tr><tr><td>88.5</td><td>The supersymmetric Higgs sector</td><td>1043</td></tr><tr><td>88.5.1</td><td>The tree-level Higgs sector</td><td>1043</td></tr><tr><td>88.5.2</td><td>The radiatively-corrected Higgs sector</td><td>1044</td></tr><tr><td>88.6</td><td>Restricting the MSSM parameter freedom</td><td>1044</td></tr><tr><td>88.6.1</td><td>Gaugino mass relations</td><td>1045</td></tr><tr><td>88.6.2</td><td>Constrained versions of the MSSM: mSUGRA, CMSSM, etc.</td><td>1046</td></tr><tr><td>88.6.3</td><td>Gauge-mediated SUSY breaking</td><td>1046</td></tr><tr><td>88.6.4</td><td>The phenomenological MSSM</td><td>1047</td></tr><tr><td>88.6.5</td><td>Simplified models</td><td>1047</td></tr><tr><td>88.7</td><td>Experimental data confronts the MSSM</td><td>1047</td></tr><tr><td>88.7.1</td><td>Naturalness constraints and the little hierarchy</td><td>1048</td></tr><tr><td>88.7.2</td><td>Indirect constraints on supersymmetric models</td><td>1049</td></tr><tr><td>88.8</td><td>Massive neutrinos in weak-scale SUSY</td><td>1050</td></tr><tr><td>88.8.1</td><td>The supersymmetric seesaw</td><td>1050</td></tr><tr><td>88.8.2</td><td>R-parity-violating SUSY</td><td>1050</td></tr><tr><td>88.9</td><td>Extensions beyond the MSSM</td><td>1051</td></tr></tbody></table>

## 88.1 Introduction

Supersymmetry (SUSY) is a generalization of the space-time symmetries of quantum field theory that transforms fermions into bosons and vice versa [1]. The existence of such a non-trivial extension of the Poincaré symmetry of ordinary quantum field theory was initially surprising, and its form is highly constrained by theoretical principles [2]. SUSY also provides a framework for the unification of particle physics and gravity [3–5] at the Planck energy scale, $M_P \sim 10^{19}$ GeV, where the gravitational interactions become comparable in strength to the gauge interactions. Moreover, supersymmetry can stabilize the hierarchy between the energy scale that characterizes electroweak symmetry breaking, $M_{EW} \sim 100$ GeV, and the Planck scale [6–9] against large radiative corrections. The stability of this large gauge hierarchy with respect to radiative quantum corrections is not possible to maintain in the Standard Model (SM) without an unnatural fine-tuning of the parameters of the fundamental theory at the Planck scale. In contrast, in a supersymmetric extension of the SM, it is possible to maintain the gauge hierarchy while providing a natural framework for elementary scalar fields.

If supersymmetry were an exact symmetry of nature, then particles and their superpartners, which differ in spin by half a unit, would be degenerate in mass. Since superpartners have not (yet) been observed, supersymmetry must be a broken symmetry. Nevertheless, the stability of the gauge hierarchy can still be maintained if the SUSY breaking is soft [10,11], and the corresponding SUSY-breaking mass parameters are no larger than a few TeV. Whether this is still plausible in light of recent SUSY searches at the LHC (see Sec. 89) will be discussed in Sec. 88.7.

In particular, soft-SUSY-breaking terms of the Lagrangian involve combinations of fields with total mass dimension of three or

less, with some restrictions on the dimension-three terms as elucidated in Ref. [10]. The impact of the soft terms becomes negligible at energy scales much larger than the size of the SUSY-breaking masses. Thus, a theory of weak-scale supersymmetry, where the effective scale of supersymmetry breaking is tied to the scale of electroweak symmetry breaking, provides a natural framework for the origin and the stability of the gauge hierarchy [6–9].

At present, there is no unambiguous experimental evidence for the breakdown of the SM at or below the TeV scale. The expectations for new TeV-scale physics beyond the SM are based primarily on three theoretical arguments. First, in a theory with an elementary scalar field of mass $m$ and interaction strength $\lambda$ (e.g., a quartic scalar self-coupling, the square of a gauge coupling or the square of a Yukawa coupling), the stability with respect to quantum corrections requires the existence of an energy cutoff roughly of order $(16\pi^2/\lambda)^{1/2}m$, beyond which new physics must enter [12]. A significantly larger energy cutoff would require an unnatural fine-tuning of parameters that govern the effective low-energy theory. Applying this argument to the SM leads to an expectation of new physics at the TeV scale [9].

Second, the unification of the three SM gauge couplings at a very high energy close to the Planck scale is possible if new physics beyond the SM (which modifies the running of the gauge couplings above the electroweak scale) is present. The minimal supersymmetric extension of the SM, where superpartner masses lie below a few TeV, provides an example of successful gauge coupling unification [13].

Third, the existence of dark matter that makes up approximately one quarter of the energy density of the universe, cannot be explained within the SM of particle physics [14]. Remarkably, a stable weakly-interacting massive particle (WIMP) whose mass and interaction rate are governed by new physics associated with the TeV-scale can be consistent with the observed density of dark matter (this is the so-called WIMP miracle, which is reviewed in Ref. [15]). The lightest supersymmetric particle (LSP), if stable, is a promising (although not the unique) candidate for the dark matter [16–20]. Further aspects of dark matter can be found in Sec. 27.

## 88.2 Structure of the MSSM

The minimal supersymmetric extension of the SM (MSSM) consists of the fields of the two-Higgs-doublet extension of the SM and the corresponding superpartner fields [21–25]. A particle and its superpartner together form a supermultiplet. The corresponding field content of the supermultiplets of the MSSM and their gauge quantum numbers are shown in Table 88.1. The electric charge $Q = T_3 + \frac{1}{2}Y$ is determined in terms of the third component of the weak isospin ($T_3$) and the U(1) weak hypercharge (Y).

The gauge supermultiplets consist of the gluons and their gluino fermionic superpartners and the SU(2)×U(1) gauge bosons and their gaugino fermionic superpartners. The matter supermultiplets consist of three generations of left-handed quarks and leptons and their scalar superpartners (squarks and sleptons, collectively referred to as sfermions), and the corresponding antiparticles. The Higgs supermultiplets consist of two complex Higgs doublets, their higgsino fermionic superpartners, and the corresponding antiparticles. The enlarged Higgs sector of the MSSM constitutes the minimal structure needed to guarantee the cancellation of gauge anomalies [27] generated by the higgsino superpartners that can appear as internal lines in triangle diagrams with three external electroweak gauge bosons. Moreover, without a second Higgs doublet, one cannot generate mass for both “up”-type and “down”-type quarks (and charged leptons) in a way consistent with the underlying SUSY [28–30].

In the most elegant treatment of SUSY, spacetime is extended to superspace which consists of the spacetime coordinates and new anticommuting fermionic coordinates $\theta$ and $\theta^\dagger$ [25, 31, 32]. Each supermultiplet is represented by a superfield that is a function of the superspace coordinates. The fields of a given supermultiplet (which are functions of the spacetime coordinates) are coefficients of the $\theta$ and $\theta^\dagger$ expansion of the corresponding superfield.

Vector superfields contain the gauge-boson fields and their gaugino partners. Chiral superfields contain the spin-0 and spin-
<table><caption>**Table 88.1:** The fields of the MSSM and their representations under the SU(3)×SU(2)×U(1) gauge group are listed. For simplicity, only one generation of quarks and leptons is exhibited. For each lepton, quark, and Higgs supermultiplet (each denoted by a hatted upper-case letter), there is a corresponding antiparticle multiplet of charge-conjugated fermions and their associated scalar partners [26].</caption><thead><tr><th colspan="7">Field Content of the MSSM</th></tr><tr><th>Super-multiplets</th><th>Super-field</th><th>Bosonic fields</th><th>Fermionic partners</th><th>SU(3)</th><th>SU(2)</th><th>U(1)</th></tr></thead><tbody><tr><td>gluon/gluino</td><td>Ṽ<sub>S</sub></td><td>g</td><td>g̃</td><td>8</td><td>1</td><td>0</td></tr><tr><td>gauge boson/ gaugino</td><td>Ṽ</td><td>W<sup>±</sup>, W<sup>0</sup><br>B</td><td>W̃<sup>±</sup>, W̃<sup>0</sup><br>B̃</td><td>1<br>1</td><td>3<br>1</td><td>0<br>0</td></tr><tr><td>slepton/ lepton</td><td>L̃</td><td>(ν̃<sub>L</sub>, ẽ<sub>L</sub>)</td><td>(ν, e<sup>-</sup>)<sub>L</sub></td><td>1<br>1</td><td>2<br>1</td><td>-1<br>2</td></tr><tr><td>squark/ quark</td><td>Q̃</td><td>(ũ<sub>L</sub>, d̃<sub>L</sub>)</td><td>(u, d)<sub>L</sub></td><td>3<br>3</td><td>2<br>1</td><td>1/3<br>-4/3</td></tr><tr><td></td><td>Ũ<sup>c</sup></td><td>ũ<sub>R</sub><sup>*</sup></td><td>u<sup>c</sup><sub>L</sub></td><td>3<br>3</td><td>1<br>1</td><td>2/3</td></tr><tr><td></td><td>D̃<sup>c</sup></td><td>d̃<sub>R</sub><sup>*</sup></td><td>d<sup>c</sup><sub>L</sub></td><td>3<br>3</td><td>1<br>1</td><td>2/3</td></tr><tr><td>Higgs boson/ higgsino</td><td>H̃<sub>d</sub></td><td>(H<sup>0</sup><sub>d</sub>, H<sup>-</sup><sub>d</sub>)</td><td>(H̃<sup>0</sup><sub>d</sub>, H̃<sup>-</sup><sub>d</sub>)</td><td>1<br>1</td><td>2<br>2</td><td>-1<br>1</td></tr></tbody></table>

1/2 fields of the matter or Higgs supermultiplets. A general supersymmetric Lagrangian is determined by three functions of the chiral superfields [4]: the superpotential, the Kähler potential, and the gauge kinetic function (which can be appropriately generalized to accommodate higher derivative terms [33]). Minimal forms for the Kähler potential and gauge kinetic function, which generate canonical kinetic energy terms for all the fields, are required for renormalizable globally supersymmetric theories. A renormalizable superpotential, which is at most cubic in the chiral superfields, yields supersymmetric Yukawa couplings and mass terms. A combination of gauge invariance and SUSY produces couplings of gaugino fields to matter (or Higgs) fields and their corresponding superpartners. The (renormalizable) MSSM Lagrangian is then constructed by including all possible supersymmetric interaction terms (of dimension four or less) that satisfy SU(3)×SU(2)×U(1) gauge invariance and B-L conservation (where B = baryon number and L = lepton number). Finally, the most general soft-supersymmetry-breaking terms consistent with these symmetries are added [10, 11, 34].

Although the MSSM is the focus of much of this review, there is some motivation for considering non-minimal supersymmetric extensions of the SM [35]. For example, extra structure is needed to generate non-zero neutrino masses as discussed in Sec. 88.8. In addition, in order to address some theoretical issues and tensions associated with the MSSM, it has been fruitful to introduce one additional singlet Higgs superfield. The resulting next-to-minimal supersymmetric extension of the Standard Model (NMSSM) [36] is briefly considered in Sec. 88.4–88.7 and 88.9. Finally, one is always free to add additional fields to the SM along with the corresponding superpartners. However, only certain choices for the new fields (e.g., the addition of complete SU(5) multiplets) will preserve the successful gauge coupling unification of the MSSM. Some examples will be briefly mentioned in Sec. 88.9.

### 88.2.1 R-parity and the lightest supersymmetric particle

The (renormalizable) SM Lagrangian possesses an accidental global B-L symmetry due to the fact that B and L-violating operators composed of SM fields must have dimension $d=5$ or larger [37]. Consequently, B and L-violating effects are suppressed by $(M_{EW}/M)^{d-4}$, where M is the characteristic mass scale of the physics that generates the corresponding higher dimensional operators. Indeed, values of M of order the grand unification scale or larger may be responsible for the observed (approximate) stability of the proton and suppression of neutrino masses. Unfortunately, these results are not guaranteed in a generic supersymmetric extension of the SM. For example, it is possible to construct gauge invariant supersymmetric dimension-four B and L-violating operators made up of fields of SM particles and their superpartners. Such operators, if simultaneously present in the theory, would typically yield a proton decay rate many orders of magnitude larger

than the current experimental bound. It is for this reason that B-L conservation is imposed on the supersymmetric Lagrangian when defining the MSSM, which is sufficient for eliminating all B and L-violating operators of dimension $d \le 4$.

As a consequence of the B-L symmetry, the MSSM possesses a multiplicative R-parity invariance, where $R = (-1)^{3(B-L)+2S}$ for a particle of spin $S$ [38]. This implies that all the particles of the SM have even R-parity, whereas the corresponding superpartners have odd R-parity. The conservation of R-parity in scattering and decay processes has a critical impact on supersymmetric phenomenology. For example, any initial state in a scattering experiment will involve ordinary (R-even) particles. Consequently, it follows that supersymmetric particles must be produced in pairs. In general, these particles are highly unstable and decay into lighter states. Moreover, R-parity invariance also implies that the LSP is absolutely stable, and must eventually be produced at the end of a decay chain initiated by the decay of a heavy unstable supersymmetric particle. In order to be consistent with cosmological constraints, a stable LSP is almost certainly electrically and color neutral [18]. Consequently, the LSP in an R-parity-conserving theory is weakly interacting with ordinary matter, i.e., it behaves like a stable heavy neutrino and will escape collider detectors without being directly observed. Thus, the canonical signature for conventional R-parity-conserving supersymmetric theories is missing (transverse) momentum, due to the escape of the LSP. Moreover, as noted in Sec. 88.1 and reviewed in Refs. [19] and [20], the stability of the LSP in R-parity-conserving SUSY makes it a promising candidate for dark matter.

The possibility of relaxing the R-parity invariance of the MSSM (which would generate new B and/or L-violating interactions) will be addressed in Sec. 88.8.2. However, note that in R-parity violating (RPV) models, the LSP is no longer stable and thus would not be a viable candidate for the dark matter (unless its lifetime was significantly longer than the age of the universe). In such scenarios, one has to look elsewhere to explain the origin of dark matter.

### 88.2.2 The goldstino and gravitino

In the MSSM, SUSY breaking is accomplished by including the most general renormalizable soft-SUSY-breaking terms consistent with the SU(3)×SU(2)×U(1) gauge symmetry and R-parity invariance. These terms parameterize our ignorance of the fundamental mechanism of supersymmetry breaking. If supersymmetry breaking occurs spontaneously, then a massless Goldstone fermion called the goldstino ($\tilde{G}_{1/2}$) must exist. The goldstino would then be the LSP, and could play an important role in supersymmetric phenomenology [39].

However, the goldstino degrees of freedom are physical only in models of spontaneously-broken global SUSY. If SUSY is a local symmetry, then the theory must incorporate gravity; the resulting
theory is called supergravity [5, 40]. In models of spontaneously-broken supergravity, the goldstino is “absorbed” by the gravitino ($\tilde{G}$), the spin-3/2 superpartner of the graviton, via the super-Higgs mechanism [41]. Consequently, the goldstino is removed from the physical spectrum and the gravitino acquires a mass (denoted by $m_{3/2}$). If $m_{3/2}$ is smaller than the mass of the lightest superpartner of the SM particles, then the gravitino is the LSP.

In processes with center-of-mass energy $E \gg m_{3/2}$, one can employ the goldstino-gravitino equivalence theorem [42], which implies that the interactions of the helicity $\pm\frac{1}{2}$ gravitino (whose properties approximate those of the goldstino) dominate those of the helicity $\pm\frac{3}{2}$ gravitino. The interactions of gravitinos with other light fields can be described by a low-energy effective Lagrangian that is determined by fundamental principles [43].

### 88.2.3 Hidden sectors and the structure of SUSY breaking

It is very difficult (perhaps impossible) to construct a realistic model of spontaneously-broken weak-scale supersymmetry where the supersymmetry breaking arises solely as a consequence of the interactions of the particles of the MSSM. A more successful scheme posits a theory with at least two distinct sectors: a visible sector consisting of the particles of the MSSM [34] and a sector where SUSY breaking is generated. It is often (but not always) assumed that particles of the hidden sector are neutral with respect to the SM gauge group. The effects of the hidden sector supersymmetry breaking are then transmitted to the MSSM by some mechanism (often involving the mediation by particles that comprise an additional messenger sector). Two theoretical scenarios that exhibit this structure are gravity-mediated and gauge-mediated SUSY breaking.

Supergravity models provide a natural mechanism for transmitting the SUSY breaking of the hidden sector to the particle spectrum of the MSSM. In models of gravity-mediated SUSY breaking, gravity is the messenger of supersymmetry breaking [44–48]. More precisely, supersymmetry breaking is mediated by effects of gravitational strength (i.e. suppressed by inverse powers of the Planck mass). The soft-SUSY-breaking parameters with dimensions of mass arise as model-dependent multiples of the gravitino mass $m_{3/2}$. In this scenario, $m_{3/2}$ is of order the electroweak-symmetry-breaking scale, while the gravitino couplings are roughly gravitational in strength [3, 49].<fnref n="1" />

Under certain theoretical assumptions that govern the structure of the Kähler potential (the so-called sequestered form introduced in Ref. [51]), SUSY breaking is due entirely to the super-conformal (super-Weyl) anomaly, which is common to all supergravity models [51]. In particular, gaugino masses are radiatively generated at one-loop, and squark and slepton squared-mass matrices are flavor-diagonal. In sequestered scenarios, sfermion squared-masses arise at two-loops, which implies that gluino and sfermion masses are of the same order of magnitude. This approach is called anomaly-mediated SUSY breaking (AMSB). Indeed, anomaly mediation is more generic than originally conceived, and provides a ubiquitous source of SUSY breaking [52]. However in the simplest formulation of AMSB as applied to the MSSM, the squared-masses of the sleptons are negative (known as the tachyonic slepton problem). It may be possible to cure this otherwise fatal flaw in non-minimal extensions of the MSSM [53]. Alternatively, one can assert that anomaly mediation is not the sole source of SUSY breaking in the sfermion sector. In non-sequestered scenarios, sfermion squared-masses can arise at tree-level, in which case squark masses would be parametrically larger than the loop-suppressed gaugino masses [54].

In gauge-mediated supersymmetry breaking (GMSB), gauge forces transmit the supersymmetry breaking to the MSSM. A typical structure of such models involves a hidden sector where SUSY is broken, a messenger sector consisting of particles (messengers) with nontrivial $SU(3) \times SU(2) \times U(1)$ quantum numbers, and the visible sector consisting of the fields of the MSSM [55–58]. The direct coupling of the messengers to the hidden sector generates a supersymmetry-breaking spectrum in the messenger sector.

Supersymmetry breaking is then transmitted to the MSSM via the virtual exchange of the messenger fields. In models of direct gauge mediation, there is no separate hidden sector. In particular, the sector in which the SUSY breaking originates includes fields that carry nontrivial SM quantum numbers, which allows for the direct transmission of SUSY breaking to the MSSM [59].

In models of gauge-mediated SUSY breaking with a minimal Kähler potential, the gravitino is the LSP [16], as its mass can range from a few eV (in the case of low SUSY breaking scales) up to a few GeV (in the case of high SUSY breaking scales). In particular, the gravitino is a potential dark matter candidate (for a review and guide to the literature, see Ref. [20]). The couplings of the helicity $\pm\frac{1}{2}$ components of $\tilde{G}$ to the particles of the MSSM (which approximate those of the goldstino as previously noted in Sec. 88.2.2) are significantly stronger than gravitational strength and amenable to experimental collider analyses.

The mass ranges of the gravitino in either gravity-mediated or gauge-mediated SUSY breaking are further constrained by cosmological considerations [60, 61]. In particular, there is a danger of over-abundance of gravitinos if it is the dark matter or, if it decays before nucleosynthesis, modifications to the successful predictions of light element abundances. Avoiding these cosmological gravitino problems imposes strong constraints on gravity-mediated and gauge-mediated SUSY breaking models.

The concept of a hidden sector is more general than SUSY. Hidden valley models [62] posit the existence of a hidden sector of new particles and interactions that are very weakly coupled to particles of the SM. The impact of a hidden valley on supersymmetric phenomenology at colliders can be significant if the LSP lies in the hidden sector [63].

### 88.2.4 SUSY and extra dimensions

Approaches to SUSY breaking have also been developed in the context of theories in which the number of spatial dimensions is greater than three. In particular, a number of SUSY-breaking mechanisms have been proposed that are inherently extra-dimensional [64]. The size of the extra dimensions can be significantly larger than $M_P^{-1}$; in some cases of order (TeV)$^{-1}$ or even larger (see, e.g., Sec. 85 and Ref. [65]).

For example, in one approach the fields of the MSSM live on some brane (a lower-dimensional manifold embedded in a higher-dimensional spacetime), while the sector of the theory that breaks SUSY lives on a second spatially-separated brane. Two examples of this approach are AMSB [51] and gaugino-mediated SUSY breaking [66]. In both cases, SUSY breaking is transmitted through fields that live in the bulk (the higher-dimensional space between the two branes). This setup has some features in common with both gravity-mediated and gauge-mediated SUSY breaking (e.g., hidden and visible sectors and messengers).

Since a higher dimensional theory must be compactified to four spacetime dimensions, one can also generate a source of SUSY breaking by employing boundary conditions on the compactified space that distinguish between fermions and bosons. This is the so-called Scherk-Schwarz mechanism [67]. The phenomenology of such models can be strikingly different from that of the usual MSSM [68].

### 88.2.5 Split-SUSY

If SUSY is not connected with the origin of the electroweak scale, it may still be possible that some remnant of the superparticle spectrum survives down to the TeV-scale or below. This is the idea of split-SUSY [69, 70], in which scalar superpartners of the quarks and leptons are significantly heavier than 1 TeV, whereas the fermionic superpartners of the gauge and Higgs bosons have masses on the order of 1 TeV or below. With the exception of a single light neutral scalar whose properties are practically indistinguishable from those of the SM Higgs boson, all other Higgs bosons are also assumed to be very heavy. Among the supersymmetric particles, only the fermionic superpartners may be kinematically accessible at the LHC.

In models of split SUSY, the top squark masses cannot be arbitrarily large, as these parameters enter in the radiative corrections to the mass of the observed Higgs boson [71–73]. In the MSSM, a Higgs boson mass of 125 GeV (see Sec. 11) implies an upper bound

<footnote n="1">¹However, such a gravitino typically plays no direct role in supersymmetric phenomenology at colliders (except perhaps indirectly in the case where the gravitino is the LSP [50]).</footnote>
on the top squark mass scale in the range of 10 to $10^8$ TeV [74–76], depending on the value of the ratio of the two neutral Higgs field vacuum expectation values, although this mass range can be somewhat extended by varying other relevant MSSM parameters. In some approaches, gaugino masses are one-loop suppressed relative to the sfermion masses, corresponding to the so-called mini-split SUSY spectrum [72,77]. The higgsino mass scale may or may not be likewise suppressed depending on the details of the model [78].

The SUSY breaking required to produce such a split-SUSY spectrum would destabilize the gauge hierarchy, and thus would not provide an explanation for the scale of electroweak symmetry breaking. Nevertheless, models of split-SUSY can account for the dark matter (which is assumed to be the LSP gaugino or higgsino) and gauge coupling unification, thereby preserving two of the desirable features of weak-scale SUSY. Finally, as a consequence of the very large squark and slepton masses, neutral flavor changing and CP-violating effects, which can be problematic in models with TeV-scale SUSY-breaking masses, are sufficiently reduced to avoid conflict with experimental observations.

## 88.3 Parameters of the MSSM

The parameters of the MSSM are conveniently described by considering separately the supersymmetry-conserving and the supersymmetry-breaking sectors. A careful discussion of the conventions used here in defining the tree-level MSSM parameters can be found in Refs. [25, 79, 80]. For simplicity, consider first the case of one generation of quarks, leptons, and their scalar superpartners.

### 88.3.1 The SUSY-conserving parameters

The parameters of the supersymmetry-conserving sector consist of: (i) gauge couplings, $g_s$, $g$, and $g'$, corresponding to the SM gauge group SU(3)×SU(2)×U(1) respectively; (ii) a supersymmetry-conserving higgsino mass parameter $\mu$; and (iii) Higgsfermion Yukawa couplings, $\lambda_u$, $\lambda_d$, and $\lambda_e$, of one generation of left- and right-handed quarks and leptons, and their superpartners to the Higgs bosons and higgsinos. Because there is no right-handed neutrino/sneutrino in the MSSM as defined here, a Yukawa coupling $\lambda_\nu$ is not included. The complex $\mu$ parameter and Yukawa couplings enter via the most general renormalizable R-parity-conserving superpotential,

$$W_{\text{MSSM}} = \lambda_d \hat{H}_d \hat{Q} \hat{D}^c - \lambda_u \hat{H}_u \hat{Q} \hat{U}^c + \lambda_e \hat{H}_d \hat{L} \hat{E}^c + \mu \hat{H}_u \hat{H}_d, \quad (88.1)$$

where the superfields are defined in Table 1 and the gauge group indices are suppressed. More explicitly, the so-called “μ-term” can be written out as $\mu e^{ab}(\hat{H}_u)_a(\hat{H}_d)_b$ with an implicit sum over repeated indices, where $\epsilon^{ab}$ is used to tie together the SU(2) weak isospin indices $a, b \in \{1, 2\}$ in a gauge-invariant way (where $\epsilon^{12} = -\epsilon^{21} = 1$ and $\epsilon^{11} = \epsilon^{22} = 0$). Likewise, the term $\hat{H}_u \hat{Q} \hat{U}^c$ can be written out as $\epsilon^{ab}(\hat{H}_u)_a \hat{Q}_{ib}(\hat{U}^c)^i$, where $i \in \{1, 2, 3\}$ is the SU(3) color index. Finally, $\hat{H}_d \hat{Q} \hat{D}^c$ can be written out as $\epsilon^{ab}(\hat{H}_d)_a \hat{Q}_{ib}(\hat{D}^c)^i$, with an analogous expression for $\hat{H}_d \hat{L} \hat{E}^c$.

### 88.3.2 The SUSY-breaking parameters

The supersymmetry-breaking sector contains the following sets of parameters: (i) three complex gaugino Majorana mass parameters, $M_3, M_2$, and $M_1$, associated with the SU(3), SU(2), and U(1) subgroups of the SM; (ii) five sfermion squared-mass parameters, $M_Q^2, M_U^2, M_D^2, M_L^2$, and $M_E^2$, corresponding to the five electroweak gauge multiplets, i.e., superpartners of the left-handed fields $(u, d)_L, u_L^c, d_L^c, (\nu, e^-)_L$, and $e_L^c$, where the superscript c indicates a charge-conjugated fermion field [26]; and (iii) three Higgs-squark-squark and Higgs-slepton-slepton trilinear interaction terms, with complex coefficients $T_U \equiv \lambda_u A_U, T_D \equiv \lambda_d A_D$, and $T_E \equiv \lambda_e A_E$ (which define the “A-parameters”), following the notation employed in Ref. [80]. It is conventional to separate out the factors of the Yukawa couplings in defining the A-parameters [3, 25] (originally motivated by a simple class of gravity-mediated SUSY-breaking models). If the A-parameters are parametrically of the same order (or smaller) relative to other SUSY-breaking mass parameters, then in most cases only the third generation A-parameters will be phenomenologically relevant.

Finally, we have (iv) two real squared-mass parameters, $m_{H_d}^2$ and $m_{H_u}^2$ (also called $m_1^2$ and $m_2^2$, respectively, in the literature), and one complex squared-mass parameter, $m_{12}^2 \equiv \mu B$ (the latter defines the “B-parameter”), which appear in the MSSM tree-level scalar Higgs potential [30],

$$V = (m_{H_d}^2 + |\mu|^2)H_d^\dagger H_d + (m_{H_u}^2 + |\mu|^2)H_u^\dagger H_u + (m_{12}^2 H_u H_d + \text{h.c.}) + \frac{1}{8}(g^2 + g'^2)(H_d^\dagger H_d - H_u^\dagger H_u)^2 + \frac{1}{2}g^2|H_d^\dagger H_u|^2, \quad (88.2)$$

where the SU(2)-invariant combination of the complex doublet scalar fields $H_u$ and $H_d$ that appears in Eq. (88.2) is given by $H_u H_d \equiv \epsilon^{ab}(H_u)_a (H_d)_b = H_u^\dagger H_d^- - H_u^0 H_d^0$. Note that the quartic Higgs couplings are related to the gauge couplings $g$ and $g'$ as a consequence of SUSY. The breaking of the SU(2)×U(1) electroweak symmetry group to U(1)$_\text{EM}$ is only possible after incorporating the SUSY-breaking Higgs squared-mass parameters $m_{H_d}^2, m_{H_u}^2$ (which can be negative) and $m_{12}^2$. After minimizing the Higgs scalar potential, these three squared-mass parameters can be re-expressed in terms of the two Higgs vacuum expectation values, $\langle H_d^0 \rangle \equiv v_d/\sqrt{2}$ and $\langle H_u^0 \rangle \equiv v_u/\sqrt{2}$, and the CP-odd Higgs mass $m_A$ [cf. Eqs. (88.4) and (88.5) below]. One is always free to rephase the Higgs doublet fields such that $v_d$ and $v_u$ (also called $v_1$ and $v_2$, respectively, in the literature) are both real and positive.

The quantity, $v_d^2 + v_u^2 = 4m_W^2/g^2 = (2G_F^2)^{-1/2} \simeq (246 \text{ GeV})^2$, is fixed by the Fermi constant, $G_F$, whereas the ratio

$$\tan \beta = v_u/v_d \quad (88.3)$$

is a free parameter such that $0 < \beta < \pi/2$. By employing the tree-level conditions resulting from the minimization of the scalar potential, one can eliminate the diagonal and off-diagonal Higgs squared-masses in favor of $m_Z^2 = \frac{1}{4}(g^2+g'^2)(v_d^2+v_u^2)$, the CP-odd Higgs mass $m_A$ and the parameter $\tan \beta$,

$$\sin 2\beta = \frac{2m_{12}^2}{m_{H_d}^2 + m_{H_u}^2 + 2|\mu|^2} = \frac{2m_{12}^2}{m_A^2}, \quad (88.4)$$

$$\frac{1}{2} m_Z^2 = -|\mu|^2 + \frac{m_{H_d}^2 - m_{H_u}^2 \tan^2 \beta}{\tan^2 \beta - 1}. \quad (88.5)$$

One must also guard against the existence of charge and/or color breaking global minima due to non-zero vacuum expectation values for the squark and charged slepton fields. This possibility can be avoided if the A-parameters are not unduly large [45, 81, 82]. Additional constraints must also be respected to avoid the possibility of directions in scalar field space in which the full tree-level scalar potential can become unbounded from below [82]. A computer program has been developed to calculate vacuum stability bounds in general models at the one-loop level [83], and has been applied to the MSSM in Ref. [84].

Note that SUSY-breaking mass terms for the fermionic superpartners of the scalar fields and non-holomorphic trilinear scalar interactions (i.e., interactions that mix scalar fields and their complex conjugates) have not been included above in the soft-SUSY-breaking sector. These terms can potentially destabilize the gauge hierarchy [10] in models with a gauge-singlet superfield. The latter is not present in the MSSM; hence as noted in Ref. [11], these so-called non-standard soft-SUSY-breaking terms are benign. The phenomenological impact of non-holomorphic soft SUSY-breaking terms has been reconsidered in Refs. [85–87]. However, in the most common approaches to constructing a fundamental theory of SUSY-breaking, the coefficients of these terms (which have dimensions of mass) are significantly suppressed compared to the TeV-scale [88]. Consequently, we follow the usual approach and omit these terms from further consideration.

### 88.3.3 MSSM-124

The total number of independent physical parameters that define the MSSM (in its most general form) is quite large, primarily due to the soft-supersymmetry-breaking sector. In particular, in the case of three generations of quarks, leptons, and their superpartners, $M_Q^2, M_U^2, M_D^2, M_L^2$, and $M_E^2$ are hermitian $3 \times 3$
matrices, and $A_U, A_D$, and $A_E$ are complex $3 \times 3$ matrices. In addition, $M_1, M_2, M_3, B$, and $\mu$ are in general complex parameters. Finally, as in the SM, the Higgs-fermion Yukawa couplings, $\lambda_f$ ($f=u, d$, and $e$), are complex $3 \times 3$ matrices that are related to the quark and lepton mass matrices via: $M_f = \lambda_f v_f / \sqrt{2}$, where $v_e = v_d$ [with $v_u$ and $v_d$ as defined above Eq. (88.3)].

However, not all these parameters are physical. Some of the MSSM parameters can be eliminated by expressing interaction eigenstates in terms of the mass eigenstates, with an appropriate redefinition of the MSSM fields to remove unphysical degrees of freedom. The analysis of Ref. [89] shows that the MSSM possesses 124 independent real degrees of freedom. Of these, 18 correspond to SM parameters (including the QCD vacuum angle $\theta_{QCD}$), one corresponds to a Higgs sector parameter (the analogue of the SM Higgs mass), and 105 are genuinely new parameters of the model. The latter include: five real parameters and three CP-violating phases in the gaugino/higgsino sector, 21 squark and slepton (sfermion) masses, 36 real mixing angles to define the sfermion mass eigenstates, and 40 CP-violating phases that can appear in sfermion interactions. The most general parameterization of the R-parity-conserving MSSM (without additional theoretical assumptions) will be denoted henceforth as MSSM-124 [90].

## 88.4 The supersymmetric-particle spectrum

The supersymmetric particles (sparticles) differ in spin by half a unit from their SM partners. The superpartners of the gauge and Higgs bosons are fermions, whose names are obtained by appending “ino” to the end of the corresponding SM particle name. The gluino is the color-octet Majorana fermion partner of the gluon with mass $M_{\tilde{g}} = |M_3|$. The superpartners of the electroweak gauge and Higgs bosons (the gauginos and higgsinos) can mix due to SU(2)×U(1) breaking effects. As a result, the physical states of definite mass are parameter dependent linear combinations of the charged or neutral gauginos and higgsinos, called charginos and neutralinos, respectively (sometimes collectively called electroweakinos). The neutralinos are Majorana fermions, which can lead to some distinctive phenomenological signatures [91,92]. The superpartners of the quarks and leptons are spin-zero bosons: the squarks, charged sleptons, and sneutrinos, respectively. A complete set of Feynman rules for the sparticles of the MSSM can be found in Refs. [93,94]. The MSSM Feynman rules also are implicitly contained in a number of amplitude generation and Feynman diagram software packages (see e.g., Refs. [95–97]).

It should be noted that all mass formulae quoted below in this Section are tree-level results. Radiative loop corrections will modify these results and must be included in any precision study of supersymmetric phenomenology [98]. Beyond tree level, the definition of the supersymmetric parameters becomes convention-dependent. For example, one can define physical couplings or running couplings, which differ beyond the tree level. This provides a challenge to any effort that attempts to extract supersymmetric parameters from data. The SUSY Les Houches Accord (SLHA) [80,99] has been adopted, which establishes a set of conventions for specifying generic file structures for supersymmetric model specifications and input parameters, supersymmetric mass and coupling spectra, and decay tables. These provide a universal interface between spectrum calculation programs, decay packages, and high energy physics event generators.

### 88.4.1 The charginos and neutralinos

The mixing of the charged gauginos ($\tilde{W}^{\pm}$) and charged higgsinos ($\tilde{H}_u^+$ and $\tilde{H}_d^-$) is described (at tree-level) by a $2 \times 2$ complex mass matrix [100, 101],

$$ M_C = \begin{pmatrix} M_2 & \frac{1}{\sqrt{2}} g v_u \\ \frac{1}{\sqrt{2}} g v_d & \mu \end{pmatrix}. \qquad (88.6) $$

To determine the physical chargino states and their masses, one must perform a singular value decomposition [102] of the complex matrix $M_C$ [25, 103]:

$$ U^* M_C V^{-1} = \mathrm{diag}(M_{\tilde{\chi}_1^+}, M_{\tilde{\chi}_2^+}), \qquad (88.7) $$

where $U$ and $V$ are unitary matrices, and the right-hand side of Eq. (88.7) is the diagonal matrix of (real non-negative) chargino masses. Explicit formulae for the singular value decomposition of $M_C$ can be found in Ref. [104]. The physical chargino states are denoted by $\tilde{\chi}_1^{\pm}$ and $\tilde{\chi}_2^{\pm}$. These are linear combinations of the charged gaugino and higgsino states determined by the matrix elements of $U$ and $V$ [100, 101]. The chargino masses correspond to the singular values [102] of $M_C$, i.e., the positive square roots of the eigenvalues of $M_C^\dagger M_C$:

$$ M_{\tilde{\chi}_1^+, \tilde{\chi}_2^+}^2 = \frac{1}{2} \left\{ \begin{aligned} & |\mu|^2 + |M_2|^2 + 2m_W^2 \\ & \mp \sqrt{\left(|\mu|^2 + |M_2|^2 + 2m_W^2\right)^2 - 4|\mu M_2 - m_W^2 \sin 2\beta|^2} \end{aligned} \right\}, \qquad (88.8) $$

in a convention where $v_u$ and $v_d$ are real and positive, and where the states are ordered such that $M_{\tilde{\chi}_1^+} \le M_{\tilde{\chi}_2^+}$. The relative phase of $\mu^*$ and $M_2$ is physical and potentially observable [105].

The mixing of the neutral gauginos ($\tilde{B}$ and $\tilde{W}^0$) and neutral higgsinos ($\tilde{H}_d^0$ and $\tilde{H}_u^0$) is described (at tree-level) by a $4 \times 4$ complex symmetric mass matrix [100, 101],

$$ M_N = \begin{pmatrix} M_1 & 0 & -\frac{1}{2}g'v_d & \frac{1}{2}g'v_u \\ 0 & M_2 & \frac{1}{2}gv_d & -\frac{1}{2}gv_u \\ -\frac{1}{2}g'v_d & \frac{1}{2}gv_d & 0 & -\mu \\ \frac{1}{2}g'v_u & -\frac{1}{2}gv_u & -\mu & 0 \end{pmatrix}. \qquad (88.9) $$

To determine the physical neutralino states and their masses, one must perform an Autonne-Takagi factorization [102,106] (also called Takagi diagonalization in Refs. [25,103,107]) of the complex symmetric matrix $M_N$:

$$ W^T M_N W = \mathrm{diag}(M_{\tilde{\chi}_1^0}, M_{\tilde{\chi}_2^0}, M_{\tilde{\chi}_3^0}, M_{\tilde{\chi}_4^0}), \qquad (88.10) $$

where $W$ is a unitary matrix (which is called $N^{-1}$ in Refs. [21,30]) and the right-hand side of Eq. (88.10) is the diagonal matrix of (real non-negative) neutralino masses. The physical neutralino states are denoted by $\tilde{\chi}_i^0$ (for $i = 1, \dots, 4$), where the states are ordered such that $M_{\tilde{\chi}_1^0} \le M_{\tilde{\chi}_2^0} \le M_{\tilde{\chi}_3^0} \le M_{\tilde{\chi}_4^0}$. The $\tilde{\chi}_i^0$ are the linear combinations of the neutral gaugino and higgsino states determined by the matrix elements of $W$. The neutralino masses correspond to the singular values of $M_N$, i.e., the positive square roots of the eigenvalues of $M_N^\dagger M_N$. Exact formulae for these masses can be found in Refs. [108, 109]. A numerical algorithm for determining the mixing matrix $W$ has been given in Ref. [110].

If a chargino or neutralino state approximates a particular gaugino or higgsino state, it is convenient to employ the corresponding nomenclature. Specifically, if $|M_1|$ and $|M_2|$ are small compared to $m_Z$ and $|\mu|$, then the lightest neutralino $\tilde{\chi}_1^0$ would be nearly a pure photino, $\tilde{\gamma}$, the superpartner of the photon. If $|M_1|$ and $m_Z$ are small compared to $|M_2|$ and $|\mu|$, then the lightest neutralino would be nearly a pure bino, $\tilde{B}$, the superpartner of the weak hypercharge gauge boson. If $|M_2|$ and $m_Z$ are small compared to $|M_1|$ and $|\mu|$, then the lightest chargino pair and neutralino would constitute a triplet of roughly mass-degenerate pure winos, $\tilde{W}^{\pm}$, and $\tilde{W}_d^0$, the superpartners of the weak SU(2) gauge bosons. Finally, if $|\mu|$ and $m_Z$ are small compared to $|M_1|$ and $|M_2|$, then the lightest chargino pair and neutralino would be nearly pure higgsino states, the superpartners of the Higgs bosons. Each of the above cases leads to a strikingly different phenomenology.

In the NMSSM, an additional Higgs singlet superfield is added to the MSSM. This superfield comprises two real Higgs scalar degrees of freedom and an associated neutral higgsino degree of freedom. Consequently, there are five neutralino mass eigenstates that are obtained by a Takagi-diagonalization of the $5 \times 5$ neutralino mass matrix. In many cases, the fifth neutralino state is dominated by its SU(2)×U(1) singlet component, and thus is very weakly coupled to the SM particles and their superpartners.