<figure><img src="image_60.png" /><figcaption>**Figure 14.10: Left:** Status of the 2+2 oscillation scenarios from Ref. [201] ($\eta_S = \sum_i |U_{is}|^2$ where $i$ runs over the two massive states mostly relevant for solar neutrino oscillations). In the figure also shown are the values of $\chi_{PC}^2$ and $\chi_{PG}^2$ relevant for parameter consistency test and parameter goodness of fit, respectively. **Right:** Present status of 3+1 oscillation scenarios from Ref. [202].</figcaption></figure>

These anomalies are under study by the experimental community with a set of follow-up measurements performed at SBL both at reactors and accelerators (see the corresponding subsections in Sec.14.6.4 and Sec.14.6.3).

As mentioned in Sec.14.1, whatever the extension of the SM we want to consider, it must contain only three light active neutrinos. Therefore if we need more than three light massive neutrinos, we must add sterile neutrinos to the particle content of the model.

The most immediate question as these anomalies were reported was whether they could all be consistently described in combination with the rest of the neutrino data – in particular with the negative results on the disappearance of $\nu_\mu$ at short distances – if one adds those additional sterile states. Quantitatively one can start by adding a fourth massive neutrino state to the spectrum and perform a global data analysis to answer this question. Although the answer is always the same, the physical reason behind it depends on ordering assumed for the states. In brief, there are six possible four-neutrino schemes that can, in principle, accommodate the results of solar+KamLAND and atmospheric+LBL neutrino experiments as well as the SBL result. They can be divided into two classes: (2+2) and (3+1). In the (3+1) schemes, there is a group of three close-by neutrino masses (as on the 3$\nu$ schemes described in the previous section) that is separated from the fourth one by a gap of the order of 1 eV, which is responsible for the SBL oscillations. In (2+2) schemes, there are two pairs of close masses (one pair responsible for solar results and the other for atmospheric [203]) separated by the $O(eV)$ gap. The main difference between these two classes is the following: if a (2+2)-spectrum is realized in nature, the transition into the sterile neutrino is a solution of either the solar or the atmospheric neutrino problem, or the sterile neutrino takes part in both. Consequently, a (2+2)-spectrum is easier to test because the required mixing of sterile neutrinos in either solar and/or atmospheric oscillations would modify their effective matter potential in the Sun and in the Earth, giving distinctive effects in the solar and/or atmospheric neutrino observables. Those distinctive effects were not observed so oscillations into sterile neutrinos did not describe well either solar or atmospheric data. Consequently, as soon as the early 2000's 2+2 spectra could be ruled out already beyond 3-4 $\sigma$ as seen in the left panel in Fig.14.10 taken from Ref. [201].

On the contrary, for a (3+1)-spectrum (and more generally for a 3+N-spectrum with an arbitrary N number of sterile states), the sterile neutrino(s) could be only slightly mixed with the active ones and mainly provide a description of the SBL results. In

this case, the oscillation probabilities for experiments working at $E/L \sim 1 \text{ eV}^2$ take a simple form:

$$P_{\alpha\alpha} = 1 - \sin^2 2\theta_{\alpha\alpha} \sin^2 \Delta, \quad P_{\mu e} = \sin^2 2\theta_{\mu e} \sin^2 \Delta, \quad (14.87)$$

where $\Delta \equiv \Delta m_{41}^2 L/4E$ and one can define effective mixing angles

$$\sin^2 2\theta_{\alpha\alpha} \equiv 4|U_{\alpha 4}|^2(1 - |U_{\alpha 4}|^2), \quad \sin^2 2\theta_{\mu e} \equiv 4|U_{\mu 4}|^2|U_{e 4}|^2. \quad (14.88)$$

In here $\alpha = e, \mu$ and $U_{\alpha 4}$ are the elements of the lepton mixing matrix describing the mixing of the 4th neutrino mass state with the electron and muon flavour. In this scenario, there is no sensitivity to CP violation in the $\Delta$ driven oscillations, so the relations above are valid for both neutrinos and antineutrinos. At linear order in the mixing elements one can derive a relation between the amplitudes of appearance and disappearance probabilities:

$$4 \sin^2 2\theta_{\mu e} \approx \sin^2 2\theta_{ee} \sin^2 2\theta_{\mu\mu}. \quad (14.89)$$

This relation implies a constraint between the possible results in disappearance and appearance experiments. Consequently, it is not trivial to find a consistent description to all the SBL anomalies. Over the years, different groups have performed a variety of such global analyses leading to different quantitative conclusions on the statistical quality of the global fit (see for example [202, 204–208], see also Refs. [209, 210] for recent reviews on the subject). Generically, the results of the global analysis show that there is significant tension between groups of different data sets – in particular between appearance and disappearance results – and Eq.(14.89) makes it difficult to obtain a good global fit as illustrated in the right panel in Fig.14.10 taken from Ref. [202] which concluded that 3+1 scenario is excluded at $4.7\sigma$ level.

A straightforward question to ask is whether the situation improves if more neutrino states at the eV scale are introduced. The simplest extension is the introduction of 2 states with eV scale mass splittings, $\nu_4$ and $\nu_5$. The ordering of the states can be such that $\Delta m_{41}^2$ and $\Delta m_{51}^2$ are both positive (“3+2”), or one of them is negative (“1+3+1”). From the point of view of the description of the data, the most important new qualitative feature is that now, non-zero CP violation at $E/L \sim \text{eV}^2$ is possibly observable [206, 211–213]. This allows some additional freedom in fitting neutrino versus antineutrino data from LSND and MiniBooNE together. However, it still holds that a non-zero $\nu_\mu \to \nu_e$ appearance at SBL necessarily predicts SBL disappearance for both $\nu_e$ and $\nu_\mu$. So, generically, the tension between appearance and
disappearance results remains, though differences in the methodology of statistical quantification of the degree of agreement/disagreement in these scenarios can lead to different conclusions on whether they can provide a successful description of all the data [202,209,210]. Cosmological observations can provide complementary information on the number of relativistic neutrino states in thermal equilibrium in the early Universe and on the sum of their masses, which sets further constraints on light sterile neutrino scenarios (see Chapter 26 of this Review, “Neutrinos in cosmology”).

## 14.9 Laboratory Probes of ν Mass Scale and its Nature

As described in Secs.14.4 and 14.5, neutrino flavour oscillations in vacuum and flavour transitions in matter only depend on the differences between the neutrino masses-squared, $\Delta m_{ij}^2$, and on the mixing matrix elements, $U_{ij}$. However, they are insensitive to the absolute mass scale for the neutrinos, $m_i$. They also give us no information on whether they are Dirac or Majorana particles.

Clearly, the observation of flavour oscillations implies a lower bound on the mass of the heavier neutrino in $\Delta m_{ij}^2$, $|m_i| \ge \sqrt{\Delta m_{ij}^2}$ for $\Delta m_{ij}^2 > 0$. However, there is no upper bound on $m_i$. In particular, oscillation results allow the neutrino spectrum to be approximately degenerate at a mass scale that is much higher than the $\sqrt{\Delta m_{ij}^2}$ that they determine. Information of the mass scale of the neutrino is provided by other types of experiments. Here we briefly summarize the most sensitive laboratory probes of the neutrino mass scale and on whether they are Dirac or Majorana particles. Cosmological observations provide, albeit indirectly, complementary information on the neutrino mass scale as it is reviewed in Chapter 26 of this Review, “Neutrinos in cosmology”.

### 14.9.1 Constraints from Kinematics of Weak Decays

The only model-independent information on the neutrino masses, rather than mass differences, can be extracted from energy-momentum conservation relation in reactions in which a neutrino or an antineutrino is involved.

Historically these bounds were labeled as limits on the mass of the flavour neutrino states corresponding to the charged flavour involved in the decay. Fermi proposed in 1933 such a kinematic search for the $\nu_e$ neutrino mass (which we will label here as $m_{\nu_e}^{\text{eff}}$) in the end part of the beta spectra in $^3\text{H}$ beta decay $^3\text{H} \to ^3\text{He} + e^- + \bar{\nu}_e$.

Because $^3\text{H}$ beta decay is a superallowed transition, the nuclear matrix elements are energy independent, so the electron spectrum is determined exclusively by the phase space

$$ \begin{aligned} \frac{dN}{dE} &= C p E (Q-T) \sqrt{(Q-T)^2 - (m_e^{\text{eff}})^2} F(E) \\ &\equiv R(E) \sqrt{(E_0 - E)^2 - (m_{\nu_e}^{\text{eff}})^2}. \end{aligned} \quad (14.90) $$

$E_0$ is the mass difference between the initial and final nucleus, $E = T + m_e$ is the total electron energy, $p$ its momentum, $Q \equiv E_0 - m_e$ is the maximum kinetic energy of the electron and Final state Coulomb interactions are contained in the Fermi function $F(E)$. $R(E)$ in the second equality contains all the $m_{\nu_e}$-independent factors.

The Kurie function is defined as $K(T) \equiv \sqrt{\frac{dN}{dE} \frac{1}{pE F(E)}}$. From Eq.(14.90), we see that if $m_{\nu_e}^{\text{eff}}=0$ $K(T)$ would depend linearly on $T$. A non-vanishing neutrino mass then provokes a distortion from the straight-line $T$-dependence at the endpoint, So for $m_{\nu_e}^{\text{eff}} = 0$, $T_{\text{max}} = Q$, while for $m_{\nu_e}^{\text{eff}} \neq 0$, $T_{\text{max}} = Q - m_{\nu_e}^{\text{eff}}$. In $^3\text{H}$ beta decay $Q = 18.6$ KeV is very small and therefore, this decay is more sensitive to this $m_{\nu_e}^{\text{eff}}$-induced distortion.

The most recent result on the kinematic search for neutrino mass in tritium decay is from KATRIN [214], an experiment which has found so far no indication of $m_{\nu_e} \neq 0$ and sets an upper limit

$$ m_{\nu_e}^{\text{eff}} < 0.8 \text{ eV}, \quad (14.91) $$

at 90% CL improving over the previous bound from the Mainz [215] and Troitsk [216] experiments which constrained

$m_{\nu_e}^{\text{eff}} < 2.2$ eV at 95% CL. KATRIN continues running with an estimated sensitivity limit of $m_{\nu_e}^{\text{eff}} \sim 0.2$ eV. Project 8 is exploring a new technique for $\beta$-spectrometry based on cyclotron radiation [217].

An alternative isotope to Tritium is $^{163}\text{Ho}$ [218] which presents the advantage of a smaller $Q = 2.8$ KeV. It decays via electron capture to $^{163}\text{Dy}$. Currently, there are three experiments exploring this decay to probe the neutrino mass: ECHO [219], HOLMES [220], and NuMECS [221]. These experiments are complementary to tritium-based searches from a technical point of view. Also, the decay of $^{163}\text{Ho}$ determines the effective electron neutrino mass as opposed to antineutrino in Tritium.

For the other flavours the present limits compiled in the listing section of the PDG read

$$ m_{\nu_{\mu}}^{\text{eff}} < 190 \text{ keV (90\% CL) from } \pi^{-} \to \mu^{-} + \bar{\nu}_{\mu}, \quad (14.92) $$

$$ m_{\nu_{\tau}}^{\text{eff}} < 18.2 \text{ MeV (95\% CL) from } \tau^{-} \to n\pi + \nu_{\tau}. \quad (14.93) $$

In the presence of mixing and for neutrinos with small mass differences, the distortion of the beta spectrum is given by the sum of the individual spectra generated incoherently by each neutrino massive state weighted with the relevant mixing matrix element squared [222]:

$$ \frac{dN}{dE} = R(E) \sum_i |U_{ei}|^2 \sqrt{(E_0 - E)^2 - m_i^2} \Theta(E_0 - E - m_i). \quad (14.94) $$

The step function $\Theta(E_0 - E - m_i)$ arises because a neutrino with a given mass $m_i$ can only be produced if the available energy is larger than its mass. Equation (14.94) shows the two main effects of the neutrino masses and mixings on the electron energy spectrum: First, kinks appear at the electron energies $E_e^{(i)} = E \sim E_0 - m_i$ with sizes that are determined by $|U_{ei}|^2$. Second, the endpoint shifts to $E_{ep} = E_0 - m_0$, where $m_0$ is the lightest neutrino mass. Corrections are induced once the energy resolution of the experiment is considered. [223,224]

In the 3-$\nu$ mixing scenario, the distortion of the spectrum can still be effectively described by a single parameter – which we will still denote as $m_{\nu_e}$ – if for all neutrino states $E_0 - E = Q - T \gg m_i$. In this case, one can expand Eq.(14.94) as:

$$ \frac{dN}{dE} \approx R(E) \sum_i |U_{ei}|^2 \sqrt{(E_0 - E)^2 - (m_{\nu_e}^{\text{eff}})^2}, \quad (14.95) $$

with

$$ (m_{\nu_e}^{\text{eff}})^2 = \frac{\sum_i m_i^2 |U_{ei}|^2}{\sum_i |U_{ei}|^2} = \sum_i m_i^2 |U_{ei}|^2, \quad (14.96) $$

where unitarity is assumed in the second equality. In this approximation, the distortion of the endpoint of the spectrum is described by a single parameter, and with the present results from KATRIN, it is bounded to be

$$ \begin{aligned} 1.1 \text{ eV} \ge m_{\nu_e}^{\text{eff}} &= \sqrt{\sum_i m_i^2 |U_{ei}|^2} \\ &= \begin{cases} \sqrt{m_0^2 + \Delta m_{21}^2 (1 - c_{13}^2 c_{12}^2) + \Delta m_{32}^2 s_{13}^2} & \text{in NO,} \\ \sqrt{m_0^2 + \Delta m_{21}^2 c_{13}^2 c_{12}^2 - \Delta m_{32}^2 c_{13}^2} & \text{in IO,} \end{cases} \end{aligned} \quad (14.97) $$

where $m_0 = m_1$ ($m_3$) is the lightest neutrino mass in the NO (IO) spectrum. Correspondingly the bounds in Eqs.(14.92) and (14.93) apply to the combinations $\sum_i m_i^2 |U_{\alpha i}|^2$ for $\alpha = \mu$ and $\tau$ respectively. So with the values known of the mixing matrix elements, the strongest constraint on the absolute value of the neutrino mass comes from Tritium beta decay.

From Eq.(14.97) we see that, given the present knowledge of the neutrino mass differences and their mixing from oscillation experiments, it is possible to translate the experimental information of $m_{\nu_e}$ on a corresponding range for the lightest neutrino mass and
<figure><img src="image_61.png" /><figcaption>**Figure 14.11:** Allowed 95% CL ranges (1 dof) for the neutrino mass observable determined in ³H beta decay (left panel) and in 0νββ (right panel) in the framework of 3ν mixing as a function of the lightest neutrino mass. The ranges are obtained by projecting the results of the global analysis of oscillation data (w/o SK-atm) in Ref. [180,181]. The region for each ordering is defined with respect to its local minimum.</figcaption></figure>

that such relation depends on the ordering of the states. We plot in Fig.14.11 the recasting of the allowed regions of the analysis in Ref. [180,181] in terms of the allowed range $m_{\nu_e}$ as a function of $m_{\text{light}} \equiv m_0$. In particular, one finds that the results of oscillation experiments imply a lower bound on $m_{\nu_e} > 0.048$ (0.0085) eV for IO (NO) at 95% CL.

<figure><img src="image_62.png" /><figcaption>**Figure 14.12:** Feynman diagram for neutrinoless double-beta decay.</figcaption></figure>

### 14.9.2 Dirac vs. Majorana: Neutrinoless Double-beta Decay

The most sensitive probe to whether neutrinos are Dirac or Majorana states is the neutrinoless double beta decay (0νββ):

$$ (A, Z) \rightarrow (A, Z + 2) + e^{-} + e^{-}. \qquad (14.98) $$

In the presence of neutrino masses and mixing, the process in Eq.(14.98) can be generated at lower order in perturbation theory by the term represented in Fig.14.12. The corresponding amplitude is proportional to the product of the two leptonic currents

$$ M_{\alpha\beta} \propto [\bar{e}\gamma_{\alpha}(1-\gamma_5)\nu_e] [\bar{e}\gamma_{\beta}(1-\gamma_5)\nu_e] \\ \propto \sum_i (U_{ei})^2 [\bar{e}\gamma_{\alpha}(1-\gamma_5)\nu_i] [\bar{e}\gamma_{\beta}(1-\gamma_5)\nu_i]. \quad (14.99) $$

$$ m_{ee} = \left| \sum_i m_i U_{ei}^2 \right| \\ = \begin{cases} \left| m_0 c_{12}^2 c_{13}^2 + \sqrt{\Delta m_{21}^2 + m_0^2 s_{12}^2 c_{13}^2} e^{2i(\eta_2 - \eta_1)} + \sqrt{\Delta m_{32}^2 + \Delta m_{21}^2 + m_0^2 s_{13}^2} e^{-2i(\delta_{CP} + \eta_1)} \right| & \text{in NO,} \quad (14.101) \\ \left| m_0 s_{13}^2 + \sqrt{m_0^2 - \Delta m_{32}^2} s_{12}^2 c_{13}^2 e^{2i(\eta_2 + \delta_{CP})} + \sqrt{m_0^2 - \Delta m_{32}^2 - \Delta m_{21}^2 c_{12}^2 c_{13}^2} e^{2i(\eta_1 + \delta_{CP})} \right| & \text{in IO,} \end{cases} $$

The neutrino propagator in Fig.14.12 can only arise from the contraction $\langle 0 | \nu_i(x) \nu_i(y)^T | 0 \rangle$. However, if the neutrino is a Dirac particle $\nu_i$ field annihilates a neutrino state and creates an antineutrino state, and neutrino and antineutrino states are different, so the contraction $\langle 0 | \nu_i(x) \nu_i(y)^T | 0 \rangle = 0$ and $M_{\alpha\beta} = 0$. On the other hand, if $\nu_i$ is a Majorana particle, neutrino and antineutrino are described by the same field and $\langle 0 | \nu_i(x) \nu_i(y)^T | 0 \rangle \neq 0$.

The conclusion is that in order to induce the 0νββ decay, neutrinos must be Majorana particles. This is consistent with the fact that the process (14.98) violates the total lepton number by two units. Conversely, if 0νββ decay is observed, massive neutrinos cannot be Dirac states [225].

It is important to stress that neutrinoless double-beta decay could be dominantly induced by other new physics effects beyond that of Majorana neutrino masses. Consequently, the connection between the observation or limitation of the neutrinoless double beta decay and the neutrino mass can only be made under some assumption about the source of total lepton number violation in the model.

The observable determined by the experiments is the half-life of the decay. Under the assumption that the Majorana neutrino mass is the only source of lepton number violation at low energies, the decay half-life is given by:

$$ (T_{1/2}^{0\nu})^{-1} = G^{0\nu} |M^{0\nu}|^2 \left(\frac{m_{ee}}{m_e}\right)^2, \qquad (14.100) $$

where $G^{0\nu}$ is the phase space integral taking into account the final atomic state, $|M^{0\nu}|$ is the nuclear matrix element of the transition, and $m_{ee}$ is the effective Majorana mass of $\nu_e$,
which, in addition to the masses and mixing parameters that affect the tritium beta decay spectrum, also depends on the lep- tonic CP-violating phases. We plot in Fig.14.11 the recasting of the allowed regions of the analysis in Ref. [180,181] in terms of the allowed range $m_{ee}$ as a function of $m_{\text{light}} \equiv m_0$ for the two orderings. As a consequence of the dependence on the unknown Majorana phases, the allowed range of $m_{ee}$ for a given value of $m_{\text{light}}$ and ordering is substantially broader than that of $m_{\nu_e}$. Nevertheless, the results of oscillation experiments imply a lower bound on the effective Majorana mass for the IO, which at 95%CL reads $m_{ee} > 0.016$ eV.

From Eq.(14.100) we see that nuclear structure details enter relation between the decay rate (or lifetime) and the effective Ma- jorana mass. As a consequence, uncertainties in the nuclear struc- ture calculations result in a spread of $m_{ee}$ values for a given $T_{1/2}^{0\nu}$ by a factor of 2–3 [226].

We present in Sec.14.9.3 a brief description of the experimental searches for neutrinoless double-beta decay. At the time of writ- ing of this review, the strongest bound on $0\nu\beta\beta$ decay lifetimes came for Xenon and Germanium from the KamLAND-Zen [227] and GERDA [228] experiments, respectively. They have set a bound on the corresponding half-lives of $T_{1/2}^{0\nu} > 2.3 \times 10^{26}$ yr, and $T_{1/2}^{0\nu} > 1.8 \times 10^{26}$ yr, at 90% CL. Using a variety of nuclear matrix element calculations, the corresponding upper bounds on the effective Majorana mass are

$$m_{ee} < 36 - 156 \text{ meV} \qquad (14.102)$$

$$m_{ee} < 79 - 180 \text{ meV}. \qquad (14.103)$$

### 14.9.3 Experimental Search for Neutrinoless Double-beta Decay

There are 35 candidate nuclei for double-beta decay. Currently, experiments using $^{136}$Xe and $^{76}$Ge have reported the most sensi- tive results of $0\nu\beta\beta$ search. Because of the uncertainties related to the nuclear matrix element, the complementarity of technologies, different backgrounds, and the investigation of the mechanism be- hind the $0\nu\beta\beta$ in case of a positive signal, it is important to pursue the searches with as many isotopes as possible.

The signature of $0\nu\beta\beta$ is that the sum of the energy of two electrons is equal to the Q-value of the nuclear transition. The energy from electrons is measured with either ionization, scintil- lation, or through phonons. In some experiments, a combination of two techniques is used to enhance the sensitivity. In the case of background-free measurement, the sensitivity to the half-life is proportional to the product of the detection efficiency of the signal $\epsilon$, the source mass $M$, and the measurement time $t$. If background exists, it is proportional to $\epsilon \sqrt{\frac{Mt}{b\Delta E}}$, where $b$ is the background rate, and $\Delta E$ is the energy resolution.

Among the experiments using ionization detection, ultra-high- purity germanium detector provides the best sensitivity thanks to high energy resolution and low background. GERDA achieved a background level of $(5.2^{+1.6}_{-1.3}) \times 10^{-4}$ counts/(keV·kg·year) [228], which enabled a background-free search. The final result from GERDA for the half-life limit for $0\nu\beta\beta$ decay of $^{76}$Ge is $T_{1/2} > 1.8 \times 10^{26}$ years at 90% CL. The Majorana Demonstrator achieved energy resolution of 2.5 keV FWHM (full width at half maxi- mum) at the Q-value (2.039 MeV) [229] and set a half-life limit of $T_{1/2} > 8.3 \times 10^{25}$ years at 90% CL. GERDA and Majorana col- laborations have united and formed, together with new groups, the LEGEND collaboration. LEGEND has adopted a phased ap- proach toward the deployment of a tonne of enriched Ge detec- tors [230]. The first phase of LEGEND based on approximately 200 kg of enriched Ge, using the GERDA infrastructure at LNGS, has started and first data are being taken. The second phase of LEGEND has been approved as a one ton experiment in a new cryostat reaching a discovery sensitivity of half-lifes beyond $10^{28}$ years. In China, CDEX-300ν using 225 kg of enriched Ge detector is under preparation [231] in the existing CDEX-DM infrastruc- ture at the China Jinping Underground Laboratory.

Liquid scintillator detectors have a simple structure and can utilize existing large detectors with low background environments. By adding an inner balloon to contain xenon-loaded liquid scintil- lator to the KamLAND detector, KamLAND-Zen uses 745 kg of

xenon with 90.9% enrichment in $^{136}$Xe. KamLAND-Zen reported the half-life limit of $2.3 \times 10^{26}$ years at 90% CL [227]. The SNO detector has also been upgraded to be filled with a liquid scin- tillator in the SNO+ experiment [232]. The SNO+ detector will be loaded with 0.5% natural tellurium, corresponding to approx- imately 1.3 t of $^{130}$Te, to search for $0\nu\beta\beta$.

With a time projection chamber, one can utilize both ioniza- tion and scintillation. EXO-200 used a liquid xenon time projec- tion chamber with enriched Xe and reported a half-life limit of $T_{1/2} > 3.5 \times 10^{25}$ years at 90% CL [233] for $^{136}$Xe. Based on the technology validated by EXO-200, nEXO plans to use about 5,000 kg of enriched xenon [234]. NEXT has been developing a high-pressure xenon gas time projection chamber with electrolu- minescent amplification and optical readouts. An energy resolu- tion of 1% FWHM at the Q-value of $^{136}$Xe $0\nu\beta\beta$ is demonstrated with NEXT-White detector [235].

CUORE uses a cryogenic bolometer to measure the energy in a calorimetric way. The detector composed of 988 TeO$_2$ bolome- ters for a total mass of 742 kg, corresponding to 206 kg of $^{130}$Te, was operated at 10 mK. CUORE set a half-life limit of $T_{1/2} > 2.2 \times 10^{25}$ years at 90% CL for $^{130}$Te [236]. For fur- ther reduction of background towards future search based on the CUORE technology, CUPID proposes to simultaneously measure the calorimetric signal and the scintillation light with a tonne- scale bolometric detector. Using the prototype the technology is demonstrated, and $0\nu\beta\beta$ is also searched for with $^{82}$Se [237] and $^{100}$Mo [238].

AMORE also uses the simultaneous detection of heat and scin- tillation. Six $^{100}$Mo-enriched and $^{48}$Ca-depleted CaMoO$_4$ crys- tals with a total mass of 1.9 kg (AMORE-Pilot) were operated in Yangyang underground laboratory located in South Korea, searching for $0\nu\beta\beta$ of $^{100}$Mo [239]. Currently, AMORE-I experi- ment [240] is under operation in Yemilab.

A tracker-calorimeter technique is employed in NEMO. Source isotopes are hosted in thin foils surrounded by a tracking detec- tor, which in turn is surrounded by a calorimeter. Full topological event reconstruction with this configuration enables background rejection and gives additional information after discovery. The NEMO-3 experiment used 7 isotopes, with the largest mass com- prised of $^{100}$Mo (7 kg) [241].

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15. Quark Model

Revised August 2023 by C. Amsler (Stefan Meyer Inst.), V. Crede (Florida State U.) and T. DeGrand (Colorado U., Boulder).

<table><tbody><tr><td>15.1</td><td>Introduction</td><td>322</td></tr><tr><td>15.2</td><td>Quantum numbers of the quarks</td><td>322</td></tr><tr><td>15.3</td><td>Mesons</td><td>323</td></tr><tr><td>15.4</td><td>Exotic mesons</td><td>326</td></tr><tr><td>15.4.1</td><td>Tetraquarks</td><td>326</td></tr><tr><td>15.4.2</td><td>Glueballs</td><td>326</td></tr><tr><td>15.4.3</td><td>Hybrids</td><td>327</td></tr><tr><td>15.5</td><td>Baryons: qqq states</td><td>327</td></tr><tr><td>15.5.1</td><td>Light baryons</td><td>327</td></tr><tr><td>15.5.2</td><td>Charmed and bottom baryons</td><td>330</td></tr><tr><td>15.6</td><td>Magnetic moments</td><td>332</td></tr><tr><td>15.7</td><td>Dynamics</td><td>333</td></tr><tr><td>15.8</td><td>Lattice Calculations of Hadronic Spectroscopy</td><td>334</td></tr><tr><td>15.8.1</td><td>Spectroscopy of low-lying states</td><td>334</td></tr><tr><td>15.8.2</td><td>Excited state spectroscopy</td><td>334</td></tr><tr><td>15.8.3</td><td>Electromagnetic effects</td><td>337</td></tr></tbody></table>

15.1 Introduction

Quantum chromodynamics (QCD) is the theory of strong interactions. QCD is a quantum field theory with an $SU(N_c)$ local “color” gauge symmetry with $N_c = 3$ colors and a collection of $N_f$ “flavors” of colored fermions, the quarks. It involves a set of $N_c^2 - 1 = 8$ non-Abelian gauge fields, the gluons. QCD is believed to confine, that is, its physical states are color singlets with internal quark and gluon degrees of freedom. This review is concerned with the description of the properties (masses and matrix elements for couplings to electromagnetism and the weak interactions) of the low lying bound states of QCD. The shorthand expression for describing this physics is called the “quark model”.

The spectrum of strongly interacting particles consists of a tower of many states, which can be either bosons (labelled as “mesons”) or fermions (labelled as “baryons”). The spectrum of baryons and mesons exhibits a high degree of regularity. The organizational principle which best categorizes this regularity is encoded in the quark model. All descriptions of strongly interacting states use the language of the quark model. At the same time, the language is not precise. The quark model exists on many levels: at the simplest level, it is an almost dynamics-free picture of strongly interacting particles as bound states of quarks and antiquarks. As one refines the description, the quark model can become a framework with more detailed descriptions of dynamics. At its most fundamental level, it might be a description of QCD. In effective field theories of QCD at low energies the hadron spectrum is not simply given by a series of states but is the manifestation of a complex dynamics involving various types of states [1].

At its heart, the quark model assumes that mesons are bound states of quark - antiquark pairs, and baryons are bound states of three quarks. These are the minimal particle content states which can be color singlets in an $SU(3)$ gauge theory. This approach cannot be justified directly from QCD; however, there is indirect evidence that this description has some fundamental validity from the version of QCD where the number of colors $N_c$ is taken to infinity [2–4]. In that limit, mesons are dominantly narrow (width proportional to $1/N_c$) bound states of a quark - antiquark pair, and baryons have a mass which scales as $N_c$.

A better justification is that this approach works. Indeed, the quark model is much older than QCD as a theory of the strong interactions (1973-1974, see the article on “50 Years of Quantum Chromodynamics” [5]). In fact, the principal issue (circa 1963-64) in strong interaction physics before QCD was to justify the success of the quark model in systemizing the properties of mesons and baryons in terms of some more fundamental dynamics (QCD).

Today one knows that this is not the whole story. There are experimentally observed states which either cannot be described, or have an uncomfortable description, as minimal quark number states. Some of them have “exotic” (non-$\bar{q}q$ or $qqq$) quantum

numbers. Given the successes of the quark model, these are classified as “tetraquarks” ($qq\bar{q}\bar{q}$), “pentaquarks” ($qqqq\bar{q}$) or “glueballs”, bound states of gluons, the gluonic degrees of freedom in QCD. Of course, such labels are imprecise: bound states with the same overall quantum numbers can mix, regardless of their internal degrees of freedom.

This review has several parts. We start by describing the properties of strongly interacting particles in terms of the properties of states made of a minimal number of quark fields which can be coupled into a color singlet – two fields (a quark and an antiquark) for the mesons, and three quarks, for a baryon. Quarks come in six flavors. We describe the properties of mesons as $\bar{q}q$ systems and baryons as $qqq$ systems. Along the way we discuss hadronic bound states which do not fit into this classification.

Finally, at the end of this review, we present results from lattice simulations of QCD, a direct approach to the solution of QCD from its Lagrangian, without reference to models. Lattice simulations interact with the quark model in (at least) two ways: first, the interpolating fields which are used in lattice simulations are usually directly based on quark model constructions. That is the simplest way to create states with the desired quantum numbers, which can then be processed by the lattice calculation. The second way that lattice calculations interact with the quark model comes when one wishes to put the lattice calculations into some context: without the quark model, there are simply the results from the lattice calculations, and the results from experiment, and no way to understand why they are similar or different. The quark model is the framework which is almost universally used to generate that context. Of course, that statement is equally valid when one tries to systemize actual experimental data: the context is always some variation of a quark model.

15.2 Quantum numbers of the quarks

As gluons carry no intrinsic quantum numbers beyond color charge, and because color is believed to be permanently confined, the quantum numbers of strongly interacting particles are given by the quantum numbers of their constituent quarks and antiquarks.

Quarks are strongly interacting fermions with spin $1/2$ and, by convention, positive parity. Antiquarks have negative parity. Quarks have the additive baryon number $1/3$, antiquarks $-1/3$. Table 15.1 gives the other additive quantum numbers (flavors) for the three generations of quarks. They are related to the charge $Q$ (in units of the elementary charge $e$) through the generalized Gell-Mann-Nishijima formula

$$Q = I_z + \frac{\mathcal{B} + \mathcal{S} + \mathcal{C} + \mathcal{B} + \mathcal{T}}{2}, \quad (15.1)$$

where $\mathcal{B}$ is the baryon number. The convention is that the quark flavor ($I_z$, $S$, $C$, $B$, or $\mathcal{T}$) has the same sign as its charge $Q$. Antiquarks have the opposite flavor signs. With this convention, any flavor carried by a charged meson has the same sign as its charge, e.g., the strangeness of the $K^+$ ($c\bar{s}$) is +1, the bottomness of the $B^+$ ($u\bar{b}$) is +1, and the charm and strangeness of the $D_s^-$ ($s\bar{c}$) are each -1.

<table><caption>Table 15.1: Quark quantum numbers.</caption><thead><tr><th></th><th>d</th><th>u</th><th>s</th><th>c</th><th>b</th><th>t</th></tr></thead><tbody><tr><td>Q - electric charge</td><td>-1/3</td><td>+2/3</td><td>-1/3</td><td>+2/3</td><td>-1/3</td><td>+2/3</td></tr><tr><td>I - isospin</td><td>1/2</td><td>1/2</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>I<sub>z</sub> - isospin z-component</td><td>-1/2</td><td>+1/2</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>S - strangeness</td><td>0</td><td>0</td><td>-1</td><td>0</td><td>0</td><td>0</td></tr><tr><td>C - charm</td><td>0</td><td>0</td><td>0</td><td>+1</td><td>0</td><td>0</td></tr><tr><td>B - bottomness</td><td>0</td><td>0</td><td>0</td><td>0</td><td>-1</td><td>0</td></tr><tr><td>T - topness</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>+1</td></tr></tbody></table>

The hypercharge is defined as

$$Y = \mathcal{B} + \mathcal{S} - \frac{\mathcal{C} - \mathcal{B} + \mathcal{T}}{3}. \quad (15.2)$$

Thus $Y$ is equal to $1/3$ for the $u$ and $d$ quarks, $-2/3$ for the $s$ quark, and $0$ for all other quarks. More details and derivations on
the quark structure of mesons and baryons can be found e.g. in Ref. [6].

The naming scheme for hadrons has been updated by the Particle Data Group in 2023, to include the heavy “exotic” states discovered recently, the tetraquark (qqq̄q̄) and pentaquark (qqqqq̄) candidates (see “Naming Scheme for Hadrons” in this Review).

## 15.3 Mesons

Mesons have baryon number $B = 0$. In the quark model, they are $q\bar{q}'$ bound states of quarks $q$ and antiquarks $\bar{q}'$ (the flavors of $q$ and $q'$ may be different).

If the orbital angular momentum of the $q\bar{q}'$ state is $\ell$, then the parity $P$ is $(-1)^{\ell+1}$. The meson spin $J$ is given by the usual relation $|\ell - s| \le J \le |\ell + s|$, where $s = 0$ (antiparallel quark spins) or $s = 1$ (parallel quark spins). The charge conjugation, or C-parity $C = (-1)^{\ell+s}$, is defined only for states made of quarks and their own antiquarks. The C-parity can be generalized to the G-parity $G = (-1)^{I+\ell+s}$ for mesons made of quarks and their own antiquarks (isospin $I_z = 0$), and for the charged $u\bar{d}$ and $d\bar{u}$ states (isospin $I = 1$).

The mesons are classified in $J^{PC}$ multiplets. The $\ell = 0$ states are the pseudoscalars ($0^{-+}$) and the vectors ($1^{--}$). The orbital excitations $\ell = 1$ are the scalars ($0^{++}$), the axial vectors ($1^{++}$) and ($1^{+-}$) aka pseudovectors, and the tensors ($2^{++}$). Assignments for many of the known mesons are given in Tables 15.2, 15.3 and 15.4. Radial excitations are denoted by the principal quantum number $n$. The very short lifetime of the $t$ quark ($\to W^+b$) makes it likely that bound-state hadrons containing $t$ quarks and/or antiquarks do not exist.

States in the natural spin-parity series $P = (-1)^J$ must, according to the above, have $s = 1$ and hence, $CP = +1$. Thus, mesons with natural spin-parity and $CP = -1$ ($0^{++}$, $1^{+-}$, $2^{+-}$, $3^{-+}$, etc.) are forbidden in the $q\bar{q}'$ model. The $J^{PC} = 0^{--}$ state is forbidden as well. Mesons with such exotic quantum numbers may exist, but would lie outside the $q\bar{q}'$ model (see section 15.4 below on exotic mesons).

Following SU(3), the nine possible $q\bar{q}'$ combinations containing the light $u, d$, and $s$ quarks are grouped into an octet and a singlet of light quark mesons:

$$3 \otimes \bar{3} = 8 \oplus 1. \qquad (15.3)$$

A fourth quark such as charm $c$ can be included by extending SU(3) to SU(4). However, SU(4) is badly broken owing to the much heavier $c$ quark. Nevertheless, in an SU(4) classification, the sixteen mesons are grouped into a 15-plet and a singlet:

$$4 \otimes \bar{4} = 15 \oplus 1. \qquad (15.4)$$

The weight diagrams for the ground-state pseudoscalar ($0^{-+}$) and vector ($1^{--}$) mesons are depicted in Fig. 15.1. The light quark mesons are members of nonets building the middle plane in Fig. 15.1(a) and (b).

Isoscalar states with the same $J^{PC}$ mix, but mixing between the two light quark isoscalar mesons, and the much heavier charmonium and bottomonium states, are generally assumed to be negligible. In the following, we shall use the generic names $a$ for the $I = 1, K$ for the $I = 1/2, f$ and $f'$ for the $I = 0$ members of the light quark nonets. Thus, the physical isoscalars are mixtures of the SU(3) wave function $\psi_8$ and $\psi_1$:

$$f' = \psi_8 \cos\theta - \psi_1 \sin\theta, \qquad (15.5)$$

$$f = \psi_8 \sin\theta + \psi_1 \cos\theta, \qquad (15.6)$$

where $\theta$ is the nonet mixing angle and

$$\psi_8 = \frac{1}{\sqrt{6}}(u\bar{u} + d\bar{d} - 2s\bar{s}), \qquad (15.7)$$

$$\psi_1 = \frac{1}{\sqrt{3}}(u\bar{u} + d\bar{d} + s\bar{s}). \qquad (15.8)$$

The mixing relations are often rewritten to exhibit the $u\bar{u} + d\bar{d}$ and $s\bar{s}$ components which decouple for the “ideal” mixing angle

<figure><img src="image_63.png" /><figcaption>**Figure 15.1:** SU(4) weight diagram showing the 16-plets for the pseudoscalar (a) and vector mesons (b) made of the $u, d, s$, and $c$ quarks as a function of isospin $I_z$, charm $C$, and hypercharge $Y = B + S - \frac{C}{3}$. The nonets of light mesons occupy the central planes to which the $c\bar{c}$ states have been added.</figcaption></figure>

$\theta_i$, such that $\tan \theta_i = 1/\sqrt{2}$ (or $\theta_i = 35.3^\circ$). Defining $\alpha = \theta + 54.7^\circ$, one obtains the physical isoscalar state in the flavor basis

$$f' = \frac{1}{\sqrt{2}}(u\bar{u} + d\bar{d}) \cos\alpha - s\bar{s} \sin\alpha, \qquad (15.9)$$

and its orthogonal partner $f$ (replace $\alpha$ by $\alpha - 90^\circ$). Thus for ideal mixing ($\alpha_i = 90^\circ$), the $f'$ becomes pure $s\bar{s}$ and the $f$ pure $u\bar{u} + d\bar{d}$. The mixing angle $\theta$ can be derived by diagonalizing the mass matrix

$$\begin{pmatrix} m_8 & m_{81} \\ m_{18} & m_1 \end{pmatrix}. \qquad (15.10)$$

The mass eigenvalues are $m_f$ and $m_f'$. The mixing angle is given by

$$\tan \theta = \frac{m_8 - m_{f'}}{m_{81}}. \qquad (15.11)$$

Calculating $m_8$ and $m_{81}$ from the wave functions Eq. (15.7) and Eq. (15.8), and expressing the quark masses as a function of the $I = 1/2$ and $I = 1$ meson masses, one obtains

$$\tan \theta = \frac{4m_K - m_a - 3m_{f'}}{2\sqrt{2}(m_a - m_K)}, \qquad (15.12)$$

which also determines the sign of $\theta$. Alternatively, one can express the mixing angle as a function of all nonet masses. The octet mass is given by

$$m_8 = m_{f'} \cos^2 \theta + m_f \sin^2 \theta \qquad (15.13)$$

whence

$$\tan^2 \theta = \frac{4m_K - m_a - 3m_{f'}}{-4m_K + m_a + 3m_f}. \qquad (15.14)$$

Eliminating $\theta$ from Eq. (15.12) and Eq. (15.14) leads to the sum rule [14]

$$(m_f + m_{f'})(4m_K - m_a) - 3m_f m_{f'} = 8m_K^2 - 8m_K m_a + 3m_a^2. \qquad (15.15)$$

This relation is verified for the ground-state vector mesons. We identify the $\phi(1020)$ with the $f'$ and the $\omega(783)$ with the $f$. Thus

$$\phi(1020) = \psi_8 \cos\theta_V - \psi_1 \sin\theta_V, \qquad (15.16)$$

$$\omega(782) = \psi_8 \sin\theta_V + \psi_1 \cos\theta_V, \qquad (15.17)$$
**Table 15.2:** Suggested $q\bar{q}$ quark-model assignments for the lightest mesons made of $u$, $d$ and $s$ quarks. Mesons in boldface are included in the Summary Table. The wave functions $f$ and $f'$ are given in the text (Eq. (15.9)) and the singlet-octet mixing angles in Table 15.5 below for the well established nonets. The classification of the $0^{++}$ mesons is controversial: (i) the scalars $a_0(980)$, $K_0^*(700)$, $f_0(980)$ and $f_0(500)$, omitted from the table, are often considered to be four-quark states, but are also proposed for the ground state scalar nonet (see the chapter “Scalar Mesons below 1 GeV” in this Review); (ii) there are three isoscalar $0^{++}$ mesons, $f_0(1370)$, $f_0(1500)$ and $f_0(1710)$, both $f_0(1500)$ and $f_0(1710)$ being proposed as glueballs. The three states are expected to mix. The isoscalar assignments in the $2^1S_0$ ($0^{-+}$) nonet are also tentative. Details and alternative schemes can be found in “Spectroscopy of Light Meson Resonances” in this Review.

a The $1^{+\pm}$ and $2^{-\pm}$ isospin $\frac{1}{2}$ states mix. In particular, the $K_{1A}$ and $K_{1B}$ are nearly equal mixtures of the $K_1(1270)$ and $K_1(1400)$ [7]. The $2^{-\pm}$ mixed partner of the $K_2(1770)$ is the established $K_2(1820)$.

b The physical vector mesons are mixtures of $1^3D_1$ and $2^3S_1$ (for a discussion see [8]).

c This state has also been proposed as a tetraquark state [9].

d The $\eta(1475)$ and $\eta(1405)$ (not shown) may be manifestations of a single state [10].

e This state has also been proposed as the ground state tensor glueball [11, 12].

<table><thead><tr><th>n<sup>2s+1</sup>ℓ<sub>J</sub></th><th>J<sup>PC</sup></th><th>l = 1</th><th>l = 1/2</th><th>l = 0</th><th>l = 0</th></tr><tr><td></td><td></td><td>u<span style="text-decoration: overline">d</span>, u<span style="text-decoration: overline">d</span>,<br>1/√2(d<span style="text-decoration: overline">d</span> - u<span style="text-decoration: overline">u</span>)</td><td>u<span style="text-decoration: overline">s</span>, d<span style="text-decoration: overline">s</span>;<br>d<span style="text-decoration: overline">s</span>, u<span style="text-decoration: overline">s</span></td><td>f'</td><td>f</td></tr></thead><tbody><tr><td>1<sup>1</sup>S<sub>0</sub></td><td>0<sup>-+</sup></td><td>π</td><td>K</td><td>η</td><td>η'(958)</td></tr><tr><td>1<sup>3</sup>S<sub>1</sub></td><td>1<sup>--</sup></td><td>ρ(770)</td><td>K<sup>*</sup>(892)</td><td>φ(1020)</td><td>ω(782)</td></tr><tr><td>1<sup>3</sup>P<sub>0</sub></td><td>0<sup>++</sup></td><td>a<sub>0</sub>(1450)</td><td>K<sub>0</sub><sup>*</sup>(1430)</td><td>f<sub>0</sub>(1370, 1500, 1710)</td><td></td></tr><tr><td>1<sup>1</sup>P<sub>1</sub></td><td>1<sup>+-</sup></td><td>b<sub>1</sub>(1235)</td><td>K<sub>1B</sub><sup>a</sup></td><td>h<sub>1</sub>(1415)</td><td>h<sub>1</sub>(1170)</td></tr><tr><td>1<sup>3</sup>P<sub>2</sub></td><td>1<sup>++</sup></td><td>a<sub>1</sub>(1260)</td><td>K<sub>1A</sub><sup>a</sup></td><td>f<sub>1</sub>(1420)</td><td>f<sub>1</sub>(1285)</td></tr><tr><td>1<sup>3</sup>D<sub>1</sub></td><td>1<sup>--</sup></td><td>a<sub>2</sub>(1320)</td><td>K<sub>2</sub><sup>*</sup>(1430)</td><td>f'<sub>2</sub>(1525)</td><td>f<sub>2</sub>(1270)</td></tr><tr><td>1<sup>1</sup>D<sub>2</sub></td><td>2<sup>-+</sup></td><td>π<sub>2</sub>(1670)</td><td>K<sub>2</sub><sup>*</sup>(1770)<sup>a</sup></td><td>φ(2170)<sup>c</sup></td><td>ω(1650)</td></tr><tr><td>1<sup>3</sup>D<sub>3</sub></td><td>3<sup>--</sup></td><td>ρ<sub>3</sub>(1690)</td><td>K<sub>3</sub><sup>*</sup>(1780)</td><td>φ<sub>3</sub>(1850)</td><td>ω<sub>3</sub>(1670)</td></tr><tr><td>1<sup>3</sup>F<sub>4</sub></td><td>4<sup>++</sup></td><td>a<sub>4</sub>(1970)</td><td>K<sub>4</sub><sup>*</sup>(2045)</td><td>f<sub>4</sub>(2300)</td><td>f<sub>4</sub>(2050)</td></tr><tr><td>1<sup>3</sup>G<sub>5</sub></td><td>5<sup>--</sup></td><td>ρ<sub>5</sub>(2350)</td><td>K<sub>5</sub><sup>*</sup>(2380)</td><td></td><td></td></tr><tr><td>2<sup>1</sup>S<sub>0</sub></td><td>0<sup>-+</sup></td><td>π(1300)</td><td>K(1460)</td><td>η(1475)<sup>d</sup></td><td>η(1295)</td></tr><tr><td>2<sup>3</sup>S<sub>1</sub></td><td>1<sup>--</sup></td><td>ρ(1450)</td><td>K<sup>*</sup>(1410)<sup>b</sup></td><td>φ(1680)</td><td>ω(1420)</td></tr><tr><td>2<sup>3</sup>P<sub>1</sub></td><td>1<sup>++</sup></td><td>a<sub>1</sub>(1640)</td><td>K<sub>1</sub>(1650)</td><td></td><td></td></tr><tr><td>2<sup>3</sup>P<sub>2</sub></td><td>2<sup>++</sup></td><td>a<sub>2</sub>(1700)</td><td>K<sub>2</sub><sup>*</sup>(1980)</td><td>f<sub>2</sub>(1950)<sup>e</sup></td><td>f<sub>2</sub>(1640)</td></tr><tr><td>2<sup>1</sup>D<sub>2</sub></td><td>2<sup>-+</sup></td><td>π<sub>2</sub>(1880)</td><td></td><td></td><td></td></tr><tr><td>3<sup>1</sup>S<sub>0</sub></td><td>0<sup>-+</sup></td><td>π(1800)</td><td>K(1830)</td><td></td><td>η(1760)</td></tr></tbody></table>

**Table 15.3:** $c\bar{c}$ quark-model assignments for the charmonium mesons with established $J^{PC}$ and their corresponding open charm mesons. Mesons in boldface are included in the Meson Summary Table. The open flavor states in the $1^{+-}$ and $1^{++}$ rows are mixtures of the $1^{+\pm}$ states.

a In unitarized chiral perturbation theory this meson splits into two states at 2105 and 2451 MeV [1, 13].

b The masses are considerably smaller than most theoretical predictions. They have also been considered as tetraquark or $D^{(*)}K$ molecular states.

c This meson splits into two states at 2247 and 2555 MeV [1].

d Mixtures of the $1^3D_1$ and $2^3S_1$ states.

<table><thead><tr><th>n<sup>2s+1</sup>ℓ<sub>J</sub></th><th>J<sup>PC</sup></th><th>l = 0</th><th>l = 1/2</th><th>l = 0</th></tr><tr><td></td><td></td><td>c<span style="text-decoration: overline">c</span></td><td>c<span style="text-decoration: overline">u</span>, c<span style="text-decoration: overline">d</span>;<br>c<span style="text-decoration: overline">u</span>, c<span style="text-decoration: overline">d</span></td><td>c<span style="text-decoration: overline">s</span>;<br>c<span style="text-decoration: overline">s</span></td></tr></thead><tbody><tr><td>1<sup>1</sup>S<sub>0</sub></td><td>0<sup>-+</sup></td><td>η<sub>c</sub>(1S)</td><td>D</td><td>D<sup>±</sup></td></tr><tr><td>1<sup>3</sup>S<sub>1</sub></td><td>1<sup>--</sup></td><td>J/ψ(1S)</td><td>D<sup>*</sup></td><td>D<sup>*</sup><sub>s</sub><sup>±</sup></td></tr><tr><td>1<sup>3</sup>P<sub>0</sub></td><td>0<sup>++</sup></td><td>χ<sub>c0</sub>(1P)</td><td>D<sub>0</sub><sup>*</sup>(2300)<sup>a</sup></td><td>D<sup>*</sup><sub>s0</sub>(2317)<sup>±b</sup></td></tr><tr><td>1<sup>3</sup>P<sub>1</sub></td><td>1<sup>++</sup></td><td>χ<sub>c1</sub>(1P)</td><td>D<sub>1</sub>(2430)<sup>c</sup></td><td>D<sub>s1</sub>(2460)<sup>±b</sup></td></tr><tr><td>1<sup>1</sup>P<sub>1</sub></td><td>1<sup>+-</sup></td><td>h<sub>c</sub>(1P)</td><td>D<sub>1</sub>(2420)</td><td>D<sub>s1</sub>(2536)<sup>±</sup></td></tr><tr><td>1<sup>3</sup>P<sub>2</sub></td><td>2<sup>++</sup></td><td>χ<sub>c2</sub>(1P)</td><td>D<sub>2</sub><sup>*</sup>(2460)</td><td>D<sup>*</sup><sub>s2</sub>(2573)<sup>±</sup></td></tr><tr><td>2<sup>1</sup>S<sub>0</sub></td><td>0<sup>-+</sup></td><td>η<sub>c</sub>(2S)</td><td>D<sub>0</sub>(2550)<sup>0</sup></td><td>D<sub>s0</sub>(2590)<sup>+</sup></td></tr><tr><td>2<sup>3</sup>S<sub>1</sub></td><td>1<sup>--</sup></td><td>ψ(2S)</td><td>D<sub>1</sub><sup>*</sup>(2600)<sup>0</sup></td><td>D<sup>*</sup><sub>s1</sub>(2700)<sup>±d</sup></td></tr><tr><td>1<sup>3</sup>D<sub>1</sub></td><td>1<sup>--</sup></td><td>ψ(3770)</td><td>D<sub>1</sub><sup>*</sup>(2760)<sup>0</sup></td><td>D<sup>*</sup><sub>s1</sub>(2860)<sup>±d</sup></td></tr><tr><td>1<sup>3</sup>D<sub>2</sub></td><td>2<sup>--</sup></td><td>ψ<sub>2</sub>(3823)</td><td>D<sub>2</sub>(2740)<sup>0</sup></td><td></td></tr><tr><td>2<sup>3</sup>P<sub>1</sub></td><td>0<sup>++</sup></td><td>χ<sub>c0</sub>(3860)</td><td></td><td></td></tr><tr><td></td><td>2<sup>++</sup></td><td>χ<sub>c2</sub>(3930)</td><td></td><td></td></tr><tr><td>3<sup>3</sup>S<sub>1</sub></td><td>1<sup>--</sup></td><td>ψ(4040)</td><td></td><td></td></tr><tr><td>2<sup>3</sup>D<sub>1</sub></td><td>1<sup>--</sup></td><td>ψ(4160)</td><td></td><td></td></tr><tr><td>4<sup>3</sup>S<sub>1</sub></td><td>1<sup>--</sup></td><td>ψ(4415)</td><td></td><td></td></tr><tr><td>1<sup>3</sup>D<sub>3</sub></td><td>3<sup>--</sup></td><td>ψ<sub>3</sub>(3842)</td><td>D<sub>3</sub><sup>*</sup>(2750)</td><td>D<sup>*</sup><sub>s3</sub>(2860)<sup>±</sup></td></tr></tbody></table>
<table><caption>**Table 15.4:** $\bar{b}\bar{b}$ quark-model assignments for the bottomonium mesons with established $J^{PC}$ and their corresponding open bottom mesons.</caption><thead><tr><th>n<sup>2s+1</sup>ℓ<sub>J</sub></th><th>J<sup>PC</sup></th><th>I = 0<br>b̄b</th><th>I = 1/2<br>b̄u, bd;<br>bu, bd</th><th>I = 0<br>b̄s;<br>bs</th><th>I = 0<br>b̄c;<br>bc</th></tr></thead><tbody><tr><td>1<sup>1</sup>S<sub>0</sub></td><td>0<sup>-+</sup></td><td>η<sub>b</sub>(1S)</td><td>B</td><td>B<sup>0</sup><sub>s</sub></td><td>B<sup>±</sup><sub>c</sub></td></tr><tr><td>1<sup>3</sup>S<sub>1</sub></td><td>1<sup>--</sup></td><td>Υ(1S)</td><td>B<sup>*</sup></td><td>B<sup>*</sup><sub>s</sub></td><td></td></tr><tr><td>1<sup>3</sup>P<sub>0</sub></td><td>0<sup>++</sup></td><td>χ<sub>b0</sub>(1P)</td><td></td><td></td><td></td></tr><tr><td>1<sup>3</sup>P<sub>1</sub></td><td>1<sup>++</sup></td><td>χ<sub>b1</sub>(1P)</td><td></td><td></td><td></td></tr><tr><td>1<sup>1</sup>P<sub>1</sub></td><td>1<sup>--</sup></td><td>h<sub>b</sub>(1P)</td><td>B<sub>1</sub>(5721)</td><td>B<sub>s1</sub>(5830)<sup>0</sup></td><td></td></tr><tr><td>1<sup>3</sup>P<sub>2</sub></td><td>2<sup>++</sup></td><td>χ<sub>b2</sub>(1P)</td><td>B<sup>*</sup><sub>2</sub>(5747)</td><td>B<sup>*</sup><sub>s2</sub>(5840)<sup>0</sup></td><td></td></tr><tr><td>2<sup>1</sup>S<sub>0</sub></td><td>0<sup>-+</sup></td><td>η<sub>b</sub>(2S)</td><td></td><td></td><td>B<sub>c</sub>(2S)<sup>±</sup></td></tr><tr><td>2<sup>3</sup>S<sub>1</sub></td><td>1<sup>--</sup></td><td>Υ(2S)</td><td></td><td></td><td></td></tr><tr><td>1<sup>3</sup>D<sub>2</sub></td><td>2<sup>--</sup></td><td>Υ<sub>2</sub>(1D)</td><td></td><td></td><td></td></tr><tr><td>2<sup>3</sup>P<sub>J</sub></td><td>0, 1, 2<sup>++</sup></td><td>χ<sub>b0,1,2</sub>(2P)</td><td></td><td></td><td></td></tr><tr><td>2<sup>1</sup>P<sub>1</sub></td><td>1<sup>--</sup></td><td>h<sub>b</sub>(2P)</td><td></td><td></td><td></td></tr><tr><td>3<sup>3</sup>S<sub>1</sub></td><td>1<sup>--</sup></td><td>Υ(3S)</td><td></td><td></td><td></td></tr><tr><td>3<sup>3</sup>P<sub>J</sub></td><td>0, 1, 2<sup>++</sup></td><td>χ<sub>b1,2</sub>(3P)</td><td></td><td></td><td></td></tr><tr><td>4<sup>3</sup>S<sub>1</sub></td><td>1<sup>--</sup></td><td>Υ(4S)</td><td></td><td></td><td></td></tr></tbody></table>

with the vector mixing angle $\theta_V = 36.5^\circ$ from Eq. (15.14), very close to ideal mixing. Thus $\phi(1020)$ is nearly pure $s\bar{s}$. For ideal mixing, Eq. (15.12) and Eq. (15.14) lead to the relations

$$m_K = \frac{m_f + m_{f'}}{2}, \quad m_a = m_f, \qquad (15.18)$$

which are satisfied for the vector mesons.

The situation for the pseudoscalar and scalar mesons is not so clear cut, either theoretically or experimentally. For the pseudoscalars, the mixing angle is small. This can be understood qualitatively via gluon-line counting of the mixing process. The size of the mixing process between the nonstrange and strange mass bases scales as $\alpha_s^2$, not $\alpha_s^3$, because of two rather than three gluon exchange as it does for the vector mesons. It may also be that the lightest isoscalar pseudoscalars mix more strongly with excited states or with states of substantial non-$\bar{q}q$ content, as will be discussed below.

In fact a large mixing from hadronic loops is expected for scalar mesons, no matter what model is assumed for $\bar{q}q$ pair production [15]. A variety of analysis methods lead to similar results: First, for these states, Eq. (15.15) is satisfied only approximately. Then Eq. (15.12) and Eq. (15.14) lead to somewhat different values for the mixing angle. Identifying the $\eta$ with the $f'$ one gets

$$\eta = \psi_8 \cos \theta_P - \psi_1 \sin \theta_P, \qquad (15.19)$$

$$\eta' = \psi_8 \sin \theta_P + \psi_1 \cos \theta_P. \qquad (15.20)$$

Following chiral perturbation theory, the meson masses in the mass formulae (Eq. (15.12) and Eq. (15.14)) might be replaced by their squares. Table 15.5 lists the mixing angle $\theta_{\text{lin}}$ from Eq. (15.14) (using the neutral members of the nonets) and the corresponding $\theta_{\text{quad}}$ obtained by replacing the meson masses by their squares throughout. The mixing angles in the $1^{--}$, $2^{++}$ and $3^{--}$ nonets are not far from ideal, while larger $s\bar{s}-(u\bar{u}+d\bar{d})$ mixing is predicted from hadronic loops in the $0^{++}$, $0^{-+}$ and $1^{+\pm}$ nonets [15].

The pseudoscalar mixing angle $\theta_P$ can also be measured by comparing the partial widths for radiative $J/\psi$ decay into a vector and a pseudoscalar [16], radiative $\phi(1020)$ decay into $\eta$ and $\eta'$ [17], radiative decays between pseudoscalar and vector mesons [18], or $\bar{p}\bar{p}$ annihilation at rest into a pair of vector and pseudoscalar or into two pseudoscalars [19, 20]. One obtains a mixing angle between $-10^\circ$ and $-20^\circ$. More recently, a lattice QCD simulation, Ref. [21], has successfully reproduced the masses of the $\eta$ and $\eta'$, and as a byproduct find a mixing angle $\theta_{\text{lin}} = -14.1(2.8)^\circ$. We return to this point in Sec. 15.8.

The nonet mixing angles can be measured in $\gamma\gamma$ collisions, e.g., for the $0^{-+}$, $0^{++}$, and $2^{++}$ nonets. In the quark model, the amplitude for the coupling of neutral mesons to two photons is proportional to $\sum_i Q_i^2$, where $Q_i$ is the charge of the $i$-th quark.

<table><caption>**Table 15.5:** Singlet-octet mixing angles for the well established nonets from the linear mass formula Eq. (15.14) and its quadratic version in which the masses are squared. The $1^{++}$ and $1^{+-}$ nonet mixing angles depend on the mixing angle $\theta_{K_1}$ between $K_{1A}$ and $K_{1B}$. The recommended values are $\sim 23^\circ$ and $\sim 28^\circ$ for $1^{++}$ and $1^{+-}$, respectively, with $\theta_{K_1} \sim 35^\circ$ [7].</caption><thead><tr><th>n<sup>2s+1</sup>ℓ<sub>J</sub></th><th>J<sup>PC</sup></th><th>θ<sub>quad</sub><br>[°]</th><th>θ<sub>lin</sub><br>[°]</th></tr></thead><tbody><tr><td>1<sup>1</sup>S<sub>0</sub></td><td>0<sup>-+</sup></td><td>-11.3</td><td>-24.5</td></tr><tr><td>1<sup>3</sup>S<sub>1</sub></td><td>1<sup>--</sup></td><td>39.2</td><td>36.5</td></tr><tr><td>1<sup>3</sup>P<sub>2</sub></td><td>2<sup>++</sup></td><td>29.6</td><td>28.0</td></tr><tr><td>1<sup>3</sup>D<sub>3</sub></td><td>3<sup>--</sup></td><td>31.8</td><td>30.8</td></tr></tbody></table>

The $2\gamma$ partial width of an isoscalar meson with mass $m$ is then given in terms of the mixing angle $\alpha$ by

$$\Gamma_{2\gamma} = C(5 \cos \alpha - \sqrt{2} \sin \alpha)^2 m^3, \qquad (15.21)$$

for $f'$ and $f$ ($\alpha \to \alpha - 90^\circ$). The coupling $C$ may depend on the meson mass. It is often assumed to be a constant in the nonet. For the isovector $a$, one finds $\Gamma_{2\gamma} = 9 C m^3$. Thus the members of an ideally mixed nonet couple to $2\gamma$ with partial widths in the ratios $f: f' : a = 25 : 2 : 9$. For tensor mesons, one finds from the ratios of the measured $2\gamma$ partial widths for the $f_2(1270)$ and $f'_2(1525)$ mesons a mixing angle $\alpha_T$ of $(81 \pm 1)^\circ$, or $\theta_T = (27 \pm 1)^\circ$, in accord with the linear mass formula. For the pseudoscalars, one finds from the ratios of partial widths $\Gamma(\eta' \to 2\gamma)/\Gamma(\eta \to 2\gamma)$ a mixing angle $\theta_P = (-18 \pm 2)^\circ$, while the ratio $\Gamma(\eta' \to 2\gamma)/\Gamma(\pi^0 \to 2\gamma)$ leads to $\sim -24^\circ$. SU(3) breaking effects for pseudoscalars are discussed in [22].

The partial width $\Gamma$ for the decay of a scalar or a tensor meson into a pair of pseudoscalar mesons is model-dependent. Following Ref. [23],

$$\Gamma = C \times \gamma^2 \times |F(q)|^2 \times q, \qquad (15.22)$$

where $C$ is a nonet constant, $q$ the momentum of the decay products, $F(q)$ a form factor, and $\gamma^2$ the SU(3) coupling. Details and explicit expressions for $\gamma^2$ and $F(q)$ are given in “Spectroscopy of Light Meson Resonances”. The decay of a $q\bar{q}$ meson into a pair of mesons involves the creation of a $q\bar{q}$ pair, and SU(3) symmetry assumes that the matrix elements for the creation of $s\bar{s}$, $u\bar{u}$, and $d\bar{d}$ pairs are equal. An excellent fit to the tensor meson decay widths is obtained assuming SU(3) symmetry and a pseudoscalar mixing angle $\theta_P \simeq -17^\circ$ [23].

The analysis of resonances is complicated by the presence of thresholds such as $K\bar{K}$ in $a_0(980) \to \eta\pi$ or $f_0(980) \to \pi\pi$, which
affect the resonance masses and widths. A particularly nasty kinematic effect is the triangle singularity in which one of the primary decay daughters in turn decays and is emitted backwards, catching up and scattering with the second primary product. This mechanism generates fake peaks in the final state and has been proposed as alternative explanation for several meson (and baryon) signals. Prominent examples are the $\eta(1475) \rightarrow K^{*}(\rightarrow K\pi)K$ and $\eta(1410) \rightarrow a_0(980)(\rightarrow \eta\pi)\pi$ signals which could be due to one single state [10] (as shown in Table 15.2).

Furthermore, to parametrize resonances, many data analyses resort to Breit-Wigner amplitudes which are not suitable to describe broad interfering mesons with the same quantum numbers, as they violate unitarity. This is in particular the case in the mass range above 1500 MeV where radial and orbital excitations, e.g. of broad scalar and tensor mesons, start to accumulate. A better approach is to determine the T-matrix poles in the complex plane, where resonances are usually located on the second Riemann sheet.

## 15.4 Exotic mesons

There are two classes of colorless exotic mesons allowed in QCD: multiquark states (section 15.4.1) and mesons with active gluons, bound states of gluons (the glueballs, section 15.4.2) and $q\bar{q}$ states with “valence” gluons (the hybrids, section 15.4.3). The first class contains the simplest system consisting of two quarks and two antiquarks (tetraquarks). Compact tetraquarks (also called “diquonium” in the older literature) are made of diquarks and antidiquarks. Hadroquarkonia consist of a pair of heavy quark and antiquark in a compact core surrounded by a light-quark cloud. In hadronic “molecules” the building blocks are color-neutral hadrons bound by meson exchanges.

### 15.4.1 Tetraquarks

The existence of a light nonet composed of four quarks (tetraquarks) with masses below 1 GeV was suggested a long time ago [24,25]. Coupling two triplets of light quarks u, d, and s, one obtains nine states, of which the six symmetric (uu, dd, ss, ud + du, us + su, ds + sd) form the six dimensional representation 6, while the three antisymmetric (ud - du, us - su, ds - sd) form the three dimensional representation $\overline{3}$ of SU(3):

$$ \mathbf{3} \otimes \mathbf{3} = \mathbf{6} \oplus \overline{3}. \qquad (15.23) $$

Hence for tetraquarks one gets the reduction

$$ \begin{aligned} \mathbf{3} \otimes \mathbf{3} \otimes \overline{3} \otimes \overline{3} \\ = \mathbf{6} \oplus \overline{3} \otimes \overline{6} \otimes \mathbf{3} \\ = \overline{3} \otimes \mathbf{3} \oplus \mathbf{6} \otimes \overline{6} \oplus \mathbf{6} \otimes \mathbf{3} \oplus \overline{3} \otimes \overline{6} \\ = \mathbf{9} \oplus \mathbf{36} \oplus \mathbf{18} \oplus \overline{18}. \end{aligned} \qquad (15.24) $$

Combining with spin and color and requiring antisymmetry for diquarks and antidiquarks, one finds for ground states (zero angular momenta) that the most deeply bound tetraquarks (and hence the lightest ones) form a nonet and are scalar mesons (see also [6]). The masses are estimated to be below 900 MeV. The strange quark determines the mass splittings and one obtains a mass inverted spectrum with a light isosinglet ($f_0(500)$), a medium heavy isodoublet ($K_0^*(700)$) and a heavy isotriplet ($a_0(980)$) + isosinglet ($f_0(980)$). In alternative schemes these states build the lightest $q\bar{q}$ scalar nonet (for details see “Scalar mesons below 1 GeV” in this “Review”).

A plethora of new heavy multiquark candidates have been reported in the charmonium spectrum (Fig. 15.2). Details and references can be found in the review “Heavy non-$q\bar{q}$ Mesons”, see also Refs. [26–28].

The most prominent one is the $\chi_{c1}(3872)$ (was X(3872)), first observed by BELLE in 2003 in B-decays in the final state $J/\psi\pi^+\pi^-$ [29] and firmly established by many experiments (Babar, BESIII, CDF, D0, LHCb) in several production modes ($e^+e^-$, $\Lambda_b$ decay, $\bar{p}\bar{p}$, $p\bar{p}$). This narrow state ($< 1$ MeV) decays mainly into $D^0\bar{D}^{0*}$ and is right at the $D^0\bar{D}^{0*}$ threshold. Its composition is believed to be mostly molecular, bound by pion exchange. Its existence was already predicted in 1991 [30]. Many non-$c\bar{c}$ candidates have been reported since then above the open charm threshold, with quantum numbers compatible with $c\bar{c}$, but which are not expected in the pure charmonium spectrum. In addition, one observes isovector (charged) mesons decaying into $c\bar{c}$ states plus a charged pion or kaon. (They are listed in this Review under “Other mesons”.) For example, the $1^{+-} T_{cc1}(3900)^{+}$ (was $Z_c(3900)^{+}$) and $T_{c\bar{c}s1}(4000)^{+}$ (was $Z_{cs}(4000)^{-}$) cannot be pure charmonia, but are compatible with $c\bar{c}ud$ and $c\bar{c}s\bar{u}$, respectively. They could belong to the same nonet, hence identifying the other multiplet members might shed light on the structure of such states.

<figure><img src="image_64.png" /><figcaption>Figure 15.2: Mass spectrum with known $I^G(J^{PC})$ observed in the charmonium region. The $D\bar{D}$ line shows the open charm threshold. The well-known $c\bar{c}$ mesons are shown in black. (The $1^1D_2$ $\eta_{c2}$, expected around 3800 MeV, has not been seen yet.) The established additional observed states are shown in red, and the states needing confirmation (i.e. omitted from the Summary Tables) are labeled in green. The blue color denotes exotic isovectors.</figcaption></figure>

Even more remarkable are the observations by LHCb of a doubly charmed state $T_{cc}(3875)^{+}$ ($c\bar{c}u\bar{d}$) [31] at the $D_0D_0^*$ threshold and of a $T_{c\bar{c}\bar{c}}(6900)$ [32] decaying into a pair of $J/\psi(1S)$. Similar states begin to emerge also in the bottomonium spectrum, such as the $1^{+-} T_{b\bar{b}1}(10610)^{+}$ (was $Z_b(10610)^{+}$) and $T_{b\bar{b}1}(10650)^{+}$ (was $Z_b(10650)^{+}$), both $b\bar{b}u\bar{d}$. Many of these states may be hadronic modules made of charge conjugated pairs of mesons such as $D^{(*)}$, $D_s^{(*)}$ or their excitations, and their B and $B^*$ counterparts. They could also be mimicked by kinematical effects such as triangle singularities.

### 15.4.2 Glueballs

QCD predicts the existence of extra isoscalar mesons which cannot be addressed by the quark model. In the pure gauge theory they contain only gluons. The ground state glueball is predicted by lattice gauge theories to be $0^{++}$, the first excited state $2^{++}$. Errors on the mass predictions are large. From Ref. [33] one obtains 1750 (50) (80) MeV for the mass of the lightest $0^{++}$ glueball from quenched QCD. As an example for the glueball mass spectrum, we show in Fig. 15.3 a calculation from [34]. A mass of 1710 MeV is predicted for the ground state, also with an error of about 100 MeV. Earlier work by other groups produced masses at 1650 MeV [35] and 1550 MeV [36] (see also [37]). The first excited charm threshold, with quantum numbers compatible with $c\bar{c}$, but which are not expected in the pure charmonium spectrum. In addition, one observes isovector (charged) mesons decaying into $c\bar{c}$ states plus a charged pion or kaon. (They are listed in this Review under “Other mesons”.) For example, the $1^{+-} T_{cc1}(3900)^{+}$ (was $Z_c(3900)^{+}$) and $T_{c\bar{c}s1}(4000)^{+}$ (was $Z_{cs}(4000)^{-}$) cannot be pure charmonia, but are compatible with $c\bar{c}ud$ and $c\bar{c}s\bar{u}$, respectively. They could belong to the same nonet, hence identifying the other multiplet members might shed light on the structure of such states.

### 15.4.3 Glueballs
state has a mass of about 2.4 GeV, and the lightest glueball with exotic quantum numbers ($2^{+ -}$) has a mass of about 4 GeV.

These calculations are made in the so-called “quenched approximation” which neglects $q\bar{q}$ loops. However, both glue and $q\bar{q}$ states couple to singlet scalar mesons. Therefore glueballs will mix with nearby $q\bar{q}$ states of the same quantum numbers. The first results from lattice calculations, which include these effects, indicate that the mass shifts are small. We return to a discussion of this point in Sec. 15.8, see also Fig. 15.15.

The existence of three singlet scalar mesons around 1.5 GeV suggests additional degrees of freedom such as glue, since only two mesons are predicted in this mass range (as indicated in Table 15.2). The $f_0(1500)$ [23, 38] or, alternatively, the $f_0(1710)$ [35], have been proposed as candidates for the scalar glueball, both states having considerable mixing also with the $f_0(1370)$. Following Ref. [39] the glue is distributed among the $0^{++}$ isoscalars $q\bar{q}$ mesons around 2 GeV. Other mixing schemes, in particular with the $f_0(500)$ and the $f_0(980)$, have also been proposed [40]. According to a holographic model of low-energy QCD scalar glueballs decay strongly into kaons and $\eta$ mesons, in good agreement with data on the $f_0(1710)$ [41]. Details can be found in the review “Spectroscopy of Light Meson Resonances” and in Ref. [42]. See also the review “Scalar Mesons below 1 GeV” in this Review.

### 15.4.3 Hybrids

Mesons made of $q\bar{q}$ pairs bound by excited gluons ($q\bar{q}q$) are also predicted. Early model estimates placed them in the 1.9 GeV mass region, according to gluon flux tube models [43]. In the bag model there are four nonets, among them an exotic $1^{-}$, around or above 1.4 GeV [44, 45].

Lattice QCD calculations show that the lightest hybrid is an exotic $1^{-+}$ state, at a mass around 2 GeV [46–48]. Fig. 15.13 will show an example of the spectroscopy from one group’s [48] simulation.

There are so far two candidates for exotic states with quantum numbers $1^{-+}$, the $\pi_1(1400)$ and $\pi_1(1600)$, which could be hybrids or four-quark states. However, a recent combined re-analysis of $\pi_1$ production in diffractive $\pi^{-}p$ interaction and low energy $\bar{p}p$ annihilation leads to a single pole at ~1560 MeV with a width of ~390 MeV, although a two-pole scenario cannot be excluded [49]. Recently BESIII has reported evidence for the existence of a $1^{-+}$ $\eta_1(1855) \to \eta\eta'$ [50] which could be one of the isoscalar partners of the $\pi_1(1600)$ in the $1^{-+}$ hybrid nonet [51]. (For details and references, see the review “Spectroscopy of Light Meson Resonances” in this Review.)

## 15.5 Baryons: $qqq$ states

Baryons are fermions with baryon number $B = 1$, i.e., in the most general case, they are composed of three quarks plus any number of quark-antiquark pairs. Until recently, all established baryons were consistent with 3-quark ($qqq$) configurations, which we mainly discuss in this section. However, in 2015 the LHCb collaboration published first evidence for charmed ‘pentaquark’ states of minimal quark content $c\bar{c}u\bar{d}$ at invariant masses close to 4.4 GeV [52]. More refined LHCb experiments have revealed evidence for four such states called $P_{c\bar{c}}(4312)^{+}$, $P_{c\bar{c}}(4380)^{+}$, $P_{c\bar{c}}(4440)^{+}$, and $P_{c\bar{c}}(4457)^{+}$ [53]. These states are located close to the thresholds of the production of ordinary baryon-meson pairs like $\Sigma_c^{+}D^0$ and $\Sigma_c^{+}D^{*0}$ and are consistent with the predictions in terms of molecular-like states [54, 55]. A nice overview on the discussion of pentaquark and tetraquark states is given in Ref. [56]. See also the article on “Pentaquarks” in this Review.

The color part of the baryon wave function is an SU(3) singlet, a completely antisymmetric state of the three colors. Since the quarks are fermions, the wave function must be antisymmetric under interchange of any two equal-mass quarks (up and down quarks in the limit of isospin symmetry). Thus, it can be written as

$$|qqq\rangle_A = |\text{color}\rangle_A \times |\text{space, spin, flavor}\rangle_S, \quad (15.25)$$

where the subscripts S and A indicate symmetry or antisymmetry under interchange of any two equal-mass quarks. Note the contrast with the state function for the three nucleons in $^3$H or $^3$He:

$$|NNN\rangle_A = |\text{space, spin, isospin}\rangle_A. \quad (15.26)$$

<figure><img src="image_65.png" /><figcaption>Figure 15.3: Predicted glueball spectrum from the lattice in quenched approximation (from [34]).</figcaption></figure>

This difference has major implications for the internal structure, magnetic moments, etc. (For a nice discussion, see [57].)

### 15.5.1 Light baryons

The “ordinary” light-flavor baryons are made up of u, d, and s quarks. The three flavors imply an approximate SU(3) flavor symmetry, which requires that baryons made of these quarks belong to the multiplets on the right side of

$$3 \otimes 3 \otimes 3 = 10_S \oplus 8_M \oplus 8_M \oplus 1_A \quad (15.27)$$

(see the section on “SU(n) Multiplets and Young Diagrams” in this Review). Here the subscripts indicate symmetric, mixed-symmetry, or antisymmetric states under interchange of any two quarks. The 1 is a $|uds\rangle$ singlet state ($\Lambda_1$), and the octet contains a similar state ($\Lambda_8$). If these have the same spin and parity, they can mix. The mechanism is the same as for the mesons (see above). In the ground state multiplet, the SU(3) flavor singlet $\Lambda_1$ is forbidden by Fermi statistics. The section on “SU(3) Isoscalar Factors and Representation Matrices,” shows how relative decay rates in, say, $10 \to 8 \otimes 8$ decays may be calculated.

For the light-flavor baryons (no c or b quark), flavor and spin may be combined in an approximate flavor-spin SU(6), in which the six basic states are $d\uparrow$, $d\downarrow$, ..., $s\downarrow$ ($\uparrow$, $\downarrow$ = spin up, down). Then the baryons belong to the multiplets on the right side of

$$6 \otimes 6 \otimes 6 = 56_S \oplus 70_M \oplus 70_M \oplus 20_A. \quad (15.28)$$

These SU(6) multiplets decompose into flavor SU(3) multiplets as follows:

$$56 = {}^410 \oplus {}^28 \quad (15.29a)$$

$$70 = {}^210 \oplus {}^48 \oplus {}^28 \oplus {}^21 \quad (15.29b)$$

$$20 = {}^28 \oplus {}^41, \quad (15.29c)$$

where the superscript $(2S+1)$ gives the net spin $S$ of the quarks for each particle in the SU(3) multiplet. The $J^P = 1/2^+$ octet containing the nucleon and the $J^P = 3/2^+$ decuplet containing the $\Delta(1232)$ together make up the “ground-state” 56-plet, in which the orbital angular momenta between the quark pairs are zero (so that the spatial part of the state function is trivially symmetric). The 70 and 20 require some excitation of the spatial part of the wave function in order to make the overall wave function symmetric. States with nonzero orbital angular momenta are classified in SU(6)$\otimes$O(3) supermultiplets.

It is useful to classify the baryons into bands that have the same number N of quanta of excitation. Each band consists of a number of supermultiplets, specified by $(D, L_N^P)$, where D is the dimensionality of the SU(6) representation, L is the total quark
orbital angular momentum, and P is the total parity. Supermultiplets contained in bands up to N = 12 are given in [58]. The N = 0 band (with positive parity), which contains the nucleon and $\Delta(1232)$, consists only of the $(56,0_0^+)$ supermultiplet. The N = 1 band (with negative parity) consists only of the $(70,1_1^-)$ multiplet and contains the negative-parity baryons with masses below about 1.9 GeV. The N = 2 band (with positive parity) contains five supermultiplets: $(56,0_2^+)$, $(70,0_2^+)$, $(56,2_2^+)$, $(70,2_2^+)$, and $(20,1_2^+)$, where the $(56,0_0^+)$ represents the multiplet that contains the first radial excitations having the same quantum numbers as the ground state.

The wave functions of the non-strange baryons in the harmonic oscillator basis are often labeled by $|X^{2S+1}L_\pi J^P\rangle$, where S, L, J, P are as above, X = N or $\Delta$, and $\pi = S, M$ or A denotes the symmetry of the spatial wave function. The possible model states for the bands with N=0,1,2 are given in Table 15.7. The assignment of experimentally observed states is only complete and well established up to the N=1 band. Some more tentative assignments for higher multiplets are suggested in [59].

In Table 15.6, quark-model assignments are given for many of the established baryons whose SU(6)⊗O(3) compositions are relatively unmixed. Apart from the mixing of the A singlet and octet states, one must, however, keep in mind that states with same $J^P$ but different L, S combinations can also mix. In the quark model with one-gluon exchange motivated interactions, the size of the mixing is determined by the relative strength of the tensor term with respect to the contact term (see below). The mixing is more important for the decay patterns of the states than for their positions. An example are the lowest lying $(70,1_1^-)$ states with $J^P=1/2^-$ and $3/2^-$. The physical states are

$$|N(1535)1/2^-\rangle = \cos(\Theta_S)|N^2P_M1/2^-\rangle - \sin(\Theta_S)|N^4P_M1/2^-\rangle \quad (15.30)$$

$$|N(1520)3/2^-\rangle = \cos(\Theta_D)|N^2P_M3/2^-\rangle - \sin(\Theta_D)|N^4P_M3/2^-\rangle \quad (15.31)$$

and the orthogonal combinations for $N(1650)1/2^-$ and $N(1700)3/2^-$. The mixing is large for the $J^P=1/2^-$ states ($\Theta_S \approx -32^\circ$), but small for the $J^P=3/2^-$ states ($\Theta_D \approx +6^\circ$) [61–63].

All baryons of the ground state multiplets are known. Many of their properties, in particular their masses, are in good agreement with even the most basic versions of the quark model, including harmonic (or linear) confinement and a spin-spin interaction, which is responsible for the octet - decuplet mass shifts. A consistent description of the ground-state electroweak properties, however, requires refined relativistic constituent quark models.

The situation for the excited states is much less clear. The assignment of some experimentally observed states with strange quarks to model configurations is only tentative and in many cases candidates are completely missing. Melde, Plessas and Sengl [60] have calculated baryon properties in relativistic constituent quark models, using one-gluon exchange and Goldstone-boson exchange for the modeling of the hyperfine interactions (see Sec. 15.7 on Dynamics). Both types of models give qualitatively comparable results and in general, underestimate experimentally observed decay widths. Nevertheless, in particular on the basis of the observed decay patterns, the authors have assigned some additional states with strangeness to the SU(3) multiplets and suggest re-assignments for a few others. Among the new assignments are states with weak experimental evidence (two or three star ratings) and partly without firm spin/parity assignments, so that further experimental efforts are necessary before final conclusions can be drawn. We have added their suggestions in Table 15.6.

In the non-strange sector there are two main problems which are illustrated in Fig. 15.4, where the experimentally observed excitation spectrum of the nucleon (N and $\Delta$ resonances) is compared to the results of a typical quark model calculation [64]. The lowest states from the N=2 band, the $N(1440)1/2^+$, and the $\Delta(1600)3/2^+$, appear lower than the negative parity states from the N=1 band (see Table 15.7) and much lower than predicted by most models. Also negative parity $\Delta$ states from the N=3 band ($\Delta(1900)1/2^-$, $\Delta(1940)3/2^-$, and $\Delta(1930)5/2^-$) are too low in energy. The low lying states show a clustering in groups of levels around 1700 MeV and 1900 MeV for $N^*$ states and around

<figure><img src="image_66.png" /><figcaption>**Figure 15.4:** Excitation spectrum of the nucleon. Compared are the positions of the excited states identified in experiment, to those predicted by a relativized quark model calculation. Left hand side: isospin $I = 1/2$ N-states, right hand side: isospin $I = 3/2$ $\Delta$-states. Experimental: (columns labeled 'exp'), three- and four-star states are indicated by full lines (two-star dashed lines, one-star dotted lines). At the very left and right of the figure, the spectroscopic notation of these states is given. Quark model [64, 65]: (columns labeled 'QM'), all states for the N=1,2 bands, low-lying states for the N=3,4,5 bands. Full lines: at least tentative assignment to observed states, dashed lines: so far no observed counterparts. Many of the assignments between predicted and observed states are highly tentative.</figcaption></figure>

1900 MeV for $\Delta$ states which is not reflected in models.

Furthermore, many more states are predicted than observed. This has been known for a long time as the 'missing resonance' problem [61]. Up to an excitation energy of 2.4 GeV, about 45 N states are predicted, but only 20 are established (four- or three-star; see Note on N and $\Delta$ Resonances for the rating of the status of resonances) and 5 are tentative (two- or one-star). However, there is some recent progress. For the N=2 band, candidates have been identified for almost all predicted states with the exception of the $(20,1_2^+)$). The total number of states has perhaps not significantly changed but the number of states with four- or three-star rating has increased from 14 to 20 compared to the 2018 PDG particle listings. Most of this progress is due to the programs concentrating on the study of meson photoproduction reactions, while the most recent partial wave analysis of elastic pion scattering and charge exchange data by Arndt and collaborators [66] found no evidence for almost half of the states listed in this review (and included in Fig. 15.4). Such analyses are of course biased against resonances which couple only weakly to the $N\pi$ channel. Quark model predictions for the couplings to other hadronic channels and to photons are given in Ref. [64]. The large experimental effort ongoing at several electron accelerators to study the baryon resonance spectrum with real and virtual photon-induced meson production reactions includes the search for as-yet-unobserved states, as well as detailed studies of the properties of the low lying states (decay patterns, electromagnetic couplings, magnetic moments, etc.) (see Ref. [67] for reviews). There are two major new aspects of this program. The investigation of single and double polarization observables allows, via the study of interference terms, access to small partial waves that do not leave a footprint in unpolarized cross sections. An example for the impact of such data is given by a comparison of results from different multipole analyses of pion photoproduction [68]. It shows clearly that with the inclusion of polarization observables the reaction model results start to converge. This will in the near future much improve the database for excited baryons in the light quark sector.

The other aspect is the study of final states with meson pairs, in particular $\pi\pi$ and $\pi\eta$ pairs, which has made much progress during the last few years. This is important for higher lying states, which in the quark model may have both possible oscillations ex-
<table><caption>**Table 15.6:** Quark-model assignments for some of the known baryons in terms of a flavor-spin SU(6) basis. Only the dominant representation is listed. Assignments for several states, especially for the Λ(1810), Λ(2350), Ξ(1620), Ξ(1690), Ξ(1820), and Ξ(2030), are merely educated guesses.
† suggestions for assignments and re-assignments from Ref. [60].</caption><thead><tr><th>J<sup>P</sup></th><th>(D, L<sub>N</sub><sup>P</sup>)</th><th>S</th><th>Octet members</th><th>Singlets</th></tr></thead><tbody><tr><td>1/2<sup>+</sup></td><td>(56,0<sub>0</sub><sup>+</sup>)</td><td>1/2</td><td>N(939) Λ(1116) Σ(1193)</td><td>Ξ(1318)</td></tr><tr><td>1/2<sup>+</sup></td><td>(56,0<sub>2</sub><sup>+</sup>)</td><td>1/2</td><td>N(1440) Λ(1600) Σ(1660)</td><td></td></tr><tr><td>1/2<sup>-</sup></td><td>(70,1<sub>1</sub><sup>-</sup>)</td><td>1/2</td><td>N(1535) Λ(1670) Σ(1620) Σ(1560)<sup>†</sup></td><td>Ξ(1620) Λ(1405)</td></tr><tr><td>3/2<sup>-</sup></td><td>(70,1<sub>1</sub><sup>-</sup>)</td><td>1/2</td><td>N(1520) Λ(1690) Σ(1670)</td><td>Ξ(1820) Λ(1520)</td></tr><tr><td>1/2<sup>-</sup></td><td>(70,1<sub>1</sub><sup>-</sup>)</td><td>3/2</td><td>N(1650) Λ(1800) Σ(1750) Σ(1620)<sup>†</sup></td><td>Ξ(1690)</td></tr><tr><td>3/2<sup>-</sup></td><td>(70,1<sub>1</sub><sup>-</sup>)</td><td>3/2</td><td>N(1700) Λ(?) Σ(1940)<sup>†</sup></td><td>Ξ(?)</td></tr><tr><td>5/2<sup>-</sup></td><td>(70,1<sub>1</sub><sup>-</sup>)</td><td>3/2</td><td>N(1675) Λ(1830) Σ(1775)</td><td>Ξ(1950)<sup>†</sup></td></tr><tr><td>1/2<sup>+</sup></td><td>(70,0<sub>2</sub><sup>+</sup>)</td><td>1/2</td><td>N(1710) Λ(1810) Σ(1880)</td><td>Ξ(?) Λ(1810)<sup>†</sup></td></tr><tr><td>3/2<sup>+</sup></td><td>(56,2<sub>2</sub><sup>+</sup>)</td><td>1/2</td><td>N(1720) Λ(1890) Σ(?)</td><td>Ξ(?)</td></tr><tr><td>5/2<sup>+</sup></td><td>(56,2<sub>2</sub><sup>+</sup>)</td><td>1/2</td><td>N(1680) Λ(1820) Σ(1915)</td><td>Ξ(2030)</td></tr><tr><td>7/2<sup>-</sup></td><td>(70,3<sub>3</sub><sup>-</sup>)</td><td>1/2</td><td>N(2190) Λ(?) Σ(?)</td><td>Ξ(?) Λ(2100)</td></tr><tr><td>9/2<sup>-</sup></td><td>(70,3<sub>3</sub><sup>-</sup>)</td><td>3/2</td><td>N(2250) Λ(?) Σ(?)</td><td>Ξ(?)</td></tr><tr><td>9/2<sup>+</sup></td><td>(56,4<sub>4</sub><sup>+</sup>)</td><td>1/2</td><td>N(2220) Λ(2350) Σ(?)</td><td>Ξ(?)</td></tr></tbody></table><table><thead><tr><th colspan="6">Decuplet members</th></tr></thead><tbody><tr><td>3/2<sup>+</sup></td><td>(56,0<sub>0</sub><sup>+</sup>)</td><td>3/2</td><td>Δ(1232) Σ(1385)</td><td>Ξ(1530)</td><td>Ω(1672)</td></tr><tr><td>3/2<sup>+</sup></td><td>(56,0<sub>2</sub><sup>+</sup>)</td><td>3/2</td><td>Δ(1600) Σ(1690)<sup>†</sup></td><td>Ξ(?)</td><td>Ω(?)</td></tr><tr><td>1/2<sup>-</sup></td><td>(70,1<sub>1</sub><sup>-</sup>)</td><td>1/2</td><td>Δ(1620) Σ(1750)<sup>†</sup></td><td>Ξ(?)</td><td>Ω(?)</td></tr><tr><td>3/2<sup>-</sup></td><td>(70,1<sub>1</sub><sup>-</sup>)</td><td>1/2</td><td>Δ(1700) Σ(?)</td><td>Ξ(?)</td><td>Ω(2012)</td></tr><tr><td>5/2<sup>+</sup></td><td>(56,2<sub>2</sub><sup>+</sup>)</td><td>3/2</td><td>Δ(1905) Σ(?)</td><td>Ξ(?)</td><td>Ω(?)</td></tr><tr><td>7/2<sup>+</sup></td><td>(56,2<sub>2</sub><sup>+</sup>)</td><td>3/2</td><td>Δ(1950) Σ(2030)</td><td>Ξ(?)</td><td>Ω(?)</td></tr><tr><td>11/2<sup>+</sup></td><td>(56,4<sub>4</sub><sup>+</sup>)</td><td>3/2</td><td>Δ(2420) Σ(?)</td><td>Ξ(?)</td><td>Ω(?)</td></tr></tbody></table>

<table><caption>**Table 15.7:** N and Δ states in the N=0,1,2 harmonic oscillator bands. $L^P$ denotes angular momentum and parity, S the three-quark spin and 'sym'=A,S,M the symmetry of the spatial wave function. Listed are all possible spin/parity combinations and assignments of experimentally observed states. Only dominant components are indicated. Assignments in the N=2 band are partly tentative.</caption><thead><tr><th>N</th><th>sym</th><th>L<sup>P</sup></th><th>S</th><th colspan="4">N(I = 1/2)</th></tr></thead><tbody><tr><td>2</td><td>A</td><td>1<sup>+</sup></td><td>1/2</td><td>1/2<sup>+</sup></td><td>3/2<sup>+</sup></td><td>-</td><td>-</td></tr><tr><td>2</td><td>M</td><td>2<sup>+</sup></td><td>3/2</td><td>1/2<sup>+</sup></td><td>3/2<sup>+</sup></td><td>5/2<sup>+</sup></td><td>7/2<sup>+</sup></td></tr><tr><td>2</td><td>M</td><td>2<sup>+</sup></td><td>1/2</td><td>-</td><td>3/2<sup>+</sup></td><td>5/2<sup>+</sup></td><td>-</td></tr><tr><td>2</td><td>M</td><td>0<sup>+</sup></td><td>3/2</td><td>-</td><td>3/2<sup>+</sup></td><td>-</td><td>-</td></tr><tr><td>2</td><td>M</td><td>0<sup>+</sup></td><td>1/2</td><td>1/2<sup>+</sup> N(1710)</td><td>-</td><td>-</td><td>-</td></tr><tr><td>2</td><td>S</td><td>2<sup>+</sup></td><td>3/2</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>2</td><td>S</td><td>2<sup>+</sup></td><td>1/2</td><td>-</td><td>3/2<sup>+</sup> N(1720)</td><td>5/2<sup>+</sup> N(1680)</td><td>-</td></tr><tr><td>2</td><td>S</td><td>0<sup>+</sup></td><td>3/2</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>2</td><td>S</td><td>0<sup>+</sup></td><td>1/2</td><td>1/2<sup>+</sup> N(1440)</td><td>-</td><td>-</td><td>-</td></tr><tr><td>1</td><td>M</td><td>1<sup>-</sup></td><td>3/2</td><td>1/2<sup>-</sup> N(1650)</td><td>3/2<sup>-</sup> N(1700)</td><td>5/2<sup>-</sup> N(1675)</td><td>-</td></tr><tr><td>1</td><td>M</td><td>1<sup>-</sup></td><td>1/2</td><td>1/2<sup>-</sup> N(1535)</td><td>3/2<sup>-</sup> N(1520)</td><td>-</td><td>-</td></tr><tr><td>0</td><td>S</td><td>0<sup>+</sup></td><td>3/2</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>0</td><td>S</td><td>0<sup>+</sup></td><td>1/2</td><td>1/2<sup>+</sup> N(938)</td><td>-</td><td>-</td><td>-</td></tr></tbody></table><table><thead><tr><th colspan="6">Δ(I = 3/2)</th></tr></thead><tbody><tr><td>2</td><td>A</td><td>1<sup>+</sup></td><td>1/2</td><td>-</td><td>-</td><td>-</td></tr><tr><td>2</td><td>M</td><td>2<sup>+</sup></td><td>3/2</td><td>-</td><td>-</td><td>-</td></tr><tr><td>2</td><td>M</td><td>2<sup>+</sup></td><td>1/2</td><td>-</td><td>3/2<sup>+</sup></td><td>5/2<sup>+</sup></td></tr><tr><td>2</td><td>M</td><td>0<sup>+</sup></td><td>3/2</td><td>-</td><td>-</td><td>-</td></tr><tr><td>2</td><td>M</td><td>0<sup>+</sup></td><td>1/2</td><td>1/2<sup>+</sup> Δ(1750)</td><td>-</td><td>-</td></tr><tr><td>2</td><td>S</td><td>2<sup>+</sup></td><td>3/2</td><td>1/2<sup>+</sup> Δ(1910)</td><td>3/2<sup>+</sup> Δ(1920)</td><td>5/2<sup>+</sup> Δ(1905)</td></tr><tr><td>2</td><td>S</td><td>2<sup>+</sup></td><td>1/2</td><td>-</td><td>-</td><td>-</td></tr><tr><td>2</td><td>S</td><td>0<sup>+</sup></td><td>3/2</td><td>-</td><td>3/2<sup>+</sup> Δ(1600)</td><td>-</td></tr><tr><td>2</td><td>S</td><td>0<sup>+</sup></td><td>1/2</td><td>-</td><td>-</td><td>-</td></tr><tr><td>1</td><td>M</td><td>1<sup>-</sup></td><td>3/2</td><td>-</td><td>-</td><td>-</td></tr><tr><td>1</td><td>M</td><td>1<sup>-</sup></td><td>1/2</td><td>1/2<sup>-</sup> Δ(1620)</td><td>3/2<sup>-</sup> Δ(1700)</td><td>-</td></tr><tr><td>0</td><td>S</td><td>0<sup>+</sup></td><td>3/2</td><td>-</td><td>3/2<sup>+</sup> Δ(1232)</td><td>-</td></tr><tr><td>0</td><td>S</td><td>0<sup>+</sup></td><td>1/2</td><td>-</td><td>-</td><td>-</td></tr></tbody></table>
cited. Such states can be expected to decay in sequential processes de-exciting the two oscillations step-by-step so that they couple strongly to multiple-meson final states but not to single-meson production. Detailed analyses of such data are for example given in [69,70] and had already significant impact on partial wave analyses.

The excitation spectrum of hyperons containing s-quarks is even less well explored. Many experimental results date back to before 2000, although parameters listed in the review are in many cases updated by more modern partial wave analyses. A recent review on the quark model interpretation of the $\Lambda$ and $\Sigma$ states is given in Ref. [71], a new $\Omega^-$ state was reported in the $\Xi K$ mass spectrum with high significance from the Belle experiment [72] with a likely $J^P = \frac{3}{2}^-$ classification, and Belle also reported on the most significant evidence so far for the $\Xi(1620)$ in the $\Xi\pi$ mass spectrum [73], which has now warranted an upgrade of this resonance from one star to two stars. In the near future, significant progress on hyperon spectroscopy is expected from JLab using the $K_L$ facility in conjunction with the GlueX experimental setup, J-PARC using the $K^-$ beam at the extended hadron facility, and the PANDA experiment at FAIR [74, 75].

In quark models, the number of excited states is determined by the effective degrees of freedom, while their ordering and decay properties are related to the residual quark-diquark interaction. An overview of quark models for baryons is given in [76], recent discussions of baryon spectroscopy are given in [59, 77]. The effective degrees of freedom in the standard nonrelativistic quark model are three equivalent valence quarks with one-gluon exchange-motivated, flavor-independent color-magnetic interactions. The QCD aspect of gluon-gluon interactions is emphasized by the hypercentral quark model [78, 79], which includes in a natural way three-body forces between the quarks. A different class of models uses interactions which give rise to a quark-diquark clustering of the baryons: for a review see [80]. If there is a tightly bound diquark, only two degrees of freedom are available at low energies, and thus fewer states are predicted. Furthermore, selection rules in the decay pattern may arise from the quantum numbers of the diquark. However, we note that the recent identification of new $N^*$ resonances seriously questions the older static diquark-quark model since these resonances are likely members of the $(70, 2_2^+)$, which requires both oscillators in the baryon to be excited. More states are predicted by collective models of the baryon like the algebraic approach in [81]. In this approach, the quantum numbers of the valence quarks are distributed over a Y-shaped string-like configuration, and additional states arise e.g., from vibrations of the strings. More states are also predicted in the framework of flux-tube models, see [82], which are motivated by lattice QCD. In addition to the quark degrees of freedom, flux-tubes responsible for the confinement of the quarks are considered as degrees of freedom. These models include hybrid baryons containing explicit excitations of the gluon fields. However, since all half integral $J^P$ quantum numbers are possible for ordinary baryons, such 'exotics' will be very hard to identify, and probably always mix with ordinary states. So far, the experimentally observed number of states is still far lower even than predicted by the quark-diquark models.

The influence of chiral symmetry on the excitation spectrum of the nucleon has been debated from a somewhat different perspective. Chiral symmetry, the fundamental symmetry of QCD, is strongly broken for the low lying states, resulting in large mass differences of parity partners like the $J^P=1/2^+$ $N(938)1/2^+$ ground state and the $J^P=1/2^-$ $N(1535)1/2^-$ excitation. However, at higher excitation energies there is some evidence for parity doublets and even some very tentative suggestions for full chiral multiplets of $N^*$ and $\Delta$ resonances. An effective restoration of chiral symmetry at high excitation energies due to a decoupling from the quark condensate of the vacuum has been discussed (see [83] for recent reviews) as a possible cause. In this case, the mass generating mechanisms for low and high lying states would be essentially different. As a further consequence, the parity doublets would decouple from pions, so that experimental bias would be worse. However, parity doublets might also arise from the spin-orbital dynamics of the 3-quark system. Presently, the status of

the data does not allow final conclusions.

The most recent developments on the theory side include progress in understanding the physics of baryons [84,85] by using the Dyson-Schwinger equations of QCD and Bethe-Salpeter equations [86, 87]. In this approach, baryons are relativistic bound states of three quarks, and the treatment of their interactions arising from QCD is non-perturbative, incorporating aspects of confinement and dynamical symmetry breaking. Two paths to solving the three-body problem are taken: direct solution of the three-body Faddeev equation, and decomposition of baryons into quark-diquark systems, with all quark pairs able to constitute the diquark. Therefore, these descriptions are different from the older static diquark-quark pictures of the baryon. Moreover, the first unquenched lattice-QCD calculations for the excitation spectrum have been presented and are discussed in Sec. 15.8. The results are basically consistent with the level counting of $SU(6) \otimes O(3)$ in the standard non-relativistic quark model and show no indication for quark-diquark structures or parity doubling. Consequently, there is as yet no indication from lattice that the mis-match between the excitation spectrum predicted by the standard quark model and experimental observations is due to inappropriate degrees of freedom in the quark model.

### 15.5.2 Charmed and bottom baryons

The naming scheme for baryons with c or b quarks follows that of the light baryons: the $\Lambda$ is an isosinglet and the $\Sigma$ an isotriplet with one heavy (s, c or b) quark. The $\Xi$ is an isodoublet which contains two heavy quarks, and the $\Omega$ an isosinglet with three heavy quarks. The number of c or b quarks is indicated by the subscripts c or b. Hyperons are baryons with at least one s quark.

For charmed baryons the addition of the c quark to the light quarks extends the flavor symmetry to $SU(4)_f$. Due to the large mass of the c quark, this symmetry is much more strongly broken than the $SU(3)_f$ of the three light quarks. Nevertheless, the $SU(4)_f$ representation is still useful for bookkeeping purposes. With the additive charm quantum number C the baryons are classified in a 3-dimensional representation with the three coordinates $(I, Z, Y, C)$. Figure 15.5 shows the $SU(4)_f$ weight diagrams.

<figure><img src="image_67.png" /><figcaption>Figure 15.5: $SU(4)_f$ multiplets of ground state baryons made of u, d, s, and c quarks. (a) The spin $\frac{1}{2}$ 20-plet extends the charmless $SU(3)_f$ octet to $C = 1, 2$; (b) the spin $\frac{3}{2}$ 20-plet extends the $SU(3)_f$ decuplet to $C = 1, 2, 3$.</figcaption></figure>

With four quarks the 64 possible configurations decompose into

$$4 \otimes 4 \otimes 4 = \bar{4}_A \oplus 20_S \oplus 20_{MS} \oplus 20_{MA}, \quad (15.32)$$

(for a review on $SU(N)$ symmetries see e.g. [88]). The subscripts S and A refer to the symmetry and antisymmetry properties of the flavor wave functions. The flavor symmetric $20_S$ multiplet, associated with spin-$\frac{3}{2}$ baryons, contains the charmless $SU(3)_f$ decuplet at the bottom level. The $20_{MS}$ and $20_{MA}$ multiplets correspond to the mixed symmetric and mixed antisymmetric flavour wave functions of the spin-$\frac{1}{2}$ baryons, with the charmless octet baryons at the bottom level. There are two $dsc$ and two $usc$ spin-$\frac{1}{2}$ states, labeled $\Xi_c^0$, $\Xi_c'^0$ and $\Xi_c^+$, $\Xi_c'^+$. This is because one of the $qq$ pairs can have spin 1 (symmetric) or spin 0 (antisymmetric), giving both the total spin $j = \frac{1}{2}$ with the third quark (see also Fig. 15.6 below).

For $C = 1$ baryons the flavor decomposition of the diquark,
made of $u, d$, or $s$ quarks, is

$$ \mathbf{3} \otimes \mathbf{3} = \bar{\mathbf{3}}_A \oplus \mathbf{6}_S. \qquad (15.33) $$

For ground-state baryons, the overall antisymmetry of baryon wave function (including color) requires the light diquark to be symmetric under the exchange of spin and flavor, hence both symmetric or both antisymmetric, that is spin 1 for the **6**$_S$ and spin 0 for the $\bar{\mathbf{3}}_A$. The $\bar{\mathbf{3}}$ then combines with the $c$ quark to form the $J^P = 1/2^+$ states, while the **6** combines to form $J^P = 1/2^+$ or $J^P = 3/2^+$. The weight diagrams of the $\bar{\mathbf{3}}$ and **6** ground-state representations are shown in Fig. 15.6. Within each multiplet the $C = 1$ baryons obey isospin and SU(3)$_f$ mass relations at the expected orders.

<figure><img src="image_68.png" /><figcaption>**Figure 15.6:** The SU(3)$_f \bar{\mathbf{3}}$ (a) and **6** (b) ground state $J^P = 1/2^+$ representations. The structure of the **6** ground state with $J^P = 3/2^+$ is identical to the one in (b).</figcaption></figure>

<figure><img src="image_69.png" /><figcaption>**Figure 15.7:** Weight diagram of the $\bar{\mathbf{4}}$ SU(4)$_f$ multiplet with the experimentally observed negative parity baryons.</figcaption></figure>

The antisymmetric quadruplet in (15.32) does not exist in the ground state but is realized for the first orbital excitations ($L = 1$). Figure 15.7 shows the weight diagram with the experimentally observed states. In the quark model the quadruplet consists of four baryons with spins coupling to $\frac{1}{2}$, leading to four $L = 1$ excitations with $J^P = \frac{1}{2}^-$ and four excitations with $\frac{3}{2}^-$. The charmed ones are the partners of the $\Lambda(1405)$ and $\Lambda(1520)$ singlets of SU(3)$_f$.

For a detailed review on charmed baryons see Ref. [77]. Quark model predictions for baryons with two heavy quarks are given in Ref. [89] and lattice results for doubly and triply charmed states are discussed in Sec. 15.8 of this Review.

The $C = 1$ ground state baryons have all been observed. Due to their relatively narrow widths the states are much easier to isolate than the light quark baryon resonances which require intricate

partial wave analyses. The production cross sections are small, but the recent measurements at the $e^+e^-$ B-factories, at the $p\bar{p}$ Tevatron collider, and at LHCb have boosted the field. The LHCb collaboration has published evidence for five new narrow $\Omega_c^0$ states (css) [90] in proton-proton collisions. Four of these states have been confirmed by the Belle experiment in $e^+e^-$ collisions [91]. Their quantum numbers are still unknown, but they could correspond to the $L = 1$ ssc orbital excitations. Constituent quark models, lattice QCD, quark-diquark models, molecular models, and pentaquark states, have been discussed to describe their structure (see e.g. [92] for references). In the meantime, also four $\Omega_b^-$ states of the bss type have been reported from LHCb [92]. They lie in a narrow range between 6316 - 6350 MeV, but since their intrinsic widths are on the order of just a few MeV, they are well separated in the invariant mass spectra. Spectroscopy and theory in this field is rapidly evolving.

LHCb has also reported a doubly charmed $\Xi_{cc}^{++}$ (ccu) baryon [93]. The quantum numbers of this state are undetermined, but its significance is already rated three stars in the current compilation. Doubly charmed baryons have a very different structure from light baryons, more resembling heavy ‘double-star’ systems with attached light ‘planets’, which opens a new window for QCD properties. The first candidate for a doubly charmed baryon $\Xi_{cc}^+$ (ccd) had been reported earlier by the SELEX experiment [94, 95]. A significance of $6.3\sigma$ was claimed for the $\Lambda_c^+ K^- \pi^+$ decay and $4.8\sigma$ for $pD^+K^-$ but searches by other experiments have not confirmed it so far (see e.g. [96] and Refs. therein) and it is not included in the particle listings. The SELEX and LHCb masses lie in the predicted 3500 - 3700 MeV mass range (see e.g. [89]). The LHCb $\Xi_{cc}^{++}$ state lies about 100 MeV above the SELEX one, with a mass splitting far too large for ucc and dcc isospin partners. However, it has also been discussed that due to the different production mechanisms this is not necessarily a contradiction [97].

Figure 15.8 shows the spectrum of the established singly-charmed baryons (3* and 4* baryons in the Listings with known quantum numbers). The parity of the $\Lambda_c^+$ is that of the $c$ quark, defined as positive. Spin and parity have not been determined experimentally for most of the states. They follow the ordering and expectation from the quark model. Candidates for $L = 2$ orbital excitations of the $\Lambda_c$ (with $J^P = \frac{3}{2}^+$ and $J^P = \frac{5}{2}^+$) have already been observed, as well as a $\frac{3}{2}^-$ state at 2940 MeV, possibly a radial $L = 1$ excitation [98].

The same SU(4)$_f$ multiplets can be constructed for the bottom baryons by replacing the $c$ quark by a $b$ quark. The established 3* and 4* bottom baryons are shown in Fig.15.8. The quadruplet $\bar{\mathbf{4}}$ contains the two negative parity candidates $\Lambda_b(5912)^0$ and $\Lambda_b(5920)^0$. It appears that the confining potential is only weakly flavour dependent. For example, the mass difference between the $\Xi_b$ and the $\Lambda_b$ is roughly the same as that between the $\Xi_c$ and the $\Lambda_c$, the mass splitting between the spin-$\frac{1}{2}$ $\Omega_b$ and the $\Lambda_b$ close to that between the spin-$\frac{1}{2}$ $\Omega_c$ and the $\Lambda_c$. The spin-$\frac{3}{2}$ states are also heavier than the spin-$\frac{1}{2}$ ones, in agreement with expectations from the spin-spin force.

The bottom hadrons can also be embedded in a larger SU(5)$_f$ group that accounts for all baryons constructed from the five quark flavors. (The existence of baryons with $t$-quarks is very unlikely due to the short lifetime of the $t$-quark.) One predicts the decomposition

$$ \mathbf{5} \otimes \mathbf{5} \otimes \mathbf{5} = \overline{\mathbf{10}}_A \oplus \mathbf{40}_{MS} \oplus \mathbf{40}_{MA} \oplus \mathbf{35}_S. \qquad (15.34) $$

The decuplet is not realized in the ground state. The two 40-plets have mixed symmetry ($\frac{1}{2}$-spin) and the 35-plet is symmetric ($\frac{3}{2}$-spin). The SU(4)$_f$ spin-$\frac{1}{2}$ multiplet in Fig. 15.5 contains 20 spin-$\frac{1}{2}$ baryons and the corresponding one with bottom quarks an additional 12, giving together 32 states. Similarly there are 30 spin-$\frac{3}{2}$ baryons. One expects 75 ground state mesons from (15.34). Thus 13 ground state baryons containing both $b$ and $c$ quarks are predicted ($8$ spin-$\frac{1}{2}$ [$4\Omega + 4\Xi$] and $5$ spin-$\frac{3}{2}$ [$3\Omega + 2\Xi$]).
<figure><img src="image_70.png" /><figcaption>Figure 15.8: Mass spectrum of the established C = 1 charmed and B = -1 bottom baryons with known $J^P$. The flavor symmetry assignments are given by the square brackets. According to their isospins the $\Sigma_{c,b}$ ($\Xi_{c,b}$) consist of three (two) charged or neutral states that are nearly degenerate.</figcaption></figure>

## 15.6 Magnetic moments

The magnetic dipole moment of a baryon is conventionally written relative to the mass of the proton:

$$ \vec{\mu}_B = g_B \mu_N \vec{s}, \qquad (15.35) $$

where $\mu_N = \frac{e}{2m_p}$ is the nuclear magneton (in natural units). The factor $g_B$ is calculated by adding the quark contributions. The magnetic moment of quark $i$ with charge $q_i e$ and constituent mass $m_i$ is $\vec{\mu}_i = 2(\frac{q_i e}{2m_i})\vec{s}$, hence for the three light quarks,

$$ \mu_u = \frac{2}{3}\kappa_u, \quad \mu_d = -\frac{1}{3}\kappa_d, \quad \mu_s = -\frac{1}{3}\kappa_s, \quad \text{with } \kappa_i \equiv \frac{e}{2m_i}. \quad (15.36) $$

The magnetic dipole moment of a baryon is then given by

$$ \mu_B = \sum_{i=1}^{3} \langle B \uparrow | \mu_i \sigma_{zi} | B \uparrow \rangle, \qquad (15.37) $$

where $|B \uparrow\rangle$ is the wavefunction for a baryon with its spin along the z-axis and $\vec{\sigma} = 2\vec{s}$. Since the quark model uses the totally symmetric SU(6) wavefunctions (flavor SU(3) × spin SU(2)) one predicts for the proton:

$$ \mu_p = \frac{4}{3}\mu_u - \frac{1}{3}\mu_d = \frac{e}{2m} \equiv \kappa, \qquad (15.38) $$

assuming $m_u = m_d \simeq m$ (for the detailed derivation see e.g. Ref. [6]). For the magnetic moment of the neutron one obtains likewise

$$ \mu_n = -\frac{1}{3}\mu_u + \frac{4}{3}\mu_d = -\frac{2}{3}\kappa, \qquad (15.39) $$

which leads to the simple prediction $\mu_n/\mu_p = g_n/g_p = -2/3$.

The famous Frisch and Stern experiment showed for the first time that $g_p \sim 5.58 \gg 2$. Modern measurements are performed in electromagnetic traps, see the *Listings*, while the magnetic moment of the neutron is measured with cold neutron beams. The

experimental ratio $\mu_n/\mu_p = -0.6849793 \pm 0.0000003$ is impres- sively close to the prediction of the quark model (note that us- ing instead the antisymmetric SU(6) wavefunctions would lead to $\mu_n/\mu_p = -2$ [6]). The constituent mass $m$ of the $u$ and $d$ quarks can be estimated from (15.35) and (15.38): $m = 2m_p/g_p \simeq 336$ MeV.

The magnetic moment of the $\Lambda$ is easy to predict: the diquark $ud$ has isospin $i=0$, since $i(s) = 0$ and $i(\Lambda) = 0$, and is therefore an antisymmetric isospin state. The symmetric quark model then also requires the spin to be zero, so that the magnetic moment of the $\Lambda$ stems from the $s$ quark:

$$ \mu_{\Lambda} = \mu_{s} = -\frac{1}{3}\kappa_{s}. \qquad (15.40) $$

The magnetic moment of the $\Lambda$ is obtained from the Larmor pre- cession frequency of the polarization vector in an homogeneous magnetic field. The latter is derived from the asymmetric distri- bution of the proton in $\Lambda \to \pi^- p$ [99]. The experimental result is listed in Table 15.8. The constituent mass of the $s$ quark is then $m_s = -m\mu_p/3\mu_\Lambda = 509$ MeV.

The magnetic moments of the other baryons (Table 15.8) are predicted from $\kappa$ and $\kappa_s$. The magnetic moment of the $\Sigma^+$ is obtained by replacing the $d$ quark in (15.38) by an $s$ quark, and for the $\Sigma^-$ the $u$ quark by an $s$ quark in (15.39). The magnetic moment of the $\Sigma^+$ has also been measured with the precession method using its decay into $p\pi^0$ [100], that of the $\Sigma^- \to n\pi^-$ likewise, or by detecting the X-rays emitted by cascading $\Sigma^-$ captured in the Coulomb shells of target atoms. The magnetic moment is then derived from the fine-structure splitting [101].

The magnetic moments of the $\Xi^0$, $\Xi^-$ and $\Omega^-$ are obtained from the polarization of the $\Lambda$ in the decays $\Lambda\pi^0$, $\Lambda\pi^-$ and $\Lambda K^-$, respectively. The SU(6) wavefunction of the (spin 3/2) $\Omega^-$ is the product of the two symmetric SU(3) flavor and SU(2) spin
<table><caption>**Table 15.8:** Quark model predictions and measured magnetic dipole moments of the ground state baryons in units of $\mu_N$; $\kappa \equiv \frac{e}{2m_s} = 2.793$ $\mu_N$ and $\kappa_s \equiv \frac{e}{2m_s} = -3\mu_\Lambda = 1.84 \mu_N$. $\dagger\Sigma^0 \to \Lambda$ transition magnetic moment.</caption><thead><tr><th>Baryon</th><th>Quark model</th><th>Experimental value</th></tr></thead><tbody><tr><td><i>p</i></td><td>$\kappa$</td><td>input 2.793</td></tr><tr><td><i>n</i></td><td>$-\frac{2}{3}\kappa$</td><td>= -1.86 -1.913</td></tr><tr><td>$\Lambda$</td><td>$-\frac{1}{3}\kappa_s$</td><td>input -0.6138 ± 0.0047</td></tr><tr><td>$\Sigma^+$</td><td>$\frac{8}{9}\kappa + \frac{1}{9}\kappa_s$</td><td>= 2.68 2.458 ± 0.010</td></tr><tr><td>$\Sigma^0$</td><td>$\frac{2}{9}\kappa + \frac{1}{9}\kappa_s$</td><td>= 0.82</td></tr><tr><td>$\Sigma^0\dagger$</td><td>$-\frac{1}{\sqrt{3}}\kappa$</td><td>= -1.61 -1.61 ± 0.08</td></tr><tr><td>$\Sigma^-$</td><td>$-\frac{4}{9}\kappa + \frac{1}{9}\kappa_s$</td><td>= -1.04 -1.160 ± 0.025</td></tr><tr><td>$\Xi^0$</td><td>$-\frac{2}{9}\kappa - \frac{4}{9}\kappa_s$</td><td>= -1.44 -1.250 ± 0.014</td></tr><tr><td>$\Xi^-$</td><td>$\frac{1}{9}\kappa - \frac{4}{9}\kappa_s$</td><td>= -0.51 -0.6507 ± 0.0025</td></tr><tr><td>$\Omega^-$</td><td>$-\kappa_s$</td><td>= -1.84 -2.024 ± 0.056</td></tr><tr><td>$\Delta^{++}$</td><td>$2\kappa$</td><td>= 5.58 4.52 ± 0.67</td></tr><tr><td>$\Delta^+$</td><td>$\kappa$</td><td>= 2.79 2.3 - 4.5</td></tr></tbody></table>

wavefunctions. Summing over the three s quarks one gets

$$ \mu_{\Omega^{-}} = \sum_{i=1}^{3} \langle \Omega^{-} \uparrow | \mu_{i} \sigma_{zi} | \Omega^{-} \uparrow \rangle = 3\mu_{s} = 3\mu_{\Lambda}. \quad (15.41) $$

The magnetic moment of the $\Omega^-$ is hard to measure, because the $\Omega^-$ is unpolarized at high energies, in contrast to the other hyperons. Polarized $\Omega^-$ hyperons have been obtained from polarized $\Lambda$ and $\Xi^0$ impinging on a nuclear target [102].

In quark models with full isospin symmetry the magnetic moments of the $\Delta$-resonances are simply related to the magnetic moment of the proton by $\mu_\Delta = Q_\Delta \cdot \mu_N$, ($Q_\Delta$ = charge) i.e. the moment of the $\Delta^+$ should equal the proton moment and the moment of the $\Delta^{++}$ should be twice as large (sum of the three u-quarks). Magnetic moments of the decuplet baryons (with the exception of the $\Omega^-$) are very difficult to measure because their lifetimes are so short that spin precession techniques cannot be used. For the $\Delta$ one can profit from an electromagnetic spin-reorientation transition inside the large width of the state. This leads to the emission of a magnetic dipole photon in reactions like $\pi^+p \to \pi^+p\gamma$ or $\gamma p \to \pi^0p\gamma'$ which is related to the magnetic moment of the resonance. The first reaction has been used to study the magnetic moment of the $\Delta^{++}$ by measuring the left-right asymmetry with polarized protons [103]. The second one was used for the $\Delta^+$ state [104, 105]. Taking all experimental and systematic uncertainties into account the result spans the range from (2.3 - 4.5)$\mu_N$, covering the quark model prediction of $\approx 2.8\mu_N$, but is not very precise.

Table 15.8 lists the current experimental values for the magnetic moments of the ground state baryons, together with the predictions from the quark model. There are significant discrepancies, but given its crudeness, the quark model performs surprisingly well.

## 15.7 Dynamics

Quantum chromodynamics (QCD) is well-established as the theory for the strong interactions. As such, one of the goals of QCD is to predict the spectrum of strongly-interacting particles. To date, the only first-principles calculations of spectroscopy from QCD use lattice methods. These are the subject of Sec. 15.8. These calculations are difficult and unwieldy, and many interesting questions do not have a good lattice-based method of solution. Therefore, it is natural to build models, whose ingredients are abstracted from QCD, or from the low-energy limit of QCD (such as chiral Lagrangians) or from the data itself.

Phenomenological models for light quark systems typically include

1. A confining interaction, which is generally spin-independent (e.g., harmonic oscillator or linear confinement);

2. A strange quark mass somewhat larger than the up and down quark masses, in order to split the SU(3) multiplets;

3. Different types of spin-dependent interactions:

a) commonly used is a color-magnetic flavor-independent interaction modeled after the effects of gluon exchange in QCD (see e.g., Ref. [106]). For example, in the S-wave states, there is a spin-spin hyperfine interaction of the form

$$ H_{HF} = -\alpha_S M \sum_{i>j} (\vec{\sigma}\lambda_a)_i (\vec{\sigma}\lambda_a)_j, \quad (15.42) $$

where M is a constant with units of energy, $\lambda_a$ ($a = 1, \dots, 8$) is the set of SU(3) unitary spin matrices, defined in the review “SU(3) Isoscalar Factors and Representation Matrices,” and the sum runs over constituent quarks or antiquarks. Spin-orbit interactions, although allowed, seem to be small in general, but a tensor term is responsible for the mixing of states with the same $J^P$ but different L, S combinations.

b) other approaches include flavor-dependent short-range quark forces from instanton effects (see e.g., [107, 108]). This interaction acts only on scalar, isoscalar pairs of quarks in a relative S-wave state:

$$ \langle q^2; S, L, T | W | q^2; S, L, T \rangle = -4g\delta_{S,0}\delta_{L,0}\delta_{I,0}W \quad (15.43) $$

where W is the radial matrix element of the contact interaction.

c) a rather different and somewhat controversial approach is based on flavor-dependent spin-spin forces arising from one-boson exchange. The interaction term is of the form:

$$ H_{HF} \propto \sum_{i<j} V(\vec{r}_{ij}) \lambda_i^F \cdot \lambda_j^F \vec{\sigma}_i \cdot \vec{\sigma}_j \quad (15.44) $$

where the $\lambda_i^F$ are in flavor space (see e.g., Ref. [109]).

4. In the case of spin-spin interactions, a flavor-symmetric interaction for mixing $q\bar{q}$ configurations of different flavors (e.g., $u\bar{u} \leftrightarrow d\bar{d} \leftrightarrow s\bar{s}$), in isoscalar channels, so as to reproduce e.g., the $\eta - \eta'$ and $\omega - \phi$ mesons.

Systems with heavy quarks have their own approaches. Besides potential models to deal with $Q\bar{Q}$ spectroscopy (which are similar to what we have just described for light quarks) there are a variety of effective field theories built on QCD. Nonrelativistic QCD or NRQCD is a nonrelativistic reduction of the QCD Lagrangian, written as an expansion in powers of the heavy quarks' velocities. Its expressions are often used in lattice calculations. Terms in the Lagrangian have obvious quark model analogs, but are derived directly from QCD. For example, the heavy quark potential is a derived quantity, extracted from simulations.

There are also many versions of heavy quark effective theory (HQET), an expansion in the inverse mass of the heavy constituent. The physics input is one of decoupling of scales: the interactions of the heavy quarks take place at short distance while the light degrees of freedom involve long distance dynamics, so to some degree they decouple. The form of terms in the expansion is fixed (often by symmetry) but the coefficients are not. These approaches allow one to interpolate from the charm sector to the bottom one, or from masses of mesons to baryons. For example, a Hamiltonian for one or more heavy quarks bound to light degrees of freedom could be written as

$$ H_l^Q = m_Q + e_l + \frac{\kappa_l}{2m_Q} + \frac{s_l}{2m_Q} \vec{S} \cdot \vec{j}_l + \dots \quad (15.45) $$

where $m_Q$ is the mass of the heavy quark (or heavy quarks where the heavy quarks are treated as a single dynamical object) and $\vec{S}$ is the angular momentum carried by the heavy degrees of freedom. $e_l$, $\kappa_l$ and $\vec{j}_l$ involve the light degrees of freedom. We will see an example of the use of this formula in the next section.
15.8 Lattice Calculations of Hadronic Spec-
troscopy

Lattice calculations are a major source of information about
QCD masses and matrix elements. The necessary theoretical
background is given in Sec. 17 of this Review. Here we confine
ourselves to some general comments and illustrations of lattice
calculations for spectroscopy.

It might seem a bit out of place to have a section about lattice
calculations in a review of the quark model, since to many readers,
the quark model is just a model while QCD could be thought of
as a construct which is something deeper than a model. But this
review is, despite its title, actually an introduction to the spec-
troscopy and related quantities of the strong interactions. From
that perspective, a presentation of lattice results is entirely ap-
propriate.

The input to lattice calculations is a discretized version of the
QCD Lagrangian. Lattice calculations measure correlation func-
tions via Monte Carlo simulation and physical observables are
determined through fits of the lattice data to theoretical expecta-
tions. It is fair to say that all lattice calculations make extensive
use of quark model ideas – for example, the operators used to cre-
ate and annihilate hadrons are almost always based on the quark
model.

There is only a sporadic literature connecting lattice results
to the quark model (one recent example is [110]) but of course
the qualitative understanding of lattice results depends as heavily
on quark model ideas as does the qualitative understanding of
experimental data.

In general, the cleanest lattice results come from computations
of processes in which there is only one particle in the simulation
volume. These quantities include masses of hadrons, simple decay
constants, like pseudoscalar meson decay constants, and semilep-
tonic form factors (such as the ones appropriate to $B \to D\nu\nu$, $K\nu\nu$, $\pi\ell\nu$). The cleanest predictions for masses are for states which are
far below any thresholds to open channels, since the effects of fi-
nal state interactions are not yet under complete control on the
lattice. As a simple corollary, the lightest state in a channel is
easier to study than the heavier ones.

Good-quality modern lattice calculations will present multi-
part error budgets with their predictions. Part of the uncertainty
is statistical, from sample size. Typically, the quoted statisti-
cal uncertainty includes uncertainty from a fit: it is rare that a
simulation computes one global quantity which is the desired ob-
servable. Simulations which include virtual quark-antiquark pairs
(also known as “dynamical quarks” or “sea quarks”) are often done
at up and down quark mass values heavier than the experimental
ones, and it is then necessary to extrapolate in these quark masses.
Simulations can be carried out at the physical values of the heavier
quarks’ masses. They are always done at nonzero lattice spacing,
and so it is necessary to extrapolate to zero lattice spacing. Some
theoretical input is needed to do this. Much of the uncertainty
in these extrapolations is systematic, from the choice of fitting
function. Other systematics include the effect of finite simulation
volume, the number of flavors of dynamical quarks actually sim-
ulated, and technical issues with how these dynamical quarks are
included. The particular choice of a fiducial mass (to normalize
other predictions) is not standardized; there are many possible
choices, each with its own set of strengths and weaknesses, and
determining it usually requires a second lattice simulation from
that used to calculate the quantity under consideration.

A systematic error of major historical interest is the “quenched approximation,” in which dynamical quarks are simply left out of the simulation. This was done because the addition of these virtual pairs presented an expensive computational problem. No generally-accepted methodology has ever allowed one to correct for quenching effects, short of redoing all calculations with dynamical quarks. (All light degrees of freedom must be included in any realistic simulation of a quantum field theory.) Advances in algorithms and computer hardware have rendered it obsolete.

With these brief remarks, we turn to examples. The field of
lattice QCD simulations is vast, and so it is not possible to give
a comprehensive review of them in a small space. The history of
lattice QCD simulations is a story of thirty years of incremental

improvements in physical understanding, algorithm development,
and ever faster computers, which have combined to bring the field
to a present state where it is possible to carry out very high qual-
ity calculations. We present a few representative illustrations, to
show the current state of the art.

15.8.1 Spectroscopy of low-lying states

<figure><img src="image_71.png" /><figcaption>**Figure 15.9:** Hadron spectrum from lattice QCD. Comprehensive results for mesons and baryons are from MILC [111, 112], PACS-CS [113], BMW [114], QCDSF [115], and ETM [116]. Results for $\eta$ and $\eta'$ are from RBC & UKQCD [21], Hadron Spectrum [117] (also the only $\omega$ mass), UKQCD [118], and Michael, Ottnad, and Urbach [119]. Results for heavy-light hadrons from Fermilab-MILC [120], HPQCD [121, 122], and Mohler and Woloshyn [123]. Circles, squares, diamonds, and triangles stand for staggered, Wilson, twisted-mass Wilson, and chiral sea quarks, respectively. Asterisks represent anisotropic lattices. Open symbols denote the masses used to fix parameters. Filled symbols (and asterisks) denote results. Red, orange, yellow, green, and blue stand for increasing numbers of ensembles (i.e., lattice spacing and sea quark mass) Black symbols stand for results with 2+1+1 flavors of sea quarks. Horizontal bars (gray boxes) denote experimentally measured masses (widths). b-flavored meson masses are offset by -4000 MeV.</figcaption></figure>

One of the first goals of large scale lattice simulations was to
compute masses of the lightest states in all of the “conventional”
$q\bar{q}$ and $qqq$ channels. We illustrate results from many groups
in Fig. 15.9, a comprehensive summary from 2012 provided by
A. Kronfeld (private communication; see also [124]).

Mesons with a valence structure of identical quark - antiquark
pairs such as the eta or eta-prime, $|\eta\rangle \sim \alpha|\bar{u}u + \bar{d}d\rangle + \beta|\bar{s}s\rangle$, present more of a challenge, because one must include the effects of “annihilation graphs” for the valence $q$ and $\bar{q}$. Many groups, among them Refs. [21, 118, 125, 126], have reported calculations of the $\eta$ and $\eta'$ mesons. The most recent calculation is that of Ref. [126], which finds masses of $554.7 \pm 9.2$ and $930 \pm 21$ MeV for the $\eta$ and $\eta'$.

The spectroscopy of mesons containing heavy quarks is truly a
high-precision endeavor. These simulations typically use NRQCD
for $Q\bar{Q}$ bound states. Figs. 15.10 and 15.11 show the low ly-
ing mass spectrum for charmonium and bottomonium states from
several different groups [122, 127–130]. Most, but not all, of the
results are for the lightest state with a given value of quantum
numbers. We return to this point below.

Fig. 15.12 shows a compilation of lattice results for doubly and
triy charmed baryons, provided by S. Meinel [138]. The posi-
tion of the observed $\Xi_{cc}^{+}$ [93] is also shown. Note that the lattice
calculations for the mass of this state were predictions, not post-
dictions.

15.8.2 Excited state spectroscopy

A close look at Fig. 15.9 reveals an issue: while many of the
states shown in it are stable under the strong interactions, some of
them (the vector mesons, the Delta) are not. They are resonances.
And so are all the excited states of hadrons listed in the PDG be
<figure><img src="image_72.png" /><figcaption>**Figure 15.10:** Selected spectroscopy of the $\bar{c}c$ spectrum, compared to lattice data from Ref. [127] in squares, from Refs. [128, 129] in diamonds, and from Ref. [130] as a fancy diamond. Particles whose masses are used to fix lattice parameters by Ref. [127] are shown with crosses; octagons label experimental values. The upper $\chi_{c1}$ state is also known as X(3872). The dotted line shows the threshold value for two charmed mesons.</figcaption></figure>

<figure><img src="image_73.png" /><figcaption>**Figure 15.11:** Spectroscopy of the $\bar{b}b$ system, adapted from Ref. [122]. Particles whose masses are used to fix lattice parameters are shown with crosses; octagons label experimental values, and lattice results are shown as squares. The dotted line shows the threshold value for two bottom mesons.</figcaption></figure>

they “ordinary’,” or “exotic.” How do lattice calculations deal with such states? This is actually a complicated business, which is an active area of research.

There are two issues involving excited state spectroscopy, which are entangled in any lattice calculation:

1. There are many states with the same quantum numbers, all of which contribute to lattice observables

2. In finite volume, states do not decay, they only mix.

To explain what is going on, we have to describe how a lattice calculation determines a mass. This is done by measuring an observable which involves the creation of a set of quark and gluon fields at some point on the lattice and their annihilation at other points on the lattice. To find the energies of zero momentum states, the operators are averaged over position and the observable is

$$C_{ij}(t) = \sum_x \langle O_i(x,t) O_j(0,0) \rangle \propto \sum_n \langle 0 | O_i | n \rangle \langle n | O_j | 0 \rangle \exp(-E_n t). \quad (15.46)$$

$E_n$ is the energy of the $n$th hadronic state with the quantum numbers of the source or sink. The reader can see that by taking $t$ large, the correlator is dominated by the contribution from the

lightest state, decaying with $t$ as $\exp(-E_{min}t)$. This is a positive thing if one is interested in the ground state, but the contribution of excited states is exponentially suppressed at large $t$. The size of a given state’s contribution, $\langle 0|O_i|n \rangle$, can provide information about the content of the state.

As we move away from hadrons which can be created by the simplest quark model operators (appropriate to the lightest meson and baryon multiplets) we encounter a host of new problems: either no good interpolating fields, or too many possible interpolating fields, and many states with the same quantum numbers. Techniques for dealing with these interrelated problems vary from collaboration to collaboration, but all share common features: typically, correlation functions from many different interpolating fields are used, and the signal is extracted in what amounts to a variational calculation using the chosen operator basis. In addition to mass spectra, wave function information can be garnered from the form of the best variational wave function.

These calculations give towers of excited states, but the states are not resonances. Lattice calculations do not see decays directly because they are done in boxes of some finite size. Generally, the energies of the single particle states do not depend strongly on the box size. Multiple particle states, such as two particle states with equal and opposite momenta, are different, though. The finite box size quantizes the allowed values of momentum and the spectrum of states, which would be a continuum in infinite volume, is just a discrete tower. As the box size is varied, some of these states can approach the energies of single particle states, and the nearby states will mix and split, participating in a quantum mechanical avoided level crossing. The mixing and splitting of single and multiparticle states is the finite volume data which is used to infer how states in the continuum interact.

As a simple example, two positive energy but interacting particles in a one dimensional box have a spatial wave function which looks like $\cos(k|z| + \delta)$ where $\delta$ is the phase shift due to their interaction. Periodic boundary conditions in a finite box of length $L$ give a quantization condition to the allowed momentum of the state ($kL + 2\delta(E) = 0 \mod 2\pi$, for example) which involves the phase shift. Simulations with many values of $L$ and operators for states carrying many values of momentum can be combined into information about phase shifts, and then to statements about the location(s) of singularities in the complex energy plane.

The techniques for studying excited states of mesons and baryons, glueballs, and exotics are similar. They all use a large basis of trial states: the progression with time started with minimum quark content states, then added more complicated states (pairs of mesons or tetraquark states), then carried out simulations with multiple volumes with the goal of extracting resonance properties. The older single volume studies typically presented results for many quantum numbers, which give a broad (though incomplete) view of the QCD spectrum.

An example of meson spectroscopy where this is done, by [48], is shown in Fig. 15.13. The quark masses are still heavier than their physical values, so the pion is at 392 MeV. The authors can assign a relative composition of nonstrange and strange quark content to their states, observing, for example, a nonstrange $\omega$ and a strange $\phi$. Some states also have a substantial component of gluonic excitation. Note especially the three exotic channels $J^{PC} = 1^{-+}, 0^{+-}$, and $2^{-+}$, with states around 2 GeV. These calculations will become more realistic as the quark masses are carried lower and resonance effects are included.

The interesting physics questions of excited baryon spectroscopy to be addressed are precisely those enumerated in the last section. An example of a calculation involving only single particle states in a single volume, due to Ref. [139], is shown in Fig. 15.14. Notice that the pion is not yet at its physical value. The lightest positive parity state is the nucleon, and the Roper resonance has not yet appeared as a light state.

Glueballs present similar issues. In Fig. 15.3 we showed a figure from [34] presenting a lattice prediction for the glueball mass spectrum in quenched approximation. A true QCD prediction of the glueball spectrum requires dynamical light quarks, some way to deal with the mixing of glue states and quark-antiquark (and beyond) states and (because glueball operators are intrinsically
<figure><img src="image_74.png" /><figcaption>**Figure 15.12:** Comparison of lattice QCD results for the doubly and triply charmed baryon masses. Labels are Liu, *et al.*, [131]; Briceno, *et al.*, [132]; Namekawa, *et al.*, [133]; Padmanath, *et al.*, [134]; Alexandrou, *et al.*, [116]; Brown, *et al.*, [135]; Perez-Rubio *et al.*, [136]; Alexandrou and Kallidonis 2017, [137]. Only calculations with dynamical light quarks are included; for the doubly charmed baryons, only calculations were performed at or extrapolated to the physical pion mass are shown. Results without estimates of systematic uncertainties are labeled "stat. only". The lattice spacing values used in the calculations are also given; *a* = 0 indicates that the results have been extrapolated to the continuum limit. In the plot of the doubly charmed baryons, the position of the experimentally observed Ξ<sub>cc</sub><sup>+</sup> state [93] is shown with a horizontal line.</figcaption></figure>

<figure><img src="image_75.png" /><figcaption>**Figure 15.13:** Isoscalar (green and black) and isovector (blue) spectrum from Ref. [48]. States are labeled *J*<sup>PC</sup>. The quark mass is heavier than its physical value; *m*<sub>π</sub> = 392 MeV. The vertical height of each box indicates the statistical uncertainty in the mass. Black and green indicate relative nonstrange and strange composition. Orange outlines show states with a large chromomagnetic component to their wave function, which the authors argue are hybrid states. Note the exotic states in the three rightmost columns.</figcaption></figure>

<figure><img src="image_76.png" /><figcaption>**Figure 15.14:** Spin-identified spectrum of nucleons and deltas, from lattices where $m_\pi = 396$ MeV, in units of the calculated $\Omega$ mass, from Ref. [139]. The colors just correspond to the different *J* assignments: grey for *J* = 1/2, red for *J* = 3/2, green for 5/2, blue for *J* = 7/2.</figcaption></figure>

noisy) high statistics. Early studies which include the 0<sup>++</sup> chan-
nel are Refs. [140,141]. Fig. 15.15 shows results from [140], done
with dynamical *u*, *d* and *s* quarks at two lattice spacings, 0.123
and 0.092 fm, with pion masses at 280 and 360 MeV respectively,
along with comparisons to the quenched lattice calculation of [33]
and to experimental isosinglet mesons. This study shows that the
effects of quenching seem to be small at its relatively heavy pion
masses.

Calculations of resonance properties are more recent. Two body
decays with a single open channel, such as the rho coupled to two
pions, have a large literature. A review [142] summarizes the
situation, and example of a calculation of the rho meson decay
width is [143]. (See also [144].) The mass and decay width of the
$f_0(500)$ have recently been computed in [145] and [146] and those

of other mesons in [147]. Ref. [148] studies the decay width of the
$\Delta(1238)$.

Studies which observe a large basis of states across many vol-
umes can probe excited state spectroscopy as resonances. These
calculations are just beginning and the ones we know about are
still performed at unphysically heavy quark masses. Examples
include the decays of an exotic $J^{PC} = 1^{-+}$ resonance [149]
and the spectroscopy of $J^{--}$ resonances [150]. The results of
Refs. [128, 129] in Fig. 15.10 come from similar analyses. We ex-
pect to see considerable progress in this area over the next few
years.

We conclude this section with some remarks about unusual
states containing heavy quarks. Lattice calculations relevant to
the extra states observed in the charmonium and bottomonium
spectrum (Sec. 15.4) are difficult, because the states sit high in
the spectrum of most channels, because of the number of nearby
multiparticle states, and because it is necessary to have the tech-
nology to study heavy and light quarks together.
<figure><img src="image_77.png" /></figure>

**Figure 15.15:** Lattice QCD predictions for glueball masses. The open and closed circles are the larger and smaller lattice spacing data of the full QCD calculation of glueball masses of Ref. [140], at pion masses of 280 and 360 MeV. Squares are the quenched data for glueball masses of Ref. [33]. The bursts labeled by particle names are experimental states with the appropriate quantum numbers.

Some lattice calculations are used as input for phenomenological studies of hybrids. The theoretical motivation is that one can imagine states made of a heavy $Q\bar{Q}$ pair along with excitations of light degrees of freedom, viewed as an excited state of the string connecting them. The analogy is to an excited state of a diatomic molecule. The heavy quark potential, the potential from orbitally excited gluon configurations, and the “string breaking” potential (from the interaction between static sources and static-light mesons) are used as inputs in Born-Oppenheimer and coupled channel calculations for systems whose constituents include heavy quarks and antiquarks. Examples of such calculations include Refs. [151–153].

The system with the largest lattice literature is a bound state of two heavy quarks and two light antiquarks, for example $b\bar{b}u\bar{d}$. The story begins with the spectroscopy of doubly and triply charmed baryons in Fig. 15.12; the lattice calculations in that figure were performed in 2010–2017. In 2017 two groups of authors [154,155] realized that one could extrapolate from the masses of $Q\bar{q}$, $Q\bar{q}q$, and $QQ\bar{q}$ states ($Q$ for a heavy quark, $\bar{q}$ for a light antiquark) to predict the mass of a $QQ\bar{q}\bar{q}$ tetraquark. The $QQ\bar{q}\bar{q}$ state would be stable if the mass of the $Q$ were heavy enough. (This basically arises because the $QQ$ diquark is in an attractive color-triplet state and the effective binding energy scales as $\alpha_s^2 m_Q$.) The $J^P = 1^+$ $b\bar{b}u\bar{d}$ state was expected to be stable. Simultaneously, a lattice calculation [156] of the mass of a $b\bar{b}u\bar{d}$ state showed that it was below threshold to fall apart into a pair of B mesons. The $I(J^P) = 0(1^+)$ state is predicted to be stable by about 128(26) MeV [157] or 189(10) MeV [156]. (These authors put the $b\bar{b}s\bar{d}$ state at 98 MeV below threshold.) These numbers are in reasonable agreement with phenomenological estimates from Refs. [154,155].

These states are easier for the lattice theorist to study than other exotics because they are the lightest single particle states in their channels. They are difficult because they are just barely bound, and the usual approximation for a lattice calculation of unphysically heavy light quarks tends to raise their mass. The calculations have to include multiple trial states: the tetraquark itself plus pairs of mesons.

The situation with regard to the recently discovered $T_{cc}(3875)^+$ [158] is still unsettled. Both lattice calculations and HQET estimates put a $c\bar{c}u\bar{d}$ state right at threshold, which is a difficult place to explore on the lattice (or by any other method, for that matter). The $T_{cc}(3875)^+$ is seen in $D^0D^0\pi^+$ and researchers studying three body decays on the lattice (an open problem, at present) mark it as an obvious research target.

### 15.8.3 Electromagnetic effects

As a final part of spectroscopy we mention electromagnetic mass splittings (such as the neutron-proton mass difference). They are interesting but difficult. These calculations are important for determining the values of the quark masses (for a discussion see Sec. 60 in this Review). Knowing that the neutron is heavier than the proton tells us that these splittings have a complicated origin. One part of the shift is because the up and down quarks have slightly different masses. The second is that the quarks have (different) charges. Phenomenologists (compare Ref. [159]) combine Coulomb forces and spin-dependent electromagnetic hyperfine interactions to model their charge effects. In order to compute hadronic mass differences on the lattice, electromagnetic interactions must be included in the simulations. This creates a host of technical issues. An important one is that electromagnetic interactions are long range, but lattice simulations are done in finite volumes. The theoretical situation is summarized in the recent Flavour Lattice Averaging Group (FLAG) review [160]. A 2014 calculation, Ref. [161], presented the first results for electromagnetic mass splittings in the baryon octet, with good agreement with observation. Refs. [162–164] have performed calculations for meson splittings.

There is a small lattice literature associated with magnetic moments of baryons. The calculations are done by including a static magnetic field in the simulation and computing the energy difference between baryons in different spin states. We are aware of no calculations at physical quark masses. A recent calculation [165] used 800 and 450 MeV pions. When expressed in units of the nucleon magneton at the simulation point ($e/(2M_N m_\pi)$) the moments are in reasonable agreement with experiment and with quark model expectations.

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# 16. Heavy-Quark and Soft-Collinear Effective Theory

Revised August 2023 by C.W. Bauer (LBNL) and M. Neubert (PRISMA, Mainz Inst. for Theor. Physics, JG U.).

## 16.1 Effective Field Theories

Quantum field theories provide the most precise computational tools for describing physics at the highest energies. One of their characteristic features is that they almost inevitably involve multiple length scales. When trying to determine the value of an observable, quantum field theory demands that all possible virtual states and hence all particles be included in the calculation. Since these particles have widely different masses, the final prediction is sensitive to many scales. This fact represents a formidable challenge from a practical point of view. No realistic quantum field theories can be solved exactly, so that one needs to resort to approximation schemes; these, however, are typically most straightforward when only a single scale is involved at a time.

Effective field theories (EFTs) provide a general theoretical framework to deal with the multi-scale problems of realistic quantum field theories. This framework aims at reducing such problems to a combination of separate and simpler single-scale problems; simultaneously, however, it provides an organization scheme whereby the other scales are not omitted but allowed to play their role in a separate step of the computation. The philosophy and basic principles of this approach are very generic, and correspondingly EFTs represent a widely used method in many different areas of high-energy physics, from the low-energy scales of atomic and nuclear physics to the high-energy scales of (partly yet unknown) elementary-particle physics, see [1–3] for some early references. EFTs can play a role both within analytic perturbative computations and in the context of non-perturbative numerical simulations. One of the simplest applications of EFTs to particle physics concerns the description of an underlying theory that is only probed at energy scales $E < \Lambda$. Any particle with mass $m > \Lambda$ cannot be produced as a real state and therefore only leads to short-distance virtual effects. Thus, one can construct an effective theory in which the quantum fluctuations of such heavy particles are “integrated out” from the generating functional for Green functions. This results in a simpler theory containing only those degrees of freedom that are relevant to the energy scales under consideration. In fact, the standard model of particle physics itself is widely viewed as an EFT of some yet unknown, more fundamental theory [4, 5].

The development of any effective theory starts by identifying the degrees of freedom that are relevant to describe the physics at a given energy (or length) scale and constructing the Lagrangian describing the interactions among these fields. Short-distance quantum fluctuations associated with much smaller length scales are absorbed into the coefficients of the various operators in the effective theory. These coefficients are determined in a matching procedure, by requiring that the EFT reproduces the matrix elements of the full theory up to power corrections. In many cases the effective Lagrangian exhibits enhanced symmetries compared with the fundamental theory, giving non-trivial relations between different observables.

## 16.2 Heavy-Quark Effective Theory

Heavy-quark systems provide prime examples for applications of the EFT technology, because the hierarchy $m_Q \gg \Lambda_{QCD}$ (with $Q = b, c$) provides a natural separation of scales. Physics at the scale $m_Q$ is of a short-distance nature and can be treated perturbatively, while for heavy-quark systems there is always also some hadronic physics governed by the confinement scale $\Lambda_{QCD}$ of the strong interaction. Being able to separate the short-distance and long-distance effects associated with these two scales is crucial for any quantitative description. For instance, if the long-distance hadronic matrix elements are obtained from lattice QCD, then it is necessary to analytically compute the effects of short-wavelength modes that do not fit on the lattice. In many other instances, the long-distance physics can be encoded in a small number of hadronic parameters.

### 16.2.1 General idea and derivation of the effective Lagrangian

The simplest effective theory for heavy-quark systems is the heavy-quark effective theory (HQET) [6–9] (see [10, 11] for detailed discussions). It provides a simplified description of the soft interactions of a single heavy quark with light partons. This includes the interactions that bind the heavy quark with other light partons inside heavy mesons and baryons.

A softly interacting heavy quark is nearly on-shell. Its momentum may be decomposed as $p_Q = m_Q v + k$, where $v$ is the 4-velocity of the hadron containing the heavy quark. The “residual momentum” $k$ results from the soft interactions of the heavy quark with its environment and satisfies $v \cdot k \sim \Lambda_{QCD}$ and $k^2 \sim \Lambda_{QCD}^2$, which in the rest frame of the heavy hadron reduces to $k^\mu \sim \Lambda_{QCD}$. In the limit $m_Q \gg \Lambda_{QCD}$, the soft interactions do not change the 4-velocity of the heavy quark, which is therefore a conserved quantum number that is often used as a label on the effective heavy-quark fields.

A nearly on-shell Dirac spinor has two large and two small components. We define

$$Q(x) = e^{-im_Q v \cdot x} [h_v(x) + H_v(x)], \qquad (16.1)$$

where

$$h_v(x) = e^{im_Q v \cdot x} \frac{1 + \not p}{2} Q(x), \qquad H_v(x) = e^{im_Q v \cdot x} \frac{1 - \not p}{2} Q(x) \qquad (16.2)$$

are the large (“upper”) and small (“lower”) components of the spinor field, respectively. The extraction of the phase factor in (16.1) implies that the fields $h_v$ and $H_v$ carry the residual momentum $k$. The field $H_v$ is $1/m_Q$ suppressed relative to $h_v$ and describes quantum fluctuations far off the mass shell. Integrating it out using its equations of motion yields the HQET Lagrangian

$$\mathcal{L}_{\text{HQET}} = \bar{h}_v i v \cdot D_s h_v + \frac{1}{2m_Q} \left[ \bar{h}_v (i D_s)^2 h_v + C_{\text{mag}}(\mu) \frac{g}{2} \bar{h}_v \sigma_{\mu\nu} G_s^{\mu\nu} h_v \right] + \dots \qquad (16.3)$$

The covariant derivative $iD_s^\mu = i\partial^\mu + gA_s^\mu$ and the field strength $G_s^{\mu\nu}$ contain only the soft gluon field. Hard gluons have been integrated out, and their effects are contained in the Wilson coefficients of the operators in the effective Lagrangian. From the leading operator one derives the Feynman rules of HQET. The new operators entering at subleading order are referred to as the “kinetic energy” and “chromo-magnetic interaction”. The kinetic-energy operator corresponds to the first correction term in the Taylor expansion of the relativistic energy $E = m_Q + \vec{p}^2/2m_Q + \dots$. Lorentz invariance, which is encoded as a reparametrization invariance of the effective Lagrangian [12], ensures that its Wilson coefficient is not renormalized ($C_{\text{kin}} \equiv 1$). The coefficient $C_{\text{mag}}$ of the chromo-magnetic operator receives corrections starting at one-loop order [6].

### 16.2.2 Spin-flavor symmetry

The leading term in the HQET Lagrangian exhibits a global spin-flavor symmetry. Its physical meaning is that, in the infinite mass limit, the properties of hadronic systems containing a single heavy quark are insensitive to the spin and flavor of the heavy quark [13, 14]. The spin symmetry results from the fact that there are no Dirac matrices in the leading term of the effective Lagrangian in (16.3), implying that the interactions of the heavy quark with soft gluons leave its spin unchanged. The flavor symmetry arises since the mass of the heavy quark does not appear at leading order. For $n_Q$ heavy quarks moving at the same velocity, one can simply extend (16.3) by summing over $n_Q$ identical terms for heavy-quark fields $h_v^i$. The result is invariant under rotations in flavor space. When combined with the spin symmetry, the symmetry group becomes promoted to $SU(2n_Q)$. These symmetries are broken by the operators at subleading power in the $1/m_Q$ expansion.

The spin-flavor symmetry leads to many interesting relations between the properties of hadrons containing a heavy quark. The
most direct consequences concern the spectroscopy of such states [15]. In the heavy-quark limit, the spin of the heavy quark and the total angular momentum j of the light degrees of freedom are separately conserved by the strong interactions. Because of heavy-quark symmetry, the dynamics is independent of the spin and mass of the heavy quark. Hadronic states can thus be classified by the quantum numbers (flavor, spin, parity, etc.) of the light degrees of freedom. The spin symmetry predicts that, for fixed j ≠ 0, there is a doublet of degenerate states with total spin J = j ± 1 $_{2}$. The flavor symmetry relates the properties of states with different heavy-quark flavor.

16.2.3 Weak decay form factors

Of particular interest are the relations between the weak decay
form factors of heavy mesons, which parametrize hadronic ma-
trix elements of currents between two mesons containing a heavy
quark. These relations have been derived by Isgur and Wise [14],
generalizing ideas developed by Nussinov and Wetzel [16] and
Voloshin and Shifman [17]. For the purpose of this discussion,
it is convenient to work with a mass-independent normalization
of meson states and use velocity rather than momentum variables.

Consider the elastic scattering of a pseudoscalar meson, $P(v) \rightarrow P(v')$, induced by an external vector current coupled to the heavy quark contained in $P$, which acts as a color source moving with the meson's velocity $v$. The action of the current is to replace instantaneously the color source by one moving at velocity $v'$. Soft gluons need to be exchanged in order to rearrange the light degrees of freedom and build up the final state meson moving at velocity $v'$. This rearrangement leads to a form-factor suppression. The important observation is that, in the $m_Q \rightarrow \infty$ limit, the form factor can only depend on the Lorentz boost $\gamma = v \cdot v'$ connecting the rest frames of the initial and final-state mesons (as long as $\gamma = O(1)$). In the effective theory the hadronic matrix element describing the scattering process can therefore be written as

$$
\langle P(v') | \bar{h}_{v'} \gamma^{\mu} h_{v} | P(v) \rangle = \xi(v \cdot v')(v + v')^{\mu}, \quad (16.4)
$$

with a form factor ξ(v · v') that is real and independent of mQ.
By flavor symmetry, the form factor remains identical when one
replaces the heavy quark Q in one of the meson states by a heavy
quark Q' of a different flavor, thereby turning P into another
pseudoscalar meson P'. At the same time, the current becomes a
flavor-changing vector current. This universal form factor is called
the Isgur-Wise function [14]. For equal velocities the vector cur-
rent Jμ = ĥvγμhν is conserved in the effective theory, irrespective
of the flavor of the heavy quarks. The corresponding conserved
charges are the generators of the flavor symmetry. It follows that
the Isgur-Wise function is normalized at the point of equal veloci-
ties: ξ(1) = 1. Since the recoil energy of the daughter meson P' in
the rest frame of the parent meson P is E<sub>recoil</sub> = mP'(v · v' - 1),
the point v · v' = 1 is referred to as the zero-recoil limit. The
heavy-quark spin symmetry leads to additional relations among
weak decay form factors. It can be used to relate matrix elements
involving vector mesons to those involving pseudoscalar mesons,
which once again can be described completely in terms of the
universal Isgur-Wise function.

The form factor relations imposed by heavy-quark symme-
try describe the semileptonic decay processes B → Dℓν and
B → D*ℓν in the limit of infinite heavy-quark masses. They are
model-independent consequences of QCD. The known normaliza-
tion of the Isgur-Wise function at zero recoil can be used to ob-
tain a model-independent measurement of the element |V<sub>cb</sub>| of the
Cabibbo-Kobayashi-Maskawa (CKM) matrix. The semileptonic
decay B → D*ℓν is particularly well suited for this purpose [18].
Experimentally this is a very clean mode, since the reconstruction
of the D* meson mass provides a powerful rejection against back-
ground. From the theoretical point of view, it is ideal since the
decay rate at zero recoil is protected by Luke's theorem against
first-order power corrections in 1/m<sub>Q</sub> [19]. This is described in
more detail in Section 12. Corrections to the heavy-quark sym-
metry relations for the B → D(*) form factors near zero recoil can
also be constrained using sum rules derived in the small-velocity
limit [20, 21].

16.2.4 Decoupling transformation

At leading order in 1/m<sub>Q</sub>, the couplings of soft gluons to heavy quarks in the effective Lagrangian (16.3) can be removed by the field redefinition h<sub>v</sub>(x) = Y<sub>v</sub>(x) h<sub>v</sub><sup>(0)</sup>(x), where Y<sub>v</sub>(x) is a soft Wilson line along the direction of v, extending from minus infinity to the point x. In terms of the new fields the leading-order HQET Lagrangian becomes L<sub>HQET</sub> = h̄<sub>v</sub><sup>(0)</sup>iv∂·h̄<sub>v</sub><sup>(0)</sup>. It describes a free theory as far as the strong interactions of heavy quarks are concerned. However, the theory is nevertheless non-trivial in the presence of external sources. Consider, e.g., the case of a weak-interaction heavy-quark current

$$
\bar{h}_{v'} \gamma^{\mu} (1 - \gamma_5) h_v = \bar{h}_{v'}^{(0)} \gamma^{\mu} (1 - \gamma_5) Y_{v'}^{\dagger} Y_v h_v^{(0)}, \quad (16.5)
$$

where v and v' are the velocities of the heavy mesons contain-
ing the heavy quarks. Unless the two velocities are equal, cor-
responding to the zero-recoil limit discussed above, the object
Y<sub>v'</sub><sup>†</sup>Y<sub>v</sub> is non-trivial, and hence the soft gluons do not decouple
from the heavy quarks inside the current operator. One may in-
terpret Y<sub>v'</sub><sup>†</sup>Y<sub>v</sub> as a Wilson loop with a cusp at the point x, where
the two paths parallel to the different velocity vectors intersect.
The presence of the cusp leads to non-trivial ultra-violet behavior
(for v ≠ v'), which is described by a cusp anomalous dimension
Γ<sub>cusp</sub>(v·v') that was calculated at two-loop order in [22]. It coin-
cides with the velocity-dependent anomalous dimension of heavy-
quark currents, which was introduced in the context of HQET
in [23]. The interpretation of heavy quarks as Wilson lines is a
useful tool, which was put forward in one of the very first papers
on the subject [6]. This technology will be useful in the study of
the interactions of heavy quarks with collinear degrees of freedom
discussed later in this review.

16.2.5 Heavy-quark expansion for inclusive decays

The theoretical description of inclusive decays of hadrons con-
taining a heavy quark exploits two observations [24–28]: bound-
state effects related to the initial state can be calculated using the
heavy-quark expansion, and the fact that the final state consists
of a sum over many hadronic channels eliminates the sensitivity
to the properties of individual final-state hadrons. The second
feature rests on the hypothesis of quark-hadron duality, i.e. the
assumption that decay rates are calculable in QCD after a smear-
ing procedure has been applied [29]. In semileptonic decays, the
integration over the lepton spectrum provides a smearing over the
invariant hadronic mass of the final state (global duality). For
nonleptonic decays, where the total hadronic mass is fixed, the
summation over many hadronic final states provides an averaging
(local duality). Since global duality is a much weaker assumption,
the theoretical control of inclusive semileptonic decays is on firmer
footing.

Using the optical theorem, the inclusive decay width of a hadron $H_b$ containing a b quark can be written in the form

$$
\Gamma(H_b) = \frac{1}{M_{H_b}} \operatorname{Im} \langle H_b | i \int d^4 x T \{ \mathcal{H}_{\text{eff}}(x), \mathcal{H}_{\text{eff}}(0) \} | H_b \rangle. \qquad (16.6)
$$

The effective weak Hamiltonian for b-quark decays consists of dimension-6 four-fermion operators and dipole operators [30]. Because of the large mass of the b quark, it follows that the separation of fields in the time-ordered product in (16.6) is small, of order $x \sim 1/m_b$. It is thus possible to construct an operator-product expansion (OPE) for the time-ordered product, in which it is represented as a series of local operators in HQET. The leading operator $\bar{h}_v h_v$ has a trivial matrix element. The next contributions arise at $O(1/m_b^2)$ and give rise to two parameters $\mu_\pi^2(H_b)$ and $\mu_G^2(H_b)$, which are defined as the matrix elements of the heavy-quark kinetic energy and chromo-magnetic interaction inside the hadron $H_b$, respectively [31]. For the ground-state heavy mesons and baryons, one has $\mu_G^2(B) = 3(m_{B^*}^2 - m_B^2)/4 \simeq 0.36 \text{ GeV}^2$ and $\mu_G^2(\Lambda_b) = 0$. Thus, the total inclusive decay rate of a hadron $H_b$
can be written as [25,26]

$$ \Gamma(H_b) = \frac{G_F^2 m_b^5 |V_{cb}|^2}{192\pi^3} \left[ c_1 + c_2 \frac{\mu_\pi^2(H_b)}{2m_b^2} + c_3 \frac{\mu_G^2(H_b)}{2m_b^2} + O\left(\frac{1}{m_b^3}\right) + \dots \right], \quad (16.7) $$

where the prefactor arises from the loop integrations and is proportional to the fifth power of the b-quark mass. The coefficient functions $c_i$ are calculable order by order in perturbation theory. While $c_1$ corresponds to the decay rate of a free heavy quark, the higher-order coefficients systematically account for bound-state effects. The coefficients of the subleading operators and of the leading operator at third order in $1/m_b$ have been calculated at NLO [32–36]. At tree level the heavy-quark expansion has been performed up to fifth order in $1/m_b$ [37].

From the fully inclusive width in (16.7) one can obtain the lifetime of a heavy hadron via $\tau(H_b) = 1/\Gamma(H_b)$. Due to the universality of the leading term in the heavy-quark expansion, lifetime ratios such as $\tau(B^-)/\tau(\bar{B}^0)$, $\tau(\bar{B}_s^0)/\tau(\bar{B}^0)$ and $\tau(\Lambda_b)/\tau(\bar{B}^0)$ are particularly sensitive to the hadronic parameters determining the power corrections in the expansion. In order to understand these ratios theoretically, it is necessary to include phase-space enhanced power corrections of order $(\Lambda_{QCD}/m_b)^3$ [38,39] as well as short-distance perturbative effects [40] in the calculation (see [41] for a discussion of the corresponding calculations).

A formula analogous to (16.7) can be derived for differential distributions in specific inclusive decay processes, assuming that these distributions are integrated over a sufficiently large region of phase space to ensure quark-hadron duality. Important examples are the distributions in the lepton energy and the lepton invariant mass, as well as moments of the invariant hadronic mass distribution in the semileptonic processes $B \to X_u \ell \bar{\nu}$ and $\bar{B} \to X_c \ell \bar{\nu}$. A global fit of semileptonic decay distributions can be used to determine the CKM matrix elements $|V_{ub}|$ and $|V_{cb}|$ along with heavy-quark parameters such as the masses $m_b, m_c$ and the hadronic parameters $\mu_\pi^2(B), \mu_G^2(B)$. These determinations provide some of the most accurate values for these parameters (see e.g. [42–45]).

### 16.2.6 Shape functions and non-local power corrections

In certain regions of phase space, in which the hadronic final state in an inclusive heavy-hadron decay is made up of light energetic partons, the local OPE for inclusive decays must be replaced by a more complicated expansion involving hadronic matrix elements of non-local light-ray operators [46,47]. Prominent examples are the radiative decay $B \to X_s \gamma$ for large photon energy $E_\gamma$ near $m_B/2$, and the semileptonic decay $\bar{B} \to X_u \ell \bar{\nu}$ at large lepton energy or small hadronic invariant mass. In these cases, the differential decay rates at leading order in the heavy-quark expansion can be written in the factorized form $d\Gamma = H J \otimes S$ [48], where the hard function H and the jet function J are calculable in perturbation theory. The characteristic scales for these functions are set by $m_b$ and $(m_b \Lambda_{QCD})^{1/2}$, respectively. The soft function

$$ S(\omega) = \int \frac{dt}{4\pi} e^{-i\omega t} \langle \bar{B}(v) | \bar{h}_v(tn) Y_n(tn) Y_n^\dagger(0) h_v(0) | \bar{B}(v) \rangle \quad (16.8) $$

is a non-perturbative object called the shape function [46, 47]. Here $Y_n$ are soft Wilson lines along a light-like direction n aligned with the momentum of the hadronic final-state jet. The jet function and the shape function share a common variable $\omega \sim \Lambda_{QCD}$, and the symbol $\otimes$ denotes a convolution in this variable.

While the hard functions are different for the decays $\bar{B} \to X_s \gamma$ and $\bar{B} \to X_u \ell \bar{\nu}$, the jet and soft functions are identical at leading order in $\Lambda_{QCD}/m_Q$. This is particularly important for the shape function, which introduces non-perturbative physics into the theoretical predictions for the decay rates in the regions of experimental interest. The fact that both processes depend on the same non-perturbative function makes it possible to use the measured shape of the $\bar{B} \to X_s \gamma$ photon spectrum to reduce the theoretical uncertainties in the determination of the CKM element $|V_{ub}|$ from semileptonic decays. In higher orders of the heavy-quark

expansion, an increasing number of subleading jet and soft functions are required to describe the decay distributions [49]. These have been analyzed in detail at order $1/m_b$ [50–52]. In the case of $B \to X_s \gamma$ (and also in the related case of $\bar{B} \to X_s \ell \bar{\ell}$), some of these non-local effects survive in the total decay rate and give rise to irreducible hadronic uncertainties [53]. The technology for deriving the corresponding factorization theorems relies on the soft-collinear effective theory, to which we now turn.

## 16.3 Soft-Collinear Effective Theory

As discussed in the previous section, soft gluons that bind a heavy quark inside a heavy meson cannot change the virtuality of that heavy quark by a significant amount. The ratio $\Lambda_{QCD}/m_Q$ provides the expansion parameter in HQET, which is a small parameter since $m_Q \gg \Lambda_{QCD}$. This obviously does not work when considering light quarks. However, if the energy $Q$ (not to be confused with the flavor of a heavy quark) of the quarks is large, the ratio $\Lambda_{QCD}/Q$ provides a small parameter, which can be used to construct an effective theory. One major difference to HQET is that light energetic quarks cannot only emit soft gluons, but they can also emit collinear gluons (an energetic gluon in the same direction as the original quark), without parametrically changing their virtuality. Thus, to fully reproduce the long-distance physics of energetic quarks requires that one includes their interactions with both soft and collinear particles. The resulting effective theory is therefore called soft-collinear effective theory (SCET) [54–57] (see [58] for a review).

A single energetic particle can always be boosted to a frame where all momentum components have similar size, in which case there is no small expansion parameter. Thus, the presence of energetic particles must refer to a reference frame defined by external kinematics. SCET has a wide range of applications; some examples are the production of energetic, light states in the decay of a heavy particle in its rest frame, the production of energetic jets in collider environments, and the scattering of energetic particles off a target at rest. In this brief review we will outline the main features of this effective theory and mention a few selected applications.

### 16.3.1 General idea of the expansion

Consider a quark with virtuality much less than its energy $Q$, moving along the direction $\bar{n}$. It is convenient to parameterize the momentum $p_n$ of this particle in terms of its light-cone components, defined by $(p_n^-, p_n^+, p_n^\perp) = (\bar{n} \cdot p_n, n \cdot p_n, p_n^\perp)$, where $n^\mu = (1, \bar{n})$ and $\bar{n}^\mu = (1, -\bar{n})$ are light-like reference vectors, and $n \cdot p_n^\perp = \bar{n} \cdot p_n^\perp = 0$. The subscript n on the momentum indicates the direction of the collinear particle. In terms of these light-cone components, the virtuality satisfies $p_n^2 = p_n^\perp p_n + p_n^\perp \bar{n}$. The individual components of the momentum obey

$$ (p_n^-, p_n^+, p_n^\perp) \sim Q(1, \lambda^2, \lambda), \quad (16.9) $$

where $\lambda^2 = p^2/Q^2$ is the expansion parameter of SCET. The virtuality of such an energetic particle remains parametrically unchanged if it interacts with energetic particles in the same direction n, or with soft particles with momentum scaling as

$$ (p_s^-, p_s^+, p_s^\perp) \sim Q(\lambda^2, \lambda^2, \lambda^2). \quad (16.10) $$

SCET is constructed in such a way as to reproduce the long-distance dynamics arising from the interactions of collinear and soft degrees of freedom.

In the above power counting the transverse momenta of soft degrees of freedom scale as $p_s^\perp \sim Q\lambda^2$, which is much smaller than the transverse momenta $p_c^\perp \sim Q\lambda$ of collinear fields. This theory is usually called SCET$_I$. If the external kinematics require that the transverse momenta of both soft and collinear fields are of the same size, $p_c^\perp \sim p_s^\perp$, then the appropriate degrees of freedom have the scaling $p_c \sim Q(1, \lambda^2, \lambda)$ and $p_s \sim Q(\lambda, \lambda, \lambda)$. This theory is usually called SCET$_{II}$ and is required, e.g., for exclusive hadronic decays such as $\bar{B} \to D\pi$, where the virtuality of both collinear and soft degrees of freedom are set by $\Lambda_{QCD}$, or for the description of transverse-momentum distributions at colliders. SCET$_I$ power counting is assumed in the following sections, while SCET$_{II}$ is discussed in more detail in 16.3.6.