<figure><img src="image_342.png" /><figcaption>Figure 77.2: World average of $\alpha$, as well as contributions from individual modes, in terms of $1-CL$.</figcaption></figure>

Fig. 77.2 between the results from $B^0 \to (\rho\pi)^0$ and the others. The combination gives a single best-fit for $\alpha$ in $[0^\circ, 180^\circ]$, but an ambiguous solution exists at $\alpha \Leftrightarrow \alpha+180^\circ$. A secondary minimum close to zero is disfavoured [30].

## 77.4 $\gamma$

The angle $\gamma$ is the weak phase between Cabibbo-favoured $b \to c$ and suppressed $b \to u$ quark transitions and can be determined by exploiting interference between them. Explicitly, the ratio of suppressed to favoured amplitudes is parameterized by

$$r_B e^{i(\delta_B \pm \gamma)} = \frac{A_{\text{sup}}}{A_{\text{fav}}}, \qquad (77.11)$$

where $r_B$ is the ratio of amplitude magnitudes, $\delta_B$ the strong phase difference and the $+$ or $-$ sign depends on whether the transition involves a $\bar{b}$ or b quark, respectively. Measurement of $\gamma$ in this way has negligible theoretical uncertainty in the Standard Model [42], and therefore this approach provides a benchmark against which determinations from other methods, typically involving loop diagrams, can be compared.

Interference between these amplitudes is realised in $B^+ \to DK^+$ decays, where D represents an admixture of $D^0$ and $\bar{D}^0$ mesons. The simplest case is that of D decays to CP-eigenstates (GLW method [43, 44]), either CP-even such as $K^+K^-$ (CP$^+$) or CP-odd such as $K_S^0\pi^0$ (CP$^-$). The normalized decay rate and CP asymmetry are given by

$$R_{CP\pm} = \frac{\Gamma(B^- \to D_{CP\pm}K^-) + \Gamma(B^+ \to D_{CP\pm}K^+)}{\Gamma(B^- \to D^0K^-) + \Gamma(B^+ \to \bar{D}^0K^+)} \\ = 1 + r_B^2 \pm 2r_B \cos(\delta_B) \cos(\gamma), \qquad (77.12a)$$

$$A_{CP\pm} = \frac{\Gamma(B^- \to D_{CP\pm}K^-) - \Gamma(B^+ \to D_{CP\pm}K^+)}{\Gamma(B^- \to D_{CP\pm}K^-) + \Gamma(B^+ \to D_{CP\pm}K^+)} \\ = \frac{\pm 2r_B \sin(\delta_B) \sin(\gamma)}{1 + r_B^2 \pm 2r_B \cos(\delta_B) \cos(\gamma)}. \qquad (77.12b)$$

These relations assume the absence of direct CP violation in the charm system; experimentally allowed deviations from this assumption are too small to cause a significant bias on $\gamma$ [7, 45]. It is convenient to determine the $R_{CP\pm}$ quantities through a double ratio, normalizing to $B^+ \to D\pi^+$ decays involving the same final states, since this cancels potential sources of systematic uncertainty due to the branching fractions of the D decays that are used; small possible effects of CP violation in $B^+ \to D\pi^+$ decays are a source of systematic uncertainty in this procedure. The GLW method can be extended to include final states that are almost CP-eigenstates [46], as is the case in $D \to \pi^+\pi^-\pi^0$ and $D \to K^+K^-\pi^0$ decays, via inclusion of a factor encoding the fraction of CP-even (or CP-odd) content, $F_\pm$, which dilutes the sensitivity to $\gamma$ by reducing the size of the interference terms (the terms linear with $r_B$) in Eq. (77.12).

For other D decays, the ratio of amplitudes for the $D^0$ and $\bar{D}^0$ decays to the final state of interest has to be accounted for in the formalism. The ADS method [47, 48] uses D decays to final states such as $K^\mp\pi^\pm$, which involve interference between Cabibbo-favoured (CF) and doubly-Cabibbo-suppressed (DCS) transitions. The observables in this case are

$$A_{\text{ADS}} = \frac{\Gamma(B^- \to [K^+\pi^-]_D K^-) - \Gamma(B^+ \to [K^-\pi^+]_D K^+)}{\Gamma(B^- \to [K^+\pi^-]_D K^-) + \Gamma(B^+ \to [K^-\pi^+]_D K^+)} \\ = \frac{2r_B r_D \sin(\delta_B + \delta_D) \sin(\gamma)}{r_B^2 + r_D^2 + 2r_B r_D \cos(\delta_B + \delta_D) \cos(\gamma)}, \qquad (77.13a)$$

$$R_{\text{ADS}} = \frac{\Gamma(B^- \to [K^+\pi^-]_D K^-) + \Gamma(B^+ \to [K^-\pi^+]_D K^+)}{\Gamma(B^- \to [K^-\pi^+]_D K^-) + \Gamma(B^+ \to [K^+\pi^-]_D K^+)} \\ = r_B^2 + r_D^2 + 2r_B r_D \cos(\delta_B + \delta_D) \cos(\gamma), \qquad (77.13b)$$

where $r_D$ and $\delta_D$ are the amplitude magnitude ratio and strong phase difference between the CF and DCS D decay. An alternative pair of observables, $(R_-, R_+)$, is also sometimes used, where $R_-$ ($R_+$) is the ratio of decay rates between the suppressed and favoured transitions for $B^-$ ($B^+$) decays. The $R_-$ and $R_+$ observables are statistically independent, while $A_{\text{ADS}}$ and $R_{\text{ADS}}$ are not (in particular, the uncertainty on $A_{\text{ADS}}$ depends on the central value of $R_{\text{ADS}}$). However, the pair $(R_-, R_+)$ has more correlated sources of systematic uncertainty compared to $(A_{\text{ADS}}, R_{\text{ADS}})$. The observables of Eq. (77.13) are therefore usually preferred once a significant signal is established. The ADS method can also be extended to include decays to multibody final states, such as $D \to K^\pm\pi^+\pi^0$ and $D \to K^\pm\pi^+\pi^-, \text{ by addition of a coherence factor [49] which appears in the interference terms of Eq. (77.13) and accounts for dilution of the sensitivity due to variation of the decay amplitude across the phase space of the final state. A similar method can be used for singly Cabibbo-suppressed D decays to non-CP eigenstates such as $K^*K$ [50].

For D decays to multibody self-conjugate final states (BPGGSZ method [51, 52]), such as $D \to K_S^0\pi^+\pi^-$, one can write the partial decay rate as a function of the position in the phase space in terms of the “Cartesian parameters” $x_\pm + iy_\pm = r_B e^{i(\delta_B \pm \gamma)}$:

$$d\Gamma(B^{\pm} \to [K_S^{0}\pi^{+}\pi^{-}]_{D}K^{\pm}) = A_{(\mp, \pm)}^{2} + r_{B}^{2}A_{(\pm, \mp)}^{2} \\ + 2A_{(\pm, \mp)}A_{(\mp, \pm)}[x_{\pm}c_{D(\pm, \mp)} + y_{\pm}s_{D(\pm, \mp)}], \qquad (77.14)$$

where the notation $(+, -)$ is shorthand for the dependence on the Dalitz-plot position — the squared invariant masses of $K_S^0\pi^+$ and $K_S^0\pi^-$ combinations, respectively. The quantities $A_{(+,-)}$ and $A_{(-,+)}$ represent the magnitudes of the $D^0$ and $\bar{D}^0$ decay amplitudes at the position $(+, -)$ and are interchangable with their CP conjugate amplitudes because CP conservation is assumed in the D decay (i.e. $A_{(-,+)} = \bar{A}_{(+,-)}$). The quantities $c_{D(\pm,\mp)}$ and $s_{D(\pm,\mp)}$ are the cosine and sine of the strong phase difference, $\delta_{D(+,-)} = \arg(\bar{A}_{(+,-)}) - \arg(A_{(+,-)})$, between the $\bar{D}^0$ and $D^0$ amplitudes. These quantities can be determined from an amplitude model, although this leads to a hard-to-quantify systematic uncertainty associated to the composition of the model. An alternative, “model-independent”, approach involves dividing the phase space into appropriate bins. In this case, the analysis benefits from external input on the values of $c_D$ and $s_D$ integrated over each bin. Measurements of these external parameters have been performed for the $D \to K_S^0\pi^+\pi^-$ decay by the CLEO-c and BES-III collaborations [53–56]. The use of common input values for these parameters in model-independent determinations of $\gamma$ with the BPGGSZ method by different experiments is a source of correlation between experiments that is currently negligible but will become more significant as the available B meson data samples increase in size.

The discussion above refers to $B^+ \to DK^+$ decays, but analogous measurements can be made also for additional channels such as $B^+ \to D^*K^+$ (with $D^* \to D\pi^0, D\gamma$) and $B^+ \to DK^*$ (with $K^{*+} \to K_S^0\pi^+, K^+\pi^0$). In the limit that these can be treated purely as two-body decays, the expressions for $B^+ \to DK^+$ are modified only by ensuring the $r_B$ and $\delta_B$ parameters are specific to each B decay. Moreover, for $B^+ \to D^*K^+$ decays an effective
shift of the strong phase by $\pi$ between $D^* \to D\pi^0$ and $D\gamma$ decays [57] has to be taken into account. In case the finite width of the decaying resonance is non-negligible, as is the case for the $K^*(892)$ state, additional amplitudes can contribute leading to a dilution of the sensitivity, which can be accounted for in the formalism through the introduction of a relevant coherence factor. For the $B^0 \to DK^{*0}$ decay, full amplitude analysis of the $B^0 \to DK^+\pi^-$ Dalitz plot provides additional sensitivity compared to the quasi-two-body approach [58, 59].

It is also possible to measure $\gamma$ using decay-time-dependent analysis of the $B_s^0$ meson [60]. The weak phase arising in the interference between direct decay of $B_s^0 \to D_s^\mp K^\pm$ and decay via mixing is $(\gamma - 2\beta_s)$, where $\beta_s$ is the angle associated with $B_s^0 \to J/\psi\phi$ decays in a similar way to the relation between $\beta$ and $B_s^0 \to J/\psi K_s^0$ decays described in Sec. 77.2. Sufficient information can be obtained from the tagged, decay-time-dependent rates of $B_s^0 \to D_s^\mp K^\pm$ decays that this weak phase can be determined, up to an ambiguity, together with the strong phase difference between, and the ratio of the magnitudes of, the suppressed and favoured amplitudes. Since $\beta_s$ is known to good precision [8], measurements of the decay-time-dependent CP-asymmetry observables in $B_s^0 \to D_s^\mp K^\pm$ decays can be used to infer constraints on $\gamma$. Alternatively, if effects of penguin pollution in $B_s^0 \to J/\psi\phi$ decays [17, 18] are a concern, as they will become in the future, results from the $B_s^0 \to D_s^\mp K^\pm$ mode can be combined with an independent precise measurement of $\gamma$ to provide a penguin-free determination of $\beta_s$.

The average for $\gamma$ requires a non-trivial combination due the complicated relations between the observables and the physics parameters of interest, such as in Eqs. (77.12), (77.13) and (77.14). Moreover, hadronic parameters such as $r_B$ and $\delta_B$ defined in Eq. (77.11) are common to all different D decay modes (but differ for each B decay mode). Thus, it is not correct to simply average results for $\gamma$ obtained by different experiments or in different channels. Instead, measurements of rate asymmetries, rate ratios and the Cartesian parameters are taken as inputs to the combination, from which results are obtained not only for $\gamma$ but also for the hadronic parameters. Independent measurements of auxiliary parameters such as $r_D$ and $\delta_D$ are also treated as inputs to the combination. In some cases the B decay data can help to reduce uncertainties on these auxiliary parameters and therefore a simultaneous fit of charm and beauty data can provide stronger constraints [61]; this approach however is not currently used for the world average.

The precision to which $\gamma$ can be measured with a particular B decay is approximately inversely proportional to the value of $r_B$. Thus, results from channels with smaller yields but larger values of $r_B$, such as $B^0 \to DK^{*0}$ and $B_s^0 \to D_s^\mp K^\pm$ ($r_B \approx 0.3-0.4$), can have a significant impact on the world average and are included in the combination. By contrast the $B^+ \to D\pi^+$ mode, for which large samples are available but $r_B \approx 0.005$, has little impact and is also more sensitive to potential systematic biases; hence it is not included. The sensitivity of the world average at present is dominated by results from $B^+ \to DK^+$, where $r_B \approx 0.1$, in particular results with the GLW [62], ADS [62] and BPGGSZ [63] methods.

The world average obtained for the angle $\gamma$, obtained by combining results from $B^+ \to DK^+$, $D^*K^+$, $DK^{*+}$, $DK^{+\pi^+\pi^-}$, $B^0 \to DK^{+\pi^-}$, $B_s^0 \to D_s^\mp K^\pm$ and $B_s^0 \to D_s^\mp K^{\pm\pi^+\pi^-}$ decays, is [4]

$$ \gamma = (65.7 \pm 3.0)^\circ, \qquad (77.15) $$

where the quoted uncertainty is at the 68.3% CL.

Effects related to charm and kaon mixing and CP violation are generally negligible at the current level of precision, in particular for modes with $r_B \gtrsim 0.1$. An exception is that a dependence of the selection efficiency on the charm decay time can induce a dependence of the observables on charm mixing parameters [64]. Such effects can be important at hadron collider experiments such as LHCb, but can be and are corrected for. Interactions of neutral kaons with detector material can also cause a bias in determination of $\gamma$ from modes with low values of $r_B$ [65], such as the BPGGSZ method applied to $B^+ \to D\pi^+$, but are negligible in modes with larger $r_B$ values. A further subtlety is that the

<figure><img src="image_343.png" /><figcaption>Figure 77.3: World average of $\gamma \equiv \phi_3$, as well as contributions from individual modes, in terms of $1-\text{CL}$.</figcaption></figure>

<figure><img src="image_344.png" /><figcaption>Figure 77.4: Constraints from the measurements of the angles of the CKM unitarity triangle in the $(\bar{\rho}, \bar{\eta})$ plane.</figcaption></figure>

identification of the weak phase between suppressed and favoured amplitudes in $B \to DK$ decays with $\gamma$, as defined in Eq. (77.2), assumes that the $2 \times 2$ submatrix of the CKM matrix is real, i.e. that $\arg[V_{ud}V_{us}^*/(V_{cd}V_{cs}^*)] = 0$. This is true to an excellent approximation in the Standard Model, and is known experimentally from independent studies of the charm system [45] to contribute negligible bias to current measurements. Nonetheless, in future it will be possible to test directly this assumption by comparing the value of $\gamma$ obtained from the $B \to DK$ and $B \to D\pi$ systems.

Effects from correlated uncertainties between amplitude models and strong phase differences in charm decays are negligible and are not explicitly accounted for in the combination, nor are effects related to charm and kaon mixing and CP violation. This world average, together with results split by decay mode, is shown in Fig. 77.3. The combination has a total of 173 experimental inputs from which 36 parameters are determined, an overall $\chi^2$ of 159.5, which corresponds to a p-value of 9% indicating reasonable agreement between the inputs. The combination gives a single solution for $\gamma$ in $[0^\circ, 180^\circ]$, but an ambiguous solution exists at $\gamma \Leftrightarrow \gamma + 180^\circ$.

## 77.5 Summary

Experimental progress has resulted in all three angles of the CKM unitarity triangle being measured with good accuracy, with $\beta$ known to subdegree precision and both $\alpha$ and $\gamma$ known to better than $5^\circ$. The constraints from these three measurements in the $(\bar{\rho}, \bar{\eta})$ plane are shown in Fig. 77.4; further discussion and comparison with constraints from independent measurements can be found in Ref. [6]. The determinations of all three angles remain statistically limited, but it will be a challenge for experiments to
ensure that this remains the case as the precision improves. Con-
sequently, the correct treatment of sources of correlation between
the measurements that go into the world average combinations is
becoming increasingly important.

References

[1] N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963).

[2] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. **49**, 652 (1973).

[3] C. Jarlskog, Phys. Rev. Lett. **55**, 1039 (1985).

[4] Y. S. Amhis et al. (HFLAV), Phys. Rev. D **107**, 5, 052008 (2023), updated averages are available online at hflav.web.cern.ch, [arXiv:2206.07501].

[5] See the review on “Semileptonic b-Hadron Decays, Determination of $V_{cb}$, $V_{ub}$” in this Review.

[6] See the review on “Cabibbo-Kobayashi-Maskawa Mixing Matrix,” in this Review.

[7] See the review on “CP Violation in the Quark Sector,” in this Review.

[8] See the review on “$B^0-\bar{B}^0$ Mixing” in this Review.

[9] A. Carter and A. Sanda, Phys. Rev. D23, 1567 (1981).

[10] I. Bigi and A. Sanda, Nucl. Phys. B193, 85 (1981).

[11] O. Long et al., Phys. Rev. D68, 034010 (2003), [hep-ex/0303030].

[12] B. Aubert et al. (BaBar), Phys. Rev. D79, 072009 (2009), [arXiv:0902.1708].

[13] I. Adachi et al., Phys. Rev. Lett. 108, 171802 (2012), [arXiv:1201.4643].

[14] R. Aaij et al. (LHCb), Phys. Rev. Lett. 132, 021801 (2024), [arXiv:2309.09728].

[15] H.-n. Li and S. Mishima, JHEP 03, 009 (2007), [hep-ph/0610120].

[16] M. Jung, Phys. Rev. D86, 053008 (2012), [arXiv:1206.2050].

[17] K. De Bruyn and R. Fleischer, JHEP 1503, 145 (2015), [arXiv:1412.6834].

[18] P. Frings, U. Nierste and M. Wiebusch, Phys. Rev. Lett. 115, 061802 (2015), [arXiv:1503.00859].

[19] M. Ciuchini, M. Pierini and L. Silvestrini, Phys. Rev. Lett. 95, 221804 (2005), [hep-ph/0507290].

[20] S. Faller et al., Phys. Rev. D79, 014030 (2009), [arXiv:0809.0842].

[21] Z. Ligeti and D. Robinson, Phys. Rev. Lett. 115, 251801 (2015), [arXiv:1507.06671].

[22] I. Adachi et al. (BaBar, Belle), Phys. Rev. Lett. 121, 261801 (2018), [arXiv:1804.06152].

[23] I. Adachi et al. (BaBar, Belle), Phys. Rev. D98, 112012 (2018), [arXiv:1804.06153].

[24] M. Antonelli et al., Phys. Rept. 494, 197 (2010), [arXiv:0907.5386].

[25] M. Gronau and D. London, Phys. Rev. Lett. 65, 3381 (1990).

[26] R. Aaij et al. (LHCb), Phys. Rev. D98, 032004 (2018), [arXiv:1805.06759].

[27] R. Aaij et al. (LHCb), JHEP 03, 075 (2021), [arXiv:2012.05319].

[28] J. P. Lees et al. (BaBar), Phys. Rev. D87, 052009 (2013), [arXiv:1206.3525].

[29] I. Adachi et al. (Belle), Phys. Rev. D88, 092003 (2013), [arXiv:1302.0551].

[30] J. Charles et al., Eur. Phys. J. C77, 574 (2017), [arXiv:1705.02981].

[31] T. Julius et al. (Belle), Phys. Rev. D96, 032007 (2017), [arXiv:1705.02083].

[32] B. Aubert et al. (BaBar), Phys. Rev. D76, 052007 (2007), [arXiv:0705.2157].

[33] P. Vanhoefer et al. (Belle), Phys. Rev. D93, 032010 (2016), [Addendum ibid: D94 099903 (2016)], [arXiv:1510.01245].

[34] B. Aubert et al. (BaBar), Phys. Rev. D78, 071104 (2008), [arXiv:0807.4977].

[35] A. E. Snyder and H. R. Quinn, Phys. Rev. D48, 2139 (1993).

[36] J. P. Lees et al. (BaBar), Phys. Rev. D88, 012003 (2013), [arXiv:1304.3503].

[37] A. Kusaka et al. (Belle), Phys. Rev. Lett. 98, 221602 (2007), [hep-ex/0701015].

[38] A. Kusaka et al. (Belle), Phys. Rev. D77, 072001 (2008), [arXiv:0710.4974].

[39] M. Gronau and J. Zupan, Phys. Rev. D71, 074017 (2005), [hep-ph/0502139].

[40] S. Gardner, Phys. Rev. D72, 034015 (2005), [hep-ph/0505071].

[41] A. Falk et al., Phys. Rev. D69, 011502 (2004), [hep-ph/0310242].

[42] J. Brod and J. Zupan, JHEP 01, 051 (2014), [arXiv:1308.5663].

[43] M. Gronau and D. London, Phys. Lett. B253, 483 (1991).

[44] M. Gronau and D. Wyler, Phys. Lett. B265, 172 (1991).

[45] See the review on “$D^0-\bar{D}^0$ Mixing” in this Review.

[46] M. Nayak et al., Phys. Lett. B740, 1 (2015), [arXiv:1410.3964].

[47] D. Atwood, I. Dunietz and A. Soni, Phys. Rev. Lett. 78, 3257 (1997), [hep-ph/9612433].

[48] D. Atwood, I. Dunietz and A. Soni, Phys. Rev. D63, 036005 (2001), [hep-ph/0008090].

[49] D. Atwood and A. Soni, Phys. Rev. D68, 033003 (2003), [hep-ph/0304085].

[50] Y. Grossman, Z. Ligeti and A. Soffer, Phys. Rev. D67, 071301 (2003), [hep-ph/0210433].

[51] A. Giri et al., Phys. Rev. D68, 054018 (2003), [hep-ph/0303187].

[52] A. Bondar, Proceedings of BINP special analysis meeting on Dalitz analysis, 24-26 Sep. 2002, unpublished.

[53] R. Briere et al. (CLEO), Phys. Rev. D80, 032002 (2009), [arXiv:0903.1681].

[54] J. Libby et al. (CLEO), Phys. Rev. D82, 112006 (2010), [arXiv:1010.2817].

[55] M. Ablikim et al. (BESIII), Phys. Rev. Lett. 124, 241802 (2020), [arXiv:2002.12791].

[56] M. Ablikim et al. (BESIII), Phys. Rev. D101, 112002 (2020), [arXiv:2003.00091].

[57] A. Bondar and T. Gershon, Phys. Rev. D70, 091503 (2004), [hep-ph/0409281].

[58] T. Gershon, Phys. Rev. D79, 051301 (2009), [arXiv:0810.2706].

[59] T. Gershon and M. Williams, Phys. Rev. D80, 092002 (2009), [arXiv:0909.1495].

[60] R. Aleksan, I. Dunietz and B. Kayser, Z. Phys. C54, 653 (1992).

[61] R. Aaij et al. (LHCb), JHEP 12, 141 (2021), [arXiv:2110.02350].

[62] R. Aaij et al. (LHCb), JHEP 04, 081 (2021), [arXiv:2012.09903].

[63] R. Aaij et al. (LHCb), JHEP 02, 169 (2021), [arXiv:2010.08483].

[64] M. Rama, Phys. Rev. D89, 014021 (2014), [arXiv:1307.4384].

[65] M. Bjørn and S. Malde, JHEP 07, 106 (2019), [arXiv:1904.01129].
# 78. Spectroscopy of Mesons Containing Two Heavy Quarks

Revised March 2024 by J. J. Hernández-Rey (IFIC, Valencia), C. Lourenço (CERN), R.E. Mitchell (Indiana U.), S. Navas (Granada U.) and C. Patrignani (Bologna U.).

A golden age for heavy quarkonium physics dawned at the turn of this century, initiated by the confluence of exciting advances in quantum chromodynamics (QCD) and an explosion of related experimental activity. The subsequent broad spectrum of breakthroughs, surprises, and continuing puzzles had not been anticipated. Indeed, CLEO-c, BESIII, and the B-factories, later joined by ATLAS, CMS and LHCb, have made a series of groundbreaking observations. For an extensive presentation of the status of heavy quarkonium physics, the reader is referred to several reviews [1–9]. This note focuses on experimental developments in heavy quarkonium spectroscopy with very few theoretical comments. Possible theoretical interpretations of the states not predicted by the quark model are presented in the review “Heavy non-$q\bar{q}$ mesons”. Note that in this review we follow the new naming scheme for hadrons (see the review “Naming scheme for hadrons” in the current edition of the RPP).

This review covers states discovered since 2003, the year that marked the unexpected discovery of the X(3872) [10]. The X(3872), now called $\chi_{c1}(3872)$, was the first of the mesons containing two heavy quarks that could not be easily accommodated by the $q\bar{q}$ quark model. Its discovery was a watershed event in meson spectroscopy. In earlier versions of this write-up the particles were sorted according to an assumed *conventional* or *unconventional* nature with respect to the quark model. However, since this classification is not always unambiguous, we here follow Ref. [11] and sort the states into three groups, namely states below (cf. Table 78.1), above (cf. Table 78.2), and near (cf. Table 78.3) the lowest open-flavor thresholds. Due to the presence of many open-flavor thresholds, we note that the division of states between “above” and “near” open-flavor thresholds is not absolute.

## 78.1 States Below Open-Flavor Threshold

Table 78.1 lists properties of recently observed heavy quarkonium states located below the lowest open-flavor thresholds. Those are expected to be (at least prominently) conventional quarkonia. The majority of charmonium ($c\bar{c}$) and bottomonium ($b\bar{b}$) states were established prior to 2003.

### 78.1.1 Charmonium

The $h_c(1P)$ is the $1^1P_1$ charmonium state, the singlet partner of the long-known $\chi_{cJ}$ triplet $1^3P_J$. After being firmly established in 2005 through the process $\psi(2S) \rightarrow \pi^0 h_c(1P)$ [12], it has since been studied extensively by BESIII using large samples of $\psi(2S)$ decays. Exclusive hadronic decays of the $h_c(1P)$, strongly suppressed relative to the dominant radiative transition $h_c(1P) \rightarrow \gamma\eta_c(1S)$, were first observed in 2019 [13] and additional decays were found in 2020 [14] and 2022 [15].

Belle reported an observation of the $\psi_2(1D)$ decaying to $\gamma\chi_{c1}(1P)$ with $J^{PC}$ presumed to be $2^{--}$ [16]. This state is listed in Table 78.1 as $\psi_2(3823)$. Its existence was confirmed with high significance by BESIII [17, 18]. While the negative C-parity is indeed established by its observed decay channel, the assignment of $J=2$ was made by matching to the closest quark model state ($1^3D_2$) and requires experimental confirmation.

The $1^1D_2$ state, or the $\eta_{c2}(1D)$, with a mass expected near 3820 MeV, has not yet been observed. Recently Belle performed a search in $B \rightarrow \eta_{c2}(1D)K(\pi)$ decays in the mass range 3795–3845 MeV and found no signal [19]. Thus, the $\eta_{c2}(1D)$ remains the only unobserved conventional charmonium state that does not have open-charm decays.

### 78.1.2 Bottomonium

The ground state of bottomonium, $\eta_b(1S)$, is well established. After the initial reports from BaBar in radiative decays of the $\Upsilon(3S)$ (observation) [20] and $\Upsilon(2S)$ (evidence) [21], Belle confirmed the existence of the $\eta_b(1S)$ with more than $5\sigma$ significance in radiative decays of the newly discovered $h_b(1P)$ [22, 23] and $h_b(2P)$ [22] (see next paragraph), as well as in $\Upsilon(2S)$ radiative decays [24]. Belle has also reported strong evidence for

the $\eta_b(2S)$ [22], but it still needs confirmation at the $5\sigma$ level. Note that there are hints of tension in the $\eta_b(1S)$ mass as measured in radiative M1 and E1 transitions. In the M1 transition $\Upsilon(2S) \rightarrow \gamma\eta_b(1S)$ Belle measures a mass of $9394.8^{+2.7+4.5}_{-3.1-2.7}$ MeV [24], while in the E1 transitions $h_b(1P, 2P) \rightarrow \gamma\eta_b(1S)$ Belle measures $9402.4 \pm 1.5 \pm 1.8$ MeV [22]. This tension may point to an incomplete understanding of the $\eta_b(1S)$ lineshape in different production mechanisms.

<figure><img src="image_345.png" /><figcaption>**Figure 78.1:** From Belle [25], the mass recoiling against $\pi^+\pi^-$ pairs, $M_{\text{miss}}$, in $e^+e^-$ collision data taken near the peak of the $\Upsilon(10860)$. The smooth combinatorial and $K_S^0 \rightarrow \pi^+\pi^-$ background contributions have been subtracted. The fit to the various labeled signal contributions is overlaid (curve). The fit is performed separately in three regions with boundaries indicated by the vertical dashed lines.</figcaption></figure>

The $h_b(1P)$, the bottomonium counterpart of the $h_c(1P)$, and the next excited state, the $h_b(2P)$, were simultaneously discovered by Belle using dipion transitions from the $\Upsilon(10860)$ [25] (Fig. 78.1). The same analysis also showed the $\Upsilon_J(1D)$, the lowest-lying D-wave triplet of the $b\bar{b}$ system, but did not resolve the $J=1, 2, 3$ components. The search for the $h_b(1P)$ was directly inspired by a CLEO result [26], which found a surprisingly copious production of $e^+e^- \rightarrow \pi^+\pi^- h_c(1P)$ as well as an indication that $\psi(4230) \rightarrow \pi^+\pi^- h_c(1P)$ occurs at a comparable rate with the signature mode $\psi(4230) \rightarrow \pi^+\pi^- J/\psi(1S)$. The presence of $\Upsilon(nS)$ peaks in Fig. 78.1 at rates two orders of magnitude larger than expected, along with separate studies with exclusive decays $\Upsilon(nS) \rightarrow \mu^+\mu^-$, allow precise calibration of the $\pi^+\pi^-$ recoil mass spectrum and very accurate measurements of the $h_b(1P)$ and $h_b(2P)$ masses. Both corresponding hyperfine splittings are consistent with zero within an uncertainty of about 1.5 MeV (lowered to 1.1 MeV for the $h_b(1P)$ in Ref. [27]). Belle later observed the transition $\Upsilon(4S) \rightarrow h_b(1P)\eta$ [23] and the corresponding 1P hyperfine splitting was also found to be compatible with zero at a similar precision level.

Just before Christmas 2011, ATLAS offered the world a beautiful gift, in the form of the discovery of the $\chi_b(3P)$ quarkonium state [28], observed by combining dimuons from $\Upsilon(1S)$ or $\Upsilon(2S)$ decays with photons emitted in the radiative $\chi_b(3P)$ decays (Fig. 78.2, bottom left panel). The new resonance, with a mass of $10\,530 \pm 5(\text{stat}) \pm 9(\text{syst})$ MeV, was soon confirmed by D0 [29]. Also LHCb observed the $\chi_b(3P)$ peak, using the full Run 1 event sample, corresponding to an integrated luminosity of $3\ \text{fb}^{-1}$ [30] (Fig. 78.2, middle left panel). Finally, CMS used $80\ \text{fb}^{-1}$ of 13 TeV pp collisions, collected in 2016 and 2017, to show two well-resolved $\chi_{b1}(3P)$ and $\chi_{b2}(3P)$ peaks [31], separated by a mass difference of $10.60 \pm 0.64(\text{stat}) \pm 0.17(\text{syst})$ MeV (Fig. 78.2, top left panel). The remarkable precision of the individual mass measurements, with relative uncertainties as small as 50 ppm, shows that the LHC experiments can provide important results in the field of hadron spectroscopy, especially in the case of heavy particles, which require very high collision energies and large event samples.

### 78.1.3 $B_c$ System

The $B_c^\pm$ family is quite special because these (charged) quarkonium states consist of two heavy quarks of different flavor. Among
<table><caption>**Table 78.1:** New states below the open-flavor thresholds in the $c\bar{c}$, $b\bar{c}$, and $\bar{b}b$ regions, ordered by mass. Masses $m$ and widths $\Gamma$ represent the PDG23 weighted averages with statistical and systematic uncertainties added in quadrature. In the Production column, the state is always denoted by $X$. Ellipses (...) indicate inclusively selected event topologies, i.e., additional particles not directly detected by experiment. A question mark (?) indicates an unmeasured value. The Discovery Year column gives the date of the first measurement cited. The Summary Table column indicates whether or not the state appears in the summary tables, usually requiring at least two independent experiments with significance of >5σ. Refer to the particle listings for references and further information.</caption><thead><tr><th>PDG Name</th><th>Former Name</th><th><i>m</i> (MeV)</th><th><i>Γ</i> (MeV)</th><th><i>I</i><sup>G</sup>(<i>J</i><sup>PC</sup>)</th><th>Production</th><th>Decay</th><th>Discovery Year</th><th>Summary Table</th></tr></thead><tbody><tr><td><i>h</i><sub>c</sub>(1P)</td><td></td><td>3525.37 ± 0.14</td><td>0.78<sup>+0.30</sup><sub>-0.27</sub></td><td>0<sup>-</sup>(1<sup>-</sup>)</td><td><i>ψ</i>(2S) → <i>π</i><sup>0</sup><i>X</i><br><i>p</i><span style="text-decoration: overline">p</span> → <i>X</i><br><i>e</i><sup>+</sup><i>e</i><sup>-</sup> → <i>π</i><i>π</i><i>X</i></td><td>γ<i>η</i><sub>c</sub>(1S)<br>hadrons<br>(see listings)</td><td>2004</td><td>YES</td></tr><tr><td><i>ψ</i><sub>2</sub>(3823)</td><td><i>X</i>(3823)</td><td>3823.51 ± 0.34</td><td>&lt; 2.9</td><td>0<sup>-</sup>(2<sup>-</sup>)</td><td><i>B</i> → <i>K</i><i>X</i><br><i>e</i><sup>+</sup><i>e</i><sup>-</sup> → <i>π</i><sup>+</sup><i>π</i><sup>-</sup><i>X</i></td><td>γ<i>χ</i><sub>c1</sub>(1P)<br><i>π</i><sup>+</sup><i>π</i><sup>-</sup><i>J</i>/<i>ψ</i>(1S)</td><td>2013</td><td>YES</td></tr><tr><td><i>B</i><sup>+</sup><sub>c</sub></td><td></td><td>6274.47 ± 0.32</td><td>stable</td><td>0(0<sup>-</sup>)</td><td><i>p</i><span style="text-decoration: overline">p</span> → <i>X</i>...<br><i>pp</i> → <i>X</i>...</td><td><i>π</i><sup>+</sup><i>J</i>/<i>ψ</i>(1S)<br>(see listings)</td><td>2007</td><td>YES</td></tr><tr><td><i>B</i><sup>+</sup><sub>c</sub>(2S)</td><td></td><td>6871.2 ± 1.0</td><td>?</td><td>0(0<sup>-</sup>)</td><td><i>pp</i> → <i>X</i>...</td><td><i>B</i><sup>+</sup><sub>c</sub><i>π</i><sup>+</sup><i>π</i><sup>-</sup></td><td>2014</td><td>YES</td></tr><tr><td><i>η</i><sub>b</sub>(1S)</td><td></td><td>9398.7 ± 2.0</td><td>10<sup>+5</sup><sub>-4</sub></td><td>0<sup>+</sup>(0<sup>-</sup>)</td><td><i>γ</i>(2S, 3S) → <i>γ</i><i>X</i><br><i>h</i><sub>b</sub>(1P, 2P) → <i>γ</i><i>X</i></td><td>hadrons<br>(see listings)</td><td>2008</td><td>YES</td></tr><tr><td><i>h</i><sub>b</sub>(1P)</td><td></td><td>9899.3 ± 0.8</td><td>?</td><td>0<sup>-</sup>(1<sup>-</sup>)</td><td><i>γ</i>(10860) → <i>π</i><sup>+</sup><i>π</i><sup>-</sup><i>X</i><br><i>γ</i>(3S) → <i>π</i><sup>0</sup><i>X</i><br><i>γ</i>(4S) → <i>η</i><i>X</i><br><i>T</i><sub>bb</sub>(10610)<sup>+</sup> → <i>π</i><sup>+</sup><i>X</i><br><i>T</i><sub>bb</sub>(10650)<sup>+</sup> → <i>π</i><sup>+</sup><i>X</i></td><td>γ<i>η</i><sub>b</sub>(1S)<br>hadrons<br>(see listings)</td><td>2011</td><td>YES</td></tr><tr><td><i>η</i><sub>b</sub>(2S)</td><td></td><td>9999.0<sup>+4.5</sup><sub>-4.0</sub></td><td>&lt; 24</td><td>0<sup>+</sup>(0<sup>-</sup>)</td><td><i>h</i><sub>b</sub>(2P) → <i>γ</i><i>X</i></td><td>hadrons</td><td>2012</td><td>NO</td></tr><tr><td><i>γ</i><sub>2</sub>(1D)</td><td></td><td>10163.7 ± 1.4</td><td>?</td><td>0<sup>-</sup>(2<sup>-</sup>)</td><td><i>γ</i>(3S) → <i>γ</i><i>γ</i><i>X</i></td><td>γγ<i>γ</i>(1S)</td><td>2004</td><td>YES</td></tr><tr><td><i>h</i><sub>b</sub>(2P)</td><td></td><td>10259.8 ± 1.2</td><td>?</td><td>0<sup>-</sup>(1<sup>-</sup>)</td><td><i>γ</i>(10860) → <i>π</i><sup>+</sup><i>π</i><sup>-</sup><i>X</i><br><i>γ</i>(10860) → <i>π</i><sup>+</sup><i>π</i><sup>-</sup><i>X</i><br><i>T</i><sub>bb</sub>(10610)<sup>+</sup> → <i>π</i><sup>+</sup><i>X</i><br><i>T</i><sub>bb</sub>(10650)<sup>+</sup> → <i>π</i><sup>+</sup><i>X</i></td><td><i>π</i><sup>+</sup><i>π</i><sup>-</sup><i>γ</i>(1S)<br>γ<i>η</i><sub>b</sub>(1S, 2S)</td><td>2011</td><td>YES</td></tr><tr><td><i>χ</i><sub>b1</sub>(3P)</td><td></td><td>10513.42 ± 0.67</td><td>?</td><td>0<sup>+</sup>(1<sup>++</sup>)</td><td><i>pp</i> → <i>X</i>...</td><td>γ<i>γ</i>(1S, 2S, 3S)</td><td>2011</td><td>YES</td></tr><tr><td><i>χ</i><sub>b2</sub>(3P)</td><td></td><td>10524.02 ± 0.78</td><td>?</td><td>0<sup>+</sup>(2<sup>++</sup>)</td><td><i>pp</i> → <i>X</i>...</td><td>γ<i>γ</i>(3S)</td><td>2011</td><td>YES</td></tr></tbody></table>

<figure><img src="image_346.png" /><figcaption>**Figure 78.2:** (Left Column) Invariant mass distributions measured by the ATLAS [28] (bottom), LHCb [30] (middle) and CMS [31] (top) experiments in their searches for the $\chi_b(3P)$ states through their radiative decays to one of the S-wave bottomonia. (Right Column) Invariant mass distributions measured by the ATLAS [32] (bottom), CMS [33] (middle) and LHCb [34] (top) experiments in their searches for $B_c^\pm$ excited states decaying to the $B_c^\pm$ ground state with the emission of two charged pions.</figcaption></figure>

other interesting properties, this means that they cannot annihi-

late into gluons, the excited states only decaying to the pseudoscalar ground state, $B_c^\pm$, via electromagnetic and pionic transitions.

On the basis of an event sample collected in the Run 1 of the LHC, corresponding to an integrated luminosity of 24 fb⁻¹, adding the 7 and 8 TeV data, the ATLAS Collaboration observed a resonance in the $B_c^+\pi^+\pi^-$ invariant mass spectrum [32] (Fig. 78.2, bottom right panel). This peak, observed with a significance of 5.2 standard deviations and a mass of 6842 ± 4(stat) ± 5(syst) MeV, was immediately recognized as the $B_c(2S)^\pm$ state, the first radial excitation in the $B_c^\pm$ family. Profiting from the much larger Run 2 event sample, collected in the 2015, 2016, 2017 and 2018 running periods and corresponding to 143 fb⁻¹ of 13 TeV pp collisions, as well as from a measurement resolution of around 6 MeV, the CMS Collaboration could observe two well-resolved peaks, separated by 29.1 ± 1.5(stat) ± 0.7(syst) MeV [33] (Fig. 78.2, middle right panel). The existence of two peaks, rather than a single one, is established with a significance of 6.5 standard deviations. The “right peak” has a mass of 6871.0 ± 1.2(stat) ± 0.8(syst) ± 0.8($B_c^+$) MeV, where the last term is the uncertainty in the $B_c^+$ mass, and is identified as the $B_c(2S)^\pm$ state, which decays directly to the $B_c^\pm$, emitting two (easy to detect) pions. The CMS observation, reported a couple of months after the end of the LHC Run 2, was soon followed by the corresponding LHCb result [34] (Fig. 78.2, top right panel), which confirmed the existence of the two states and reported a second measurement of the $B_c(2S)^\pm$ mass, 6872.1 ± 1.3(stat) ± 0.1(syst) ± 0.8($B_c^+$) MeV.

The “left peak” is interpreted as being the $B_c^*(2S)^\pm$ signal. It is observed at a mass lower than the real value because the experiments are unable to detect the low-energy photon emitted in the decay chain, $B_c^*(2S)^\pm \to B_c^{\pm\pi^+\pi^-}$ followed by $B_c^{\pm\pi^+\pi^-} \to B_c^{\pm\gamma}$ (Fig. 78.3). Its energy, expected to be in the range 40–80 MeV, leads to a very small probability that the photon converts into an $e^+e^-$ pair and the two electrons are reconstructed. The relative ordering of the two peaks is based on a generally-agreed assumption: the $M(B_c^{\pm\pi^+\pi^-}) - M(B_c^\pm)$ mass difference is larger than the $M(B_c^*(2S)^\pm) - M(B_c(2S)^\pm)$ differ-
ence. Naturally, these observations provide evidence for the existence of the $B_c^*(1S)^{\pm}$ state. They also provide measurements of two interesting mass differences, between the masses of the pseudoscalar mesons, $M(B_c(2S)^{\pm}) - M(B_c(1S)^{\pm}) = 596.1$ MeV, and of the vector mesons, $M(B_c^*(2S)^{\pm}) - M(B_c^*(1S)^{\pm}) = 567.0$ MeV (Fig. 78.3).

<figure><img src="image_347.png" /><figcaption>Figure 78.3: Diagram showing the decays mentioned in the text.</figcaption></figure>

## 78.2 States Above Open-Flavor Threshold

Many states have been discovered both above and near the lowest open-flavor thresholds. They are displayed in Tables 78.2 and 78.3, respectively. With the exception of the $\psi_3(3842)$ and the tensor state located at 3930 MeV (now called $\chi_{c2}(3930)$), which have properties consistent with those expected for the $\psi_3(1^3D_3)$ and $\chi_{c2}(2^3P_2)$, respectively, none of these states can easily be assigned a place in the quark model spectrum of the charmonium or bottomonium families. The theoretical interpretation of these states remains under discussion.

### 78.2.1 Charmonium

Using proton-proton collisions, LHCb observed a narrow state, the $\psi_3(3842)$ resonance, in the decay modes $\psi_3(3842) \to D^0\bar{D}^0$ and $D^+D^-$ [35]. The mass and width of this state are measured to be $3842.71 \pm 0.16 \pm 0.12$ MeV and $2.79 \pm 0.51 \pm 0.35$ MeV, respectively. The observed mass and narrow width are consistent with expectations for the spin-3 $\psi_3(1^3D_3)$ charmonium. Accordingly, the state is given the name $\psi_3(3842)$ in the listings, with the remark that the quantum numbers are fixed from the quark model and need to be confirmed.

The $\chi_{c2}(3930)$, which is a natural candidate for the $\chi_{c2}(2^3P_2)$ quark model state, was originally seen by Belle [36] and later confirmed by BaBar [37] in the $\gamma\gamma$ process $e^+e^- \to e^+e^-D\bar{D}$. This interpretation was strengthened by the more recent LHCb observation of the $\chi_{c2}(3930)$ alongside the $\psi_3(3842)$ in proton-proton collisions [35].

Unlike the $\chi_{c2}(2^3P_2)$, the identification of the $\chi_{c0}(2^3P_0)$ quark model state remains controversial. The original candidate was the $\chi_{c0}(3915)$, discovered by Belle in the $\gamma\gamma$ process $e^+e^- \to e^+e^-\omega J/\psi(1S)$ [38]. In a subsequent measurement by BaBar, its quantum numbers were determined to be $J^{PC} = 0^{++}$ [39]. However, its identification as the $\chi_{c0}(2^3P_0)$ quark model state was soon challenged [40, 41]. In addition, it was pointed out in Ref. [42] that if the assumption of helicity-2 dominance is abandoned and, instead, one allows for a sizeable helicity-0 component, a $J^{PC} = 2^{++}$ assignment is possible. This could imply that it is the same as the $\chi_{c2}(3930)$, but to explain the large helicity-0 component a sizable portion of non-$q\bar{q}$ is necessary [42]. A more recent LHCb amplitude analysis of the process $B^+ \to D^+D^-K^+$ finds distinct $0^{++}$ and $2^{++}$ components decaying to $D^+D^-$ [43],

which are currently identified in the listings as the $\chi_{c0}(3915)$ and $\chi_{c2}(3930)$, respectively.

An alternative candidate for the $\chi_{c0}(2^3P_0)$ (here referred to as the $\chi_{c0}(3860)$) was reported in Ref. [44] with properties more consistent with expectation: its mass is close to the potential model expectations, it decays to $D\bar{D}$, and the preferred quantum numbers are $J^{PC} = 0^{++}$ (this hypothesis is favored over the $2^{++}$ one with a $2.5\sigma$ significance).

In the excited vector charmonium spectrum, the $\psi(4040)$, $\psi(4160)$, and $\psi(4415)$ are prominent in the inclusive $e^+e^-$ hadronic cross section and are naturally identified as the $3^3S_1$, $2^3D_1$, and $4^3S_1 c\bar{c}$ quark model states, respectively. In addition to these long-established states, however, another set of peaks has been found in exclusive $e^+e^-$ cross sections. Unlike conventional vector charmonia, they do not appear in the inclusive hadronic cross section and they apparently do not decay to $D\bar{D}$. The PDG summary table currently lists the $\psi(4230)$, $\psi(4360)$, and $\psi(4660)$ within this category. The first of these to be discovered was originally known as the $Y(4260)$ (now the $\psi(4230)$), seen by BaBar [45] and Belle [46, 47] in $e^+e^- \to \pi^+\pi^-J/\psi(1S)$ using initial state radiation. In a more recent high-statistics scan of $e^+e^- \to \pi^+\pi^-J/\psi(1S)$, BESIII demonstrated that the lineshape in this mass range is highly non-trivial [48]. The latter observation was interpreted by the authors as the presence of two states. However, this lineshape is also consistent with other possible interpretations, such as one assuming a molecular structure for the $\psi(4230)$ [49]. The data of Ref. [48] also called for a significant downward shift of the mass of what was originally called the $Y(4260)$, making it consistent with peaks in other exclusive cross sections, such as $h_c(1P)\pi\pi$ [50]. We thus merged the original $Y(4260)$ (or, more formally, the $\psi(4260)$) with the $\psi(4230)$ in the listings.

BESIII observed the $\chi_{c1}(3872)$, also known as $X(3872)$, in $e^+e^- \to \gamma\chi_{c1}(3872)$ in the $\psi(4230)$ mass range [51], which could allow for additional insight into the structure of both states (see the review on heavy non-$q\bar{q}$ mesons). BESIII also performed a recent study of the process $e^+e^- \to \pi^+\pi^-\psi(2S)$ and found evidence for a lower mass state, possibly the $\psi(4230)$, in addition to the more dominant $\psi(4360)$ [52].

Another interesting question is whether a heavier $\pi^+\pi^-\psi(2S)$ state, the $\psi(4660)$, discovered by Belle [53, 54] and confirmed by BaBar [55], is identical to the $\Lambda_c^+\bar{\Lambda}_c^-$ resonance observed by Belle with a nearby mass and width [56]. Most probably it is, the $\Lambda_c^+\bar{\Lambda}_c^-$ being one more decay mode of the $\psi(4660)$ (see the review on heavy non-$q\bar{q}$ mesons for more details). Note that this is the interpretation adopted in the particle listings. In addition, Belle reported the first observation of a vector charmonium-like state decaying to $D_s^+D_s^-(2536)$ with a significance of $5.9\sigma$ [57]. Its measured mass and width are $4625.9_{-6.0}^{+6.2} \pm 0.4$ MeV and $49.8_{-11.5}^{+13.9} \pm 4.0$ MeV, respectively, consistent with those of the $\psi(4660)$. Therefore, $D_s^+D_s^-(2536)$ appears as an additional decay mode of the $\psi(4660)$ in the listings.

A series of isovector states¹ containing $c\bar{c}$ have been found in B decays to $K\pi(c\bar{c})$, where the isovector state decays to $\pi(c\bar{c})$ and $(c\bar{c})$ stands for $J/\psi(1S)$, $\psi(2S)$, or $\chi_{c1}(1P)$. They are manifestly non-$q\bar{q}$. The $T_{c\bar{c}1}(4430)$ (originally called the $Z_c(4430)$), decaying to $\pi\psi(2S)$, is the most well established. Based on a full amplitude analysis of $B^0 \to K^+\pi^-\psi(2S)$ decays, Belle determined the spin-parity of the $T_{c\bar{c}1}(4430)$ to be $J^P = 1^+$ [58]. From their study of $B^0 \to K^+\pi^-J/\psi(1S)$ decays, Belle also found evidence for the decay mode $T_{c\bar{c}1}(4430) \to \pi J/\psi(1S)$ [59], which has an order of magnitude lower branching fraction than the discovery mode $T_{c\bar{c}1}(4430) \to \pi\psi(2S)$. In the same analysis, Belle reported evidence for one more charged state, dubbed $T_{c\bar{c}1}(4200)$, decaying to $\pi J/\psi(1S)$. The observation of the $T_{c\bar{c}1}(4430)$ in $\pi\psi(2S)$, as well as its quantum number assignments, were confirmed by LHCb [60] with a much larger data sample, leading to improved mass and width values, consistent with earlier measurements; the experiment even reports a resonant behavior of the $T_{c\bar{c}1}(4430)$ amplitude. The $T_{c\bar{c}1}(4430)$ was not confirmed (or excluded) by BaBar [61].
<table><caption>**Table 78.2:** As in Table 78.1, but for states above the first open-flavor thresholds in the $c\bar{c}$ and $b\bar{b}$ regions, ordered by mass.</caption><thead><tr><th>PDG Name</th><th>Former Name(s)</th><th>m (MeV)</th><th>Γ (MeV)</th><th>I<sup>G</sup>(J<sup>PC</sup>)</th><th>Production</th><th>Decay</th><th>Discovery Year</th><th>Summary Table</th></tr></thead><tbody><tr><td>ψ<sub>3</sub>(3842)</td><td></td><td>3842.71 ± 0.20</td><td>2.79 ± 0.62</td><td>0<sup>-</sup>(3<sup>--</sup>)<sup>*</sup></td><td>pp → X...</td><td>DD</td><td>2019</td><td>YES</td></tr><tr><td>χ<sub>c0</sub>(3860)</td><td></td><td>3862<sup>+48</sup><sub>-35</sub></td><td>201<sup>+177</sup><sub>-106</sub></td><td>0<sup>+</sup>(0<sup>++</sup>)</td><td>e<sup>+</sup>e<sup>-</sup> → J/ψ(1S)X</td><td>D<span style="text-decoration: overline">D</span></td><td>2017</td><td>NO</td></tr><tr><td>χ<sub>c0</sub>(3915)</td><td>X(3915),<br>Y(3940)</td><td>3922.1 ± 1.8</td><td>20 ± 4</td><td>0<sup>+</sup>(0/2<sup>++</sup>)</td><td>B → KX</td><td>ωJ/ψ(1S)</td><td>2004</td><td>YES</td></tr><tr><td>χ<sub>c2</sub>(3930)</td><td>χ<sub>c2</sub>(2P),<br>Z(3930)</td><td>3922.5 ± 1.0</td><td>35.2 ± 2.2</td><td>0<sup>+</sup>(2<sup>++</sup>)</td><td>e<sup>+</sup>e<sup>-</sup> → e<sup>+</sup>e<sup>-</sup>X</td><td>D<span style="text-decoration: overline">D</span></td><td>2005</td><td>YES</td></tr><tr><td>X(3940)</td><td></td><td>3942<sup>+9</sup><sub>-8</sub></td><td>37<sup>+27</sup><sub>-17</sub></td><td>??(???)</td><td>e<sup>+</sup>e<sup>-</sup> → J/ψ(1S)X</td><td>D<span style="text-decoration: overline">D</span>*</td><td>2007</td><td>NO</td></tr><tr><td>T<sub>c<span style="text-decoration: overline">c</span></sub>(4050)</td><td>Z<sub>1</sub>(4050)<br>X(4050)</td><td>4051<sup>+24</sup><sub>-43</sub></td><td>82<sup>+51</sup><sub>-28</sub></td><td>1<sup>-</sup>(??<sup>+</sup>)</td><td>B<sup>0</sup> → K<sup>-</sup>X</td><td>π<sup>+</sup>χ<sub>c1</sub>(1P)</td><td>2008</td><td>NO</td></tr><tr><td>T<sub>c<span style="text-decoration: overline">c</span></sub>(4055)</td><td>Z<sub>c</sub>(4055)<br>X(4055)</td><td>4054 ± 3</td><td>45 ± 13</td><td>1<sup>+</sup>(??<sup>-</sup>)</td><td>e<sup>+</sup>e<sup>-</sup> → π<sup>-</sup>X</td><td>π<sup>+</sup>ψ(2S)</td><td>2015</td><td>NO</td></tr><tr><td>T<sub>c<span style="text-decoration: overline">c</span></sub>(4100)</td><td>X(4100)</td><td>4096<sup>+27</sup><sub>-30</sub></td><td>152<sup>+83</sup><sub>-98</sub></td><td>1<sup>-</sup>(???)</td><td>B<sup>0</sup> → K<sup>-</sup>X</td><td>π<sup>+</sup>η<sub>c</sub>(1S)</td><td>2018</td><td>NO</td></tr><tr><td>χ<sub>c1</sub>(4140)</td><td>Y(4140)</td><td>4146.5 ± 3.0</td><td>19<sup>+9</sup><sub>-5</sub></td><td>0<sup>+</sup>(1<sup>++</sup>)</td><td>B<sup>+</sup> → K<sup>+</sup>X</td><td>φJ/ψ(1S)</td><td>2009</td><td>YES</td></tr><tr><td>X(4160)</td><td></td><td>4153<sup>+23</sup><sub>-21</sub></td><td>136<sup>+60</sup><sub>-35</sub></td><td>??(???)</td><td>e<sup>+</sup>e<sup>-</sup> → J/ψ(1S)X</td><td>D<sup>*</sup><span style="text-decoration: overline">D</span>*</td><td>2007</td><td>NO</td></tr><tr><td>T<sub>c<span style="text-decoration: overline">c</span></sub>(4200)</td><td>Z<sub>c</sub>(4200)</td><td>4196<sup>+35</sup><sub>-32</sub></td><td>370<sup>+99</sup><sub>-149</sub></td><td>1<sup>+</sup>(1<sup>-</sup>)</td><td>B<sup>+</sup> → K<sup>+</sup>X</td><td>φJ/ψ(1S)</td><td>2014</td><td>NO</td></tr><tr><td>ψ(4230)</td><td>Y(4230)<br>Y(4260)</td><td>4222.1 ± 2.3</td><td>49 ± 7</td><td>0<sup>-</sup>(1<sup>--</sup>)</td><td>B<sup>0</sup> → K<sup>-</sup>X</td><td>J/ψ(1S)π<sup>+</sup></td><td>2015</td><td>YES</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>e<sup>+</sup>e<sup>-</sup> → X</td><td>π<sup>+</sup>π<sup>-</sup>J/ψ(1S)<br>ωχ<sub>c0</sub>(1P)<br>π<sup>+</sup>π<sup>-</sup>h<sub>c</sub>(1P)<br>(see listings)</td><td></td><td></td></tr><tr><td>T<sub>c<span style="text-decoration: overline">c</span></sub>(4240)</td><td>Z<sub>c</sub>(4240)<br>R<sub>c0</sub>(4240)</td><td>4239<sup>+48</sup><sub>-21</sub></td><td>220<sup>+118</sup><sub>-88</sub></td><td>1<sup>+</sup>(0<sup>--</sup>)</td><td>B<sup>0</sup> → K<sup>-</sup>X</td><td>π<sup>+</sup>ψ(2S)</td><td>2014</td><td>NO</td></tr><tr><td>T<sub>c<span style="text-decoration: overline">c</span></sub>(4250)</td><td>Z<sub>2</sub>(4250)<br>X(4250)</td><td>4248<sup>+185</sup><sub>-45</sub></td><td>177<sup>+321</sup><sub>-72</sub></td><td>1<sup>-</sup>(??<sup>+</sup>)</td><td>B<sup>0</sup> → K<sup>-</sup>X</td><td>π<sup>+</sup>χ<sub>c1</sub>(1P)</td><td>2008</td><td>NO</td></tr><tr><td>χ<sub>c1</sub>(4274)</td><td>Y(4274)</td><td>4286<sup>+8</sup><sub>-9</sub></td><td>51 ± 7</td><td>0<sup>+</sup>(1<sup>++</sup>)</td><td>B<sup>+</sup> → K<sup>+</sup>X</td><td>φJ/ψ(1S)</td><td>2011</td><td>YES</td></tr><tr><td>X(4350)</td><td></td><td>4350.6<sup>+4.7</sup><sub>-5.1</sub></td><td>13<sup>+18</sup><sub>-10</sub></td><td>0<sup>+</sup>(??<sup>+</sup>)</td><td>e<sup>+</sup>e<sup>-</sup> → e<sup>+</sup>e<sup>-</sup>X</td><td>φJ/ψ(1S)</td><td>2009</td><td>NO</td></tr><tr><td>ψ(4360)</td><td>Y(4360)</td><td>4374 ± 7</td><td>118 ± 12</td><td>0<sup>-</sup>(1<sup>--</sup>)</td><td>e<sup>+</sup>e<sup>-</sup> → X</td><td>π<sup>+</sup>π<sup>-</sup>ψ(2S)<br>π<sup>+</sup>π<sup>-</sup>J/ψ(1S)</td><td>2007</td><td>YES</td></tr><tr><td>T<sub>c<span style="text-decoration: overline">c</span></sub>(4430)</td><td>Z<sub>c</sub>(4430)</td><td>4478<sup>+15</sup><sub>-18</sub></td><td>181 ± 31</td><td>1<sup>+</sup>(1<sup>-</sup>)</td><td>B<sup>0</sup> → K<sup>-</sup>X</td><td>π<sup>+</sup>ψ(2S)<br>π<sup>+</sup>J/ψ(1S)</td><td>2007</td><td>YES</td></tr><tr><td>χ<sub>c0</sub>(4500)</td><td>X(4500)</td><td>4474 ± 4</td><td>77<sup>+12</sup><sub>-10</sub></td><td>0<sup>+</sup>(0<sup>++</sup>)</td><td>B<sup>+</sup> → K<sup>+</sup>X</td><td>φJ/ψ(1S)</td><td>2017</td><td>NO</td></tr><tr><td>X(4630)</td><td></td><td>4626<sup>+24</sup><sub>-111</sub></td><td>174<sup>+137</sup><sub>-78</sub></td><td>0<sup>+</sup>(??<sup>+</sup>)</td><td>B<sup>+</sup> → K<sup>+</sup>X</td><td>φJ/ψ(1S)</td><td>2021</td><td>NO</td></tr><tr><td>ψ(4660)</td><td>Y(4660),<br>X(4660)</td><td>4641 ± 10</td><td>72<sup>+13</sup><sub>-11</sub></td><td>0<sup>-</sup>(1<sup>--</sup>)</td><td>e<sup>+</sup>e<sup>-</sup> → X</td><td>π<sup>+</sup>π<sup>-</sup>ψ(2S)<br>Λ<sub>c</sub><sup>+</sup>Λ<sub>c</sub><sup>-</sup><br>D<sub>s</sub><sup>+</sup>D<sub>s1</sub>(2536)</td><td>2007</td><td>YES</td></tr><tr><td>χ<sub>c1</sub>(4685)</td><td></td><td>4684<sup>+15</sup><sub>-17</sub></td><td>126<sup>+40</sup><sub>-44</sub></td><td>0<sup>+</sup>(1<sup>++</sup>)</td><td>B<sup>+</sup> → K<sup>+</sup>X</td><td>φJ/ψ(1S)</td><td>2021</td><td>NO</td></tr><tr><td>χ<sub>c0</sub>(4700)</td><td>X(4700)</td><td>4694<sup>+16</sup><sub>-5</sub></td><td>87<sup>+18</sup><sub>-10</sub></td><td>0<sup>+</sup>(0<sup>++</sup>)</td><td>B<sup>+</sup> → K<sup>+</sup>X</td><td>φJ/ψ(1S)</td><td>2017</td><td>NO</td></tr><tr><td>T<sub>c<span style="text-decoration: overline">c</span></sub><sub>c<span style="text-decoration: overline">c</span></sub>(6900)</td><td>X(6900)</td><td>6899 ± 12</td><td>153 ± 29</td><td>0<sup>+</sup>(??<sup>+</sup>)</td><td>pp → X...</td><td>J/ψ(1S)J/ψ(1S)</td><td>2020</td><td>NO</td></tr><tr><td>Υ(10753)</td><td></td><td>10752.7<sup>+5.9</sup><sub>-6.0</sub></td><td>36<sup>+18</sup><sub>-12</sub></td><td>??(1<sup>--</sup>)</td><td>e<sup>+</sup>e<sup>-</sup> → X</td><td>ππΥ(1S, 2S, 3S)</td><td>2019</td><td>NO</td></tr><tr><td>Υ(10860)</td><td>Υ(5S)</td><td>10885.2<sup>+2.6</sup><sub>-1.6</sub></td><td>37 ± 4</td><td>0<sup>-</sup>(1<sup>--</sup>)</td><td>e<sup>+</sup>e<sup>-</sup> → X</td><td>B<sup>(s)</sup><span style="text-decoration: overline">B</span><sup>(s)</sup>(π)<br>ππΥ(1S, 2S, 3S)<br>π<sup>+</sup>π<sup>-</sup>h<sub>b</sub>(1P, 2P)<br>ηΥ(1S, 2S)<br>π<sup>+</sup>π<sup>-</sup>Υ(1D)<br>(see listings)</td><td>1985</td><td>YES</td></tr><tr><td>Υ(11020)</td><td>Υ(6S)</td><td>11000 ± 4</td><td>24<sup>+8</sup><sub>-6</sub></td><td>0<sup>-</sup>(1<sup>--</sup>)</td><td>e<sup>+</sup>e<sup>-</sup> → X</td><td>B<sup>(s)</sup><span style="text-decoration: overline">B</span><sup>(s)</sup>(π)<br>ππΥ(1S, 2S, 3S)<br>π<sup>+</sup>π<sup>-</sup>h<sub>b</sub>(1P, 2P)<br>(see listings)</td><td>1985</td><td>YES</td></tr></tbody></table>

Belle also reported an observation of two charged states decaying to $\pi\chi_{c1}(1P)$ in an analysis of $B^0 \to K^+\pi^-\chi_{c1}(1P)$ decays [62]. These were originally called $Z_1(4050)^\pm$ and $Z_2(4250)^\pm$, but are referred to in Table 78.2 as $T_{c\bar{c}}(4050)$ and $T_{c\bar{c}}(4250)$. These states were not confirmed by BaBar [63]. Belle observes signals with 5.0$\sigma$ significance for both the $T_{c\bar{c}}(4050)$ and $T_{c\bar{c}}(4250)$,

whereas BaBar reports 1.1$\sigma$ and 2.0$\sigma$ effects, respectively, setting upper limits that are not inconsistent with Belle's measured rates. The situation remains unresolved.

The decay $B^+ \to K^+\phi J/\psi(1S)$ appears to be especially rich in resonant substructure. The $Y(4140)$ (now the $\chi_{c1}(4140)$), decaying to $\phi J/\psi(1S)$, was first observed in 2008 by CDF [64, 65], and confirmed by D0 and CMS [66, 67]. However, a second structure, the $Y(4274)$ (now the $\chi_{c1}(4274)$), could not be established

<footnote>*Quantum numbers fixed from the quark model and need confirmation.</footnote>

<footnote n="1">¹These isovector states were originally called $Z_c$. They are now referred to as $T_{c\bar{c}}$. See the naming scheme review.</footnote>
unambiguously. Neither of the two states was seen in B decays at Belle [68], LHCb [69] and BaBar [70], or in γγ collisions at Belle [71]. The real breakthrough happened when LHCb performed a full amplitude analysis of $B^+ \to K^+\phi J/\psi(1S)$ with $J/\psi(1S) \to \mu^+\mu^-$, $\phi \to K^+K^-$ decays and showed that the data cannot be described in a model that contains only excited kaon states decaying into $K^+\phi$ [72, 73]. They observe two $1^{++}$ states with masses close to those originally reported by CDF (the $\chi_{c1}(4140)$ and $\chi_{c1}(4274)$), but the width of the one at 4140 MeV is much larger. In addition, they find two significant $0^{++}$ structures at 4500 and 4700 MeV (the $\chi_{c0}(4500)$ and $\chi_{c0}(4700)$). The LHCb analysis was extended even further in Ref. [74] with a factor of six increase in statistics: the $\chi_{c1}(4140)$, $\chi_{c1}(4274)$, $\chi_{c0}(4500)$, and $\chi_{c0}(4700)$ were confirmed, and two new states, the $X(4630)$ and $\chi_{c1}(4685)$ were reported, also decaying to $\phi J/\psi(1S)$.

### 78.2.2 Resonances in the $J/\psi(1S)$-pair mass spectrum

Based on proton-proton event samples collected at $\sqrt{s} = 7, 8$ and 13 TeV, corresponding to an integrated luminosity of $9 \text{ fb}^{-1}$, the LHCb Collaboration reported the observation of a narrow structure in the invariant mass distribution of $J/\psi(1S)$ pairs [75]. The new resonance, denoted as $T_{cc\bar{c}\bar{c}}(6900)$ (also $X(6900)$), is clearly seen at a mass of 6.9 GeV in the bottom panel of Fig. 78.4, as indicated by the red curve. The ATLAS Collaboration has also

<figure><img src="image_348.png" /><figcaption>**Figure 78.4:** $J/\psi(1S)$-pair invariant mass distributions measured by the LHCb [75] (bottom), ATLAS [76] (middle), and CMS [77] (top) experiments.</figcaption></figure>

observed the $T_{cc\bar{c}\bar{c}}(6900)$ resonance in the di-$J/\psi(1S)$ mass spectrum, using $140 \text{ fb}^{-1}$ of pp collisions at $\sqrt{s} = 13$ TeV [76], as can be seen in the middle panel of Fig. 78.4. More recently, also the CMS Collaboration reported the observation of the $T_{cc\bar{c}\bar{c}}(6900)$ in the $J/\psi(1S)$-pair mass distribution, using $135 \text{ fb}^{-1}$ of 13 TeV pp collisions [77]. Thanks to its good mass resolution and relatively large event yields, CMS determined the mass and width of the $T_{cc\bar{c}\bar{c}}(6900)$ with the best precision among the three measurements.

Furthermore, another resonance is seen with a significance exceeding 5 standard deviations, at a mass of 6.55 GeV, as shown in the top panel of Fig. 78.4. The measured values of the masses and widths of the new states depend on whether the model used to fit the distributions includes interference effects between the resonances and the underlying continuum (and also among the resonances). The larger event samples presently being collected by the LHC experiments will surely lead to significantly improved measurements, as well as to complementary results (including determinations of spin-parity quantum numbers), so that one can expect near-future progress in our understanding of the nature of these “narrow structures”.

### 78.2.3 Bottomonium

Belle reported a new measurement of the $e^+e^- \to \Upsilon(nS)\pi^+\pi^-$ ($n=1,2,3$) cross sections at energies from 10.52 to 11.02 GeV [78]. They observed, with a $5.2\sigma$ significance, a new structure in the energy dependence of the cross sections. If described by a Breit-Wigner function, its mass and width are $10752.7 \pm 5.9^{+0.7}_{-1.1} \text{ MeV}$ and $35.5^{+17.6}_{-11.3} \pm 3.9 \text{ MeV}$, respectively. The new structure could have a resonant origin and correspond to the not yet observed $\Upsilon(3D)$ state, provided $S-D$ mixing is enhanced, or an exotic state, e.g., a compact tetraquark or hadrobottomonium. It could also be a non-resonant effect due to rescattering. The $\Upsilon(10750)$ was confirmed in a global $K$-matrix fit to exclusive and inclusive $e^+e^- \to b\bar{b}$ cross sections in Ref. [79].

We no longer mention a hypothetical $Y_b(10888)$ state since a new analysis of the $\Upsilon(10860)$ energy range does not show evidence for an additional state with a mass different from that of the $\Upsilon(10860)$ [80]. After the mass of the $\eta_b(1S)$ was shifted upwards by about 10 MeV based on the Belle measurements [22, 23], all of the bottomonium states mentioned above fit into their respective spectroscopies, roughly where expected. An independent experimental confirmation of the shifted masses came from the Belle observation of $\Upsilon(4S) \to \eta h_b(1P)$ decays [23]. This process turns out to be the strongest observed transition of the $\Upsilon(4S)$ to lower bottomonium states.

## 78.3 States Near Open-Flavor Threshold

A number of states, listed in Table 78.3, appear near open-flavor thresholds, which is likely an important factor in their theoretical interpretation [81].

### 78.3.1 Charmonium

The $\chi_{c1}(3872)$, also known as $X(3872)$, is widely studied and seen in many transitions — see Table 78.3. Yet its interpretation remains unsettled (see the heavy non-$q\bar{q}$ review). Its unique experimental features include: it has $J^{PC} = 1^{++}$ [82, 83], yet it is too light to be the $\chi_{c1}(2^3P_1)$ quark model state; its mass is within 200 keV of the $D^0\bar{D}^0*$ threshold; it shows substantial isospin-breaking in its decays to $\rho J/\psi(1S)$ and $\pi^0\chi_{c1}(1P)$; and it is extremely narrow. Using a large sample of inclusively produced $\chi_{c1}(3872)$ decaying to $\pi^+\pi^- J/\psi(1S)$, LHCb recently determined the decay width of the $\chi_{c1}(3872)$ under two different assumptions [84]. Assuming a Flatté-inspired line shape and exploiting the strong coupling of the $\chi_{c1}(3872)$ to $D^0\bar{D}^0$, LHCb performed the first exploration of the pole structure of the $\chi_{c1}(3872)$, finding a FWHM width of $0.22^{+0.06}_{-0.08} \pm 0.25$ MeV. On the other hand, assuming a Breit-Wigner line shape, its width was found to be $1.39 \pm 0.24 \pm 0.10$ MeV. While the former analysis has a more firm theoretical foundation, the LHCb detector resolution did not allow for a distinction between the different line shapes.

In addition to the $T_{c\bar{c}}$ (also known as $Z_c$) states found in B decays, discussed above, several isovector states with masses near $D\bar{D}^*$ and $D^*\bar{D}^*$ thresholds appear to be unique to $e^+e^-$ annihilation. In 2013, a state named $T_{c\bar{c}1}(3900)$ (originally $Z_c(3900)$) was unearthed in the charmonium region at BESIII [86] and Belle [47] in the process $e^+e^- \to \pi^\mp T_{c\bar{c}1}(3900)^\pm$ with $T_{c\bar{c}1}(3900)^\pm \to \pi^\pm J/\psi(1S)$. The corresponding spectrum from BESIII is shown in Fig. 78.5. An analysis of CLEO data [87] confirmed this finding and also provided evidence for a neutral partner. A nearby signal was also seen in the $DD^*$ channel [88] whose quantum numbers were fixed to $1^{+-}$. BESIII reported its neutral partner in both $J/\psi(1S)\pi^0$ [89] and $D\bar{D}^*$ [90] decay modes. The masses extracted
<table><caption>**Table 78.3:** As in Table 78.1, but for states near the first open-flavor thresholds in the $c\bar{c}$ and $b\bar{b}$ regions, ordered by mass. Updated from Ref. [85] with kind permission, copyright (2011), Springer, and from Ref. [11] with kind permission from the authors.</caption><thead><tr><th>PDG Name</th><th>Former Name</th><th><i>m</i> (MeV)</th><th><i>Γ</i> (MeV)</th><th><i>I</i><sup>G</sup>(<i>J</i><sup>PC</sup>)</th><th>Production</th><th>Decay</th><th>Discovery Year</th><th>Summary Table</th></tr></thead><tbody><tr><td>χ<sub>c1</sub>(3872)</td><td>X̄(3872)</td><td>3871.64±0.06</td><td>1.19 ± 0.21</td><td>0<sup>+</sup>(1<sup>++</sup>)</td><td>B → KX<br>p<span style="text-decoration: overline">p</span> → X...<br>pp → X...<br>e<sup>+</sup>e<sup>−</sup> → γX</td><td>π<sup>+</sup>π<sup>−</sup>J/ψ(1S)<br>3πJ/ψ(1S)<br>D<sup>*</sup>D<sup>0</sup><br>γJ/ψ(1S)<br>γψ(2S)<br>π<sup>0</sup>χ<sub>c1</sub>(1P)</td><td>2003</td><td>YES</td></tr><tr><td>T<sub>cc</sub>(3875)</td><td></td><td>3874.83 ± 0.11</td><td>0.410<sup>+0.172</sup><sub>−0.175</sub></td><td>?(??)</td><td>pp → X...</td><td>D<sup>0</sup>D<sup>0</sup>π</td><td>2022</td><td>NO</td></tr><tr><td>T<sub>c<span style="text-decoration: overline">c</span>1</sub>(3900)</td><td>Z<sub>c</sub>(3900)</td><td>3887.1 ± 2.6</td><td>28.4 ± 2.6</td><td>1<sup>+</sup>(1<sup>+-</sup>)</td><td>ψ(4230) → π<sup>−</sup>X<br>ψ(4230) → π<sup>0</sup>X</td><td>π<sup>+</sup>J/ψ(1S)<br>π<sup>0</sup>J/ψ(1S)<br>(D<span style="text-decoration: overline">D</span><sup>*</sup>)<sup>+</sup><br>(D<span style="text-decoration: overline">D</span><sup>*</sup>)<sup>0</sup></td><td>2013</td><td>YES</td></tr><tr><td>T<sub>c<span style="text-decoration: overline">c</span>c<span style="text-decoration: overline">s</span>1</sub>(4000)</td><td>Z<sub>cs</sub>(4000)</td><td>3980 − 4010</td><td>5 − 150</td><td><sup>1</sup>&frasl;<sub>2</sub>(1<sup>+</sup>)</td><td>e<sup>+</sup>e<sup>−</sup> → KX<br>B<sup>+</sup> → φX</td><td>D<sub>s</sub>D<sup>*</sup> + D<sub>s</sub><sup>*</sup>D<br>K<sup>+</sup>J/ψ(1S)</td><td>2021</td><td>NO</td></tr><tr><td>T<sub>c<span style="text-decoration: overline">c</span>c</sub>(4020)</td><td>Z<sub>c</sub>(4020)<br>X(4020)</td><td>4024.1 ± 1.9</td><td>13 ± 5</td><td>1<sup>+</sup>(?−?)</td><td>ψ(4230, 4360) → π<sup>−</sup>X<br>ψ(4230, 4360) → π<sup>0</sup>X</td><td>π<sup>+</sup>h<sub>c</sub>(1P)<br>π<sup>0</sup>h<sub>c</sub>(1P)<br>(D<sup>*</sup><span style="text-decoration: overline">D</span><sup>*</sup>)<sup>+</sup><br>(D<sup>*</sup><span style="text-decoration: overline">D</span><sup>*</sup>)<sup>0</sup></td><td>2013</td><td>YES</td></tr><tr><td>T<sub>c<span style="text-decoration: overline">c</span>c<span style="text-decoration: overline">s</span>1</sub>(4220)</td><td>Z<sub>cs</sub>(4220)</td><td>4216<sup>+49</sup><sub>−38</sub></td><td>233<sup>+110</sup><sub>−90</sub></td><td><sup>1</sup>&frasl;<sub>2</sub>(1<sup>+</sup>)</td><td>B<sup>+</sup> → φX</td><td>K<sup>+</sup>J/ψ(1S)</td><td>2021</td><td>NO</td></tr><tr><td>T<sub>b<span style="text-decoration: overline">b</span>1</sub>(10610)</td><td>Z<sub>b</sub>(10610)</td><td>10607.2 ± 2.0</td><td>18.4 ± 2.4</td><td>1<sup>+</sup>(1<sup>+-</sup>)</td><td>Υ(10860) → π<sup>−</sup>X<br>Υ(10860) → π<sup>0</sup>X</td><td>π<sup>+</sup>Υ(1S, 2S, 3S)<br>π<sup>0</sup>Υ(1S, 2S, 3S)<br>π<sup>+</sup>h<sub>b</sub>(1P, 2P)<br>(B<span style="text-decoration: overline">B</span><sup>*</sup>)<sup>+</sup></td><td>2011</td><td>YES</td></tr><tr><td>T<sub>b<span style="text-decoration: overline">b</span>1</sub>(10650)</td><td>Z<sub>b</sub>(10650)</td><td>10652.2 ± 1.5</td><td>11.5 ± 2.2</td><td>1<sup>+</sup>(1<sup>+-</sup>)</td><td>Υ(10860) → π<sup>−</sup>X</td><td>π<sup>+</sup>Υ(1S, 2S, 3S)<br>π<sup>+</sup>h<sub>b</sub>(1P, 2P)<br>(B<sup>*</sup><span style="text-decoration: overline">B</span><sup>*</sup>)<sup>+</sup></td><td>2011</td><td>YES</td></tr></tbody></table>

<figure><img src="image_349.png" /><figcaption>**Figure 78.5:** The $\pi^{\pm}J/\psi(1S)$ invariant mass distribution from BESIII [86] $e^{+}e^{-}$ collision data taken at a center-of-mass energy near 4260 MeV.</figcaption></figure>

from these experiments in different decay modes agree within $2\sigma$. However, since the extraction of the mass and width parameters did not allow for an interference with the background and Breit-Wigner line shapes are used, which is not justified near thresholds, there might be some additional systematic uncertainty in the mass values. Therefore, in the RPP listings as well as in Table 78.3 both structures appear under the name $T_{c\bar{c}1}(3900)$. BESIII also reported the observation of another charged state, the $T_{c\bar{c}}(4020)^{\pm}$ (originally called $Z_c(4020)^{\pm}$), in two decay modes: $h_c(1P)\pi^{\pm}$ [91] and $(D^*\bar{D}^*)^{\pm}$ [92]. The neutral partners have also been observed by BESIII in the $h_c(1P)\pi^0$ [93] and $(D^*\bar{D}^*)^0$ [94] final states. The $T_{c\bar{c}}$ states show some remarkable similarities to the $T_{b\bar{b}}$ states (discussed below), e.g. they decay dominantly to $D^*(\bar{D}^*)$ channels. However, current analyses suggest that the mass of the $T_{c\bar{c}1}(3900)$ might be somewhat above the $D\bar{D}^*$ threshold. If confirmed, this feature would challenge a possible $D\bar{D}^*$-molecular interpretation with S-wave interactions only — prominent D-waves can shift molecular poles above threshold (see the discussion in Sec. 78.3.3).

Finally, $3.5\sigma$ evidence for one more charged charmonium-like state at 4055 MeV decaying into $\psi(2S)\pi^{\pm}$ was reported by Belle in their analysis of the process $e^{+}e^{-} \to \psi(2S)\pi^{+}\pi^{-}$ [54]. The same process was studied by BESIII, where there appears to be complications in the Dalitz plot requiring further investigation [52].

An isospin-1/2 state with open strangeness, the $T_{c\bar{c}s1}(4000)$, has been reported near $D_sD^*$ threshold by both BESIII [95, 96] and LHCb [74]. BESIII observes the charged $T_{c\bar{c}s1}(4000)$ in the process $e^{+}e^{-} \to K^{+}(D_{s}^{-}D^{*0} + D_{s}^{*}D^{0})$ [95] and the neutral $T_{c\bar{c}s1}(4000)$ in the process $e^{+}e^{-} \to K_S(D_{s}^{-}D^{*+} + D_{s}^{*}D^{+})$ [96]. LHCb observes the charged version in $B^{+} \to \phi(K^{+}J/\psi(1S))$ [74], although with a much larger decay width. The same LHCb analysis also finds a higher mass $T_{c\bar{c}s}$ candidate, the $T_{c\bar{c}s1}(4220)$. A search by BESIII for a heavier partner $T'_{c\bar{c}s}$ near $D_s^*D^*$ threshold was negative [97].

### 78.3.2 Double Charm

The most striking recent result, however, is due to the observation by LHCb [98, 99] of a doubly charmed state, the $T_{cc}(3875)^{+}$, in the inclusive cross section of $D^{*+}D^{0} \to D^{0}D^{0}\pi^{+}$ (see Fig. 78.6). The statistical significance is overwhelming, the minimal quark content is $cc\bar{u}\bar{d}$, and the width is approximately what would be expected for an isoscalar ground state having $J^{P} = 1^{+}$. The integrated luminosity used in this study was 9 fb$^{-1}$. The larger integrated luminosity of the event sample to be collected by the end of the LHC Run 3, together with improved analysis techniques, might bring the $T_{bc}$ states within reach.

### 78.3.3 Bottomonium

New results on the $\eta_b$, $h_b$, and $T_{b\bar{b}}$ mostly come from Belle [22, 23, 25, 27, 80, 100–106], all from analyses of 121.4 fb$^{-1}$ of $e^{+}e^{-}$ collision data collected near the peak of the $\Upsilon(10860)$ resonance, as well as from an additional 25 fb$^{-1}$ of data collected during the scans of the c.m. energy range 10.63–11.05 GeV. The $\eta_b$, $h_b$, and $T_{b\bar{b}}$ appear in the decay chains $\Upsilon(10860) \to \pi^{-}T_{b\bar{b}}^{+}$, $T_{b\bar{b}}^{+} \to \pi^{+}(b\bar{b})$, and, when the $b\bar{b}$ forms an $h_b(1P)$, frequently decaying as $h_b(1P) \to \gamma\eta_b$.

Belle soon noticed that, for events in the peaks of Fig. 78.1 corresponding to the processes $e^{+}e^{-} \to \pi^{+}\pi^{-}\Upsilon(1S, 2S, 3S)$ and $\pi^{+}\pi^{-}h_b(1P, 2P)$, there seemed to be two intermediate charged
[26] T. K. Pedlar et al. (CLEO), Phys. Rev. Lett. 107, 041803 (2011), [arXiv:1104.2025].

[27] I. Adachi (Belle), in "8th International Workshop On Heavy Quarkonium (QWG2011) Darmstadt, Germany, October 3-7, 2011," (2011), [arXiv:1110.3934].

[28] G. Aad et al. (ATLAS), Phys. Rev. Lett. 108, 152001 (2012), [arXiv:1112.5154].

[29] V. M. Abazov et al. (D0), Phys. Rev. D 86, 031103 (2012), [arXiv:1203.6034].

[30] R. Aaij et al. (LHCb), JHEP 10, 088 (2014), [arXiv:1409.1408].

[31] A. M. Sirunyan et al. (CMS), Phys. Rev. Lett. 121, 092002 (2018), [arXiv:1805.11192].

[32] G. Aad et al. (ATLAS), Phys. Rev. Lett. 113, 212004 (2014), [arXiv:1407.1032].

[33] A. M. Sirunyan et al. (CMS), Phys. Rev. Lett. 122, 132001 (2019), [arXiv:1902.00571].

[34] R. Aaij et al. (LHCb), Phys. Rev. Lett. 122, 232001 (2019), [arXiv:1904.00081].

[35] R. Aaij et al. (LHCb), JHEP 07, 035 (2019), [arXiv:1903.12240].

[36] S. Uehara et al. (Belle), Phys. Rev. Lett. 96, 082003 (2006), [hep-ex/0512035].

[37] B. Aubert et al. (BaBar), Phys. Rev. D 81, 092003 (2010), [arXiv:1002.0281].

[38] K. Abe et al. (Belle), Phys. Rev. Lett. 94, 182002 (2005), [hep-ex/0408126].

[39] J. P. Lees et al. (BaBar), Phys. Rev. D86, 072002 (2012), [arXiv:1207.2651].

[40] F.-K. Guo and U.-G. Meissner, Phys. Rev. D86, 091501 (2012), [arXiv:1208.1134].

[41] S. L. Olsen, Phys. Rev. D91, 5, 057501 (2015), [arXiv:1410.6534].

[42] Z.-Y. Zhou, Z. Xiao and H.-Q. Zhou, Phys. Rev. Lett. 115, 2, 022001 (2015), [arXiv:1501.00879].

[43] R. Aaij et al. (LHCb), Phys. Rev. D 102, 112003 (2020), [arXiv:2009.00026].

[44] K. Chilikin et al. (Belle), Phys. Rev. D95, 112003 (2017), [arXiv:1704.01872].

[45] B. Aubert et al. (BaBar), Phys. Rev. Lett. 95, 142001 (2005), [hep-ex/0506081].

[46] C. Z. Yuan et al. (Belle), Phys. Rev. Lett. 99, 182004 (2007), [arXiv:0707.2541].

[47] Z. Q. Liu et al. (Belle), Phys. Rev. Lett. 110, 252002 (2013), [arXiv:1304.0121].

[48] M. Ablikim et al. (BESIII), Phys. Rev. Lett. 118, 9, 092001 (2017), [arXiv:1611.01317].

[49] M. Cleven et al., Phys. Rev. D90, 7, 074039 (2014), [arXiv:1310.2190].

[50] M. Ablikim et al. (BESIII), Phys. Rev. Lett. 118, 9, 092002 (2017), [arXiv:1610.07044].

[51] M. Ablikim et al. (BESIII), Phys. Rev. Lett. 112, 9, 092001 (2014), [arXiv:1310.4101].

[52] M. Ablikim et al. (BESIII), Phys. Rev. D96, 3, 032004 (2017), [erratum: Phys. Rev.D99,no.1,019903(2019)], [arXiv:1703.08787].

[53] X. L. Wang et al. (Belle), Phys. Rev. Lett. 99, 142002 (2007), [arXiv:0707.3699].

[54] X. L. Wang et al. (Belle), Phys. Rev. D91, 112007 (2015), [arXiv:1410.7641].

[55] J. P. Lees et al. (BaBar), Phys. Rev. D89, 11, 111103 (2014), [arXiv:1211.6271].

[56] G. Pakhlova et al. (Belle), Phys. Rev. Lett. 101, 172001 (2008), [arXiv:0807.4458].

[57] S. Jia et al. (Belle), Phys. Rev. D100, 11, 111103 (2019), [arXiv:1911.00671].

[58] K. Chilikin et al. (Belle), Phys. Rev. D88, 7, 074026 (2013), [arXiv:1306.4894].

[59] K. Chilikin et al. (Belle), Phys. Rev. D90, 11, 112009 (2014), [arXiv:1408.6457].

[60] R. Aaij et al. (LHCb), Phys. Rev. Lett. 112, 22, 222002 (2014), [arXiv:1404.1903].

[61] B. Aubert et al. (BaBar), Phys. Rev. D79, 112001 (2009), [arXiv:0811.0564].

[62] R. Mizuk et al. (Belle), Phys. Rev. D78, 072004 (2008), [arXiv:0806.4098].

[63] J. P. Lees et al. (BaBar), Phys. Rev. D85, 052003 (2012), [arXiv:1111.5919].

[64] T. Aaltonen et al. (CDF), Phys. Rev. Lett. 102, 242002 (2009), [arXiv:0903.2229].

[65] T. Aaltonen et al. (CDF), Mod. Phys. Lett. A32, 26, 1750139 (2017), [arXiv:1101.6058].

[66] V. M. Abazov et al. (D0), Phys. Rev. D89, 1, 012004 (2014), [arXiv:1309.6580].

[67] S. Chatrchyan et al. (CMS), Phys. Lett. B734, 261 (2014), [arXiv:1309.6920].

[68] J. Brodzicka, Conf. Proc. C0908171, 299 (2009), [,299(2009)].

[69] R. Aaij et al. (LHCb), Phys. Rev. D85, 091103 (2012), [arXiv:1202.5087].

[70] J. P. Lees et al. (BaBar), Phys. Rev. D91, 1, 012003 (2015), [arXiv:1407.7244].

[71] C. P. Shen et al. (Belle), Phys. Rev. Lett. 104, 112004 (2010), [arXiv:0912.2383].

[72] R. Aaij et al. (LHCb), Phys. Rev. D95, 1, 012002 (2017), [arXiv:1606.07898].

[73] R. Aaij et al. (LHCb), Phys. Rev. Lett. 118, 2, 022003 (2017), [arXiv:1606.07895].

[74] R. Aaij et al. (LHCb), Phys. Rev. Lett. 127, 8, 082001 (2021), [arXiv:2103.01803].

[75] R. Aaij et al. (LHCb), Sci. Bull. 65, 1983 (2020), [arXiv:2006.16957].

[76] G. Aad et al. (ATLAS) (2023), [arXiv:2304.08962].

[77] A. Hayrapetyan et al. (CMS) (2023), [arXiv:2306.07164].

[78] R. Mizuk et al. (Belle), JHEP 10, 220 (2019), [arXiv:1905.05521].

[79] N. Hüsken, R. E. Mitchell and E. S. Swanson, Phys. Rev. D 106, 9, 094013 (2022), [arXiv:2204.11915].

[80] D. Santel et al. (Belle), Phys. Rev. D93, 1, 011101 (2016), [arXiv:1501.01137].

[81] F.-K. Guo, X.-H. Liu and S. Sakai, Prog. Part. Nucl. Phys. 112, 103757 (2020), [arXiv:1912.07030].

[82] R. Aaij et al. (LHCb), Phys. Rev. Lett. 110, 222001 (2013), [arXiv:1302.6269].

[83] R. Aaij et al. (LHCb), Phys. Rev. D92, 1, 011102 (2015), [arXiv:1504.06339].

[84] R. Aaij et al. (LHCb), Phys. Rev. D 102, 9, 092005 (2020), [arXiv:2005.13419].

[85] N. Brambilla et al., Eur. Phys. J. C71, 1534 (2011), [arXiv:1010.5827].

[86] M. Ablikim et al. (BESIII), Phys. Rev. Lett. 110, 252001 (2013), [arXiv:1303.5949].

[87] T. Xiao et al., Phys. Lett. B727, 366 (2013), [arXiv:1304.3036].

[88] M. Ablikim et al. (BESIII), Phys. Rev. Lett. 112, 2, 022001 (2014), [arXiv:1310.1163].

[89] M. Ablikim et al. (BESIII), Phys. Rev. Lett. 115, 11, 112003 (2015), [arXiv:1506.06018].
[90] M. Ablikim et al. (BESIII), Phys. Rev. Lett. 115, 22, 222002 (2015), [arXiv:1509.05620].

[91] M. Ablikim et al. (BESIII), Phys. Rev. Lett. 111, 24, 242001 (2013), [arXiv:1309.1896].

[92] M. Ablikim et al. (BESIII), Phys. Rev. Lett. 112, 13, 132001 (2014), [arXiv:1308.2760].

[93] M. Ablikim et al. (BESIII), Phys. Rev. Lett. 113, 21, 212002 (2014), [arXiv:1409.6577].

[94] M. Ablikim et al. (BESIII), Phys. Rev. Lett. 115, 18, 182002 (2015), [arXiv:1507.02404].

[95] M. Ablikim et al. (BESIII), Phys. Rev. Lett. 126, 10, 102001 (2021), [arXiv:2011.07855].

[96] M. Ablikim et al. (BESIII), Phys. Rev. Lett. 129, 11, 112003 (2022), [arXiv:2204.13703].

[97] M. Ablikim et al. ((BESIII), BESIII), Chin. Phys. C 47, 3, 033001 (2023), [arXiv:2211.12060].

[98] R. Aaij et al. (LHCb), Nature Phys. 18, 7, 751 (2022), [arXiv:2109.01038].

[99] R. Aaij et al. (LHCb), Nature Commun. 13, 1, 3351 (2022), [arXiv:2109.01056].

[100] A. Bondar et al. (Belle), Phys. Rev. Lett. 108, 122001 (2012), [arXiv:1110.2251].

[101] A. Garmash et al. (Belle), Phys. Rev. D91, 7, 072003 (2015), [arXiv:1403.0992].

[102] P. Krokovny et al. (Belle), Phys. Rev. D88, 5, 052016 (2013), [arXiv:1308.2646].

[103] I. Adachi et al. (Belle) (2012), [arXiv:1209.6450].

[104] K. F. Chen et al. (Belle), Phys. Rev. Lett. 100, 112001 (2008), [arXiv:0710.2577].

[105] P. Krokovny (Belle Collab.), talk given at Les Rencontres de Physique de la Vallée d'Aoste, La Thuile, Aosta Valley, Italy, 2012.

[106] A. Abdesselam et al. (Belle), Phys. Rev. Lett. 117, 14, 142001 (2016), [arXiv:1508.06562].

[107] M. Cleven et al., Eur. Phys. J. A47, 120 (2011), [arXiv:1107.0254].

[108] F. K. Guo et al., Phys. Rev. D93, 7, 074031 (2016), [arXiv:1602.00940].

[109] Q. Wang et al., Phys. Rev. D 98, 7, 074023 (2018), [arXiv:1805.07453].

[110] A. Garmash et al. (Belle), Phys. Rev. Lett. 116, 21, 212001 (2016), [arXiv:1512.07419].
79. Heavy Non-qq Mesons

Revised March 2024 by T. Gutsche (Tübingen U.), C. Hanhart (FZ Jülich) and R.E. Mitchell (Indiana U.).

The constituent quark model describes the observed meson
spectrum as bound **q̄q** states grouped into SU(N) flavor multi-
plets (see the ‘Quark Model’ in this issue of the *Review of Particle
Physics*). However, the self coupling of gluons in QCD suggests
that additional mesons made of bound gluons (glueballs), or **q̄q**-
pairs with an excited gluon (hybrids), may exist. Furthermore,
multiquark color singlet states such as **qq̄qq** (tetraquarks as com-
pact diquark-antidiquark systems and ‘molecular’ bound states
of two mesons) or **qq̄qq̄q** (six-quark, ‘baryonium’ or quasinuclear
baryon-antibaryon bound states) have also been predicted. The
focus of this review is on the current understanding of exotic states
that apparently do not fit into the constituent quark model and
contain at least one heavy quark (charm or bottom — the life-
time of the top quark is too short to allow it to hadronise). Light
non-**q̄q** candidates (made of *u*, *d* and *s* quarks) are discussed in
‘Scalar Mesons below 1 GeV’ and ‘Spectroscopy of Light Meson
Resonances’ in this *Review*.

There are candidates for exotic states with both quantum num-
bers not allowed and quantum number allowed for ḡqq states. For
the latter class it is up to now unclear to what extend a mixing of
conventional and unconventional states occurs. While Refs. [1-4]
report a significant mixing, Refs. [5, 6] describe a scenario where
the mixing especially between molecular structures and ḡqq states
is suppressed.

The review is split into 3 parts, discussing separately heavy–
light systems, heavy–heavy systems, as well as systems with more
than two heavy quarks. For a more detailed discussion of the ex-
perimental and/or theoretical status on exotic meson candidates
in the doubly heavy meson sector we refer to Refs. [7–13]. Re-
views with main focus on tetraquarks and molecular states are
presented in Ref. [14] and in Ref. [15], respectively.

The naming scheme for hadrons has been updated by the Part-
ticle Data Group in 2023, and now covers the heavy “exotic”
tetraquark (qq̄qq̄) and pentaquark (qq̄qq̄q) candidates recently dis-
covered. We use the new naming scheme in the following. See
“Naming Scheme for Hadrons” in this *Review* for details and a
table mapping old and new particle names.

79.1 Heavy-light systems

79.1.1 Positive parity open charm ground states

Two very narrow states, $D_{s0}^*(2317)^{\pm}$ and $D_{s1}(2460)^{\pm}$, were observed at B factories [16, 17]. They lie far below the predicted masses for the two expected broad P-wave $c\bar{s}$ mesons. However, strong cusp effects, due to the nearby $DK(DK^*)$ thresholds, could shift their masses downwards and quench the observed widths, still allowing for a conventional explanation of these states [18]. Such an effect was also claimed for the $a_0(980)$ and $f_0(980)$ mesons, which lie just below the $K\bar{K}$ threshold. In contrast to this picture, many authors favour exotic explanations for these lightest positive parity charmed states, such as four-quark states [19–22] or $DK(DK^*)$ molecules [23–28]. Both states are isoscalars and thus strong isospin-violating decays are possible only to the isovector channels $D_s^{(*)\pm}\pi^0$, leading to very small widths. For compact structures the hadronic decay is driven by $\eta - \pi^0$ mixing and one finds widths below 10 keV [18, 29]. In contrast to this a hadronic width typically larger than 100 keV would be the unequivocal signature for a prominent molecular nature of $D_{s0}^*(2317)^{\pm}$ [25, 27, 28, 30, 31], since then meson loops generating enhanced isospin violation due to the energy difference between the $D^{*0}K^+$ and the $D^{*+}K^0$ thresholds would contribute significantly. The most refined analysis in the molecular scenario [30], where the parameters of the formalism are constrained by a large bulk of lattice QCD data on singly charmed systems, reports a prediction for the width in the range (104–116) keV. The currently measured upper bound for the width is 3.8 MeV.

It should be stressed that – akin to q̄q mesons – multiquark
states also appear in multiplets. An important probe for the
structure of hadrons is therefore the spectra, and in particular
the SU(3) flavor multiplets within the different scenarios. For
example the tetraquark model advocated in Refs. [22, 32] finds

that the $D_{s0}^*(2317)^{\pm}$ should be degenerate with its non-strange
partner state $D_0^*(2300)$ in line with the currently available experi-
mental analyses. However, Ref. [33] showed that the recent LHCb
data for $B \to D\pi\pi$ [34, 35] call for a significantly lighter lightest
non-strange scalar charmed meson as soon as constraints from
the chiral symmetry of QCD are employed in the analysis. This
claim is in line with the recent lattice QCD study of Ref. [36]. In
fact, experimental as well as lattice data turn out to be consistent
with the predictions from unitarised chiral perturbation theory
pioneered for these systems in Ref. [24]. For more recent calcula-
tions see Refs. [30, 37, 38]. Moreover, all these latter calculations
predict that there should be in addition an about 300 MeV heav-
ier flavor exotic state, formally a member of a flavor sixtet and
absent for c̄q states, that indeed shows up in the above mentioned
LHCb data [38] — why this state was not reported in the lattice
study of Ref. [39] is explained in Ref. [40] by its being located on
a hidden Rieman-sheet. Thus, in this picture the structure that
appears as $D_0^*(2300)$ in the *Listings* originates from the interplay
of two poles, similar to the $\Lambda(1405)$ in the baryon sector, see ‘Pole
structure of the $\Lambda(1405)$ region’ in this *Review*. Two poles in the
non-strange scalar sector are also generated in the tetraquark pic-
ture of Ref. [22], but in this work the real parts of the poles are
located at 2308 MeV and 2666 MeV, in conflict with the data
mentioned above. In the refined analysis of Ref. [32] it is shown,
however, that the exact pole locations and even the ordering of
the higher multiplets depend on the details of the model.

Lattice QCD allows for predictions of the flavor partner states in
the *B*-sector. The most recent analysis can be found in Ref. [41].
In this work also a comparison with theoretical predictions can
be found. Unfortunately an experimental confirmation for those
states is difficult with existing facilities. Radiative decays are
predicted to be the most prominent decay channels [31], since the
above mentioned isospin violation from meson loops is suppressed
in the *B* sector as a consequence of the tiny $B^{\pm} - B^0$ mass dif-
ference.

79.1.2 The $T_{cs0}^*(2870)$ and $T_{cs1}^*(2900)$

Manifestly exotic candidates around 2.9 GeV were reported in
Refs. [42, 43] by LHCb in the $\bar{D}K$ system. Since the strong in-
teraction preserves the quark flavor and the final state contains
an $\bar{s}$ as well as a $\bar{c}$, color neutrality calls for a minimum quark
content of $u\bar{d}\bar{s}$. The explanation for those states include com-
pact tetraquarks [44–46], hadronic molecules with a prominent
$D^*\bar{K}^*$ component [47–51], or kinematic effects such as cusps or
triangle singularities without nearby poles [52, 53]. The different
pictures are contrasted and observable differences between them
are discussed in Ref. [54].

79.1.3 The $T_{c\bar{s}0}^*(2900)^0$ and $T_{c\bar{s}0}^*(2900)^{++}$

There is a very recent report by LHCb [55] on the observation of a doubly charged open-charm tetraquark candidate together with a neutral partner in the $D_s\pi$ channel, with masses near 2.9 GeV. The results imply two states with $c\bar{s}u\bar{d}$ or $c\bar{s}u\bar{d}$ flavor structures. Although masses and widths are consistent with the $T_{cs}$ states of Refs. [42, 43], the strangeness quantum numbers do not match. The first explanation for those states in coupled-channel studies is based on a kinematic effect such as a cusp [56], while a molecular structure with a dominant $D^*K^*$ component is claimed to be excluded [57].

79.2 Heavy-heavy systems

With the discovery of the $\chi_{c1}(3872)^1$ in $B^{\pm} \to K^{\pm}X$ ($X \to J/\psi(1S)\pi^+\pi^-$) by Belle [58] in 2003, soon confirmed by BaBar [59], many searches for states beyond the standard q̄q quark model were initiated in the charm and bottom sectors. For an updated collection of the currently available experimental information on multiquark states we refer to ‘Spectroscopy of mesons containing two heavy quarks’ in this *Review*, and in particular to Table 78.2. Moreover, in the decay $\Lambda_b^0 \to J/\psi(1S) K^-p$ the LHCb collaboration has recently reported the observation of new baryons decaying into $J/\psi(1S)p$, which are candidates for heavy pentaquark

<footnote n="1">¹ The χ<sub>c1</sub>(3872) is also known as the X(3872). According to the PDG naming scheme, the primary name for a meson expresses its quantum numbers.</footnote>
states [60,61]. They are discussed in some depth in 'Pentaquarks' in this Review.

When restricting ourselves to confirmed states we are faced with several states that do not seem to fit into the mass and quantum number schemes of $q\bar{q}$ quark models. This is clear for the five established charged states ($T_{c\bar{c}1}(3900)^{\pm}$, $T_{c\bar{c}2}(4020)^{\pm}$ and $T_{c\bar{c}1}(4430)^{\pm}$ in the charmonium sector, and $T_{b\bar{b}1}(10610)^{\pm}$ and $T_{b\bar{b}1}(10650)^{\pm}$ in the bottomonium sector). The neutral states ($\chi_{c1}(3872)$, $\psi(4230)$, $\psi(4360)$, and $\psi(4660)^2$ also challenge the quark model since their masses and decay properties are in conflict with expectations. Although so far seen just by a single experiment in a single channel [62,63], the $T_{cc}(3875)$ should not remain unmentioned.

### 79.2.1 The $\chi_{c1}(3872)$

The quantum numbers of the $\chi_{c1}(3872)$ have been determined by LHCb to be $J^{PC} = 1^{++}$, first by assuming zero angular momentum between the $J/\psi(1S)$ and the dipion [64] and then by relaxing this constraint [65]. The $\chi_{c1}(3872)$ cannot be identified with the $\chi_{c1}(2^3P_1)$ since the latter is predicted to lie about 100 MeV higher in mass [66]. Instead, the $X(3940)$ reported by Belle in $e^+e^- \to J/\psi(1S)X$, decaying into $D^*\bar{D}$ but not into $D\bar{D}$ [67] could be the $\chi_{c1}(2^3P_1)$. The $\chi_{c2}(3930)\ 2^3P_2$ tensor partner was reported by Belle at 3931 MeV in $\gamma\gamma$ interactions [68].

The $\chi_{c1}(3872)$ lies within 200 keV of the $D^0\bar{D}^{*0}$ threshold and its width is very small — recent Breit-Wigner analyses by LHCb [69, 70] result in an average value of $\Gamma = (1.19 \pm 0.21)$ MeV as indicated in the *Listings*; employing the Flatté formalism of Ref. [71] under the constraint that the $D^*\bar{D}$ decay channel dominates, results in $\Gamma = 0.22^{+0.18}_{-0.19}$ MeV, where the standard convention is applied to use twice the imaginary part of the pole location as width. In the future those values can be refined more by either a direct measurement of the width that should be possible at the planned PANDA experiment [72, 73] or by exploiting the interplay of a triangle singularity and the $\chi_{c1}(3872)$ pole [74–76]. Therefore the most natural explanation for this state is a $1^{++}\bar{D}D^*$ molecule [77], for which strong isospin breaking is predicted [77, 78], since the distance of the pole of the $\chi_{c1}(3872)$ to the $D^0\bar{D}^{*0}$ threshold is significantly smaller than to the $D^+D^{*-}$ threshold. Indeed, the comparable rates for $\omega J/\psi(1S)$ and $\rho^0 J/\psi(1S)$ are consistent with an interpretation of $\chi_{c1}(3872)$ as an isoscalar $D\bar{D}^*$ molecule, when the different widths of the $\rho$ and $\omega$ are taken into account [79]. Similarly, a dominant molecular $D^0\bar{D}^{*0}$ structure in the $\chi_{c1}(3872)$ with further subleading hadronic components is used to explain strong and radiative decays involving $J/\psi(1S)$ and $\psi(2S)$ in the final states [80, 81]. From an effective field theory point of view these radiative transitions are not well suited to identify the molecular admixture in the $\chi_{c1}(3872)$, since there appears a counter term at leading order, diminishing the sensitivity to the long ranged molecular component [82].

A four-quark state $cq\bar{c}q'$ is another possible interpretation of the $\chi_{c1}(3872)$ [21] but a charged partner of the $\chi_{c1}(3872)$ has not been observed (e.g. not in $B^- \to K^0X^-$ nor in $B^0 \to K^+X^-$, where $X^- \to J/\psi(1S)\pi^-\pi^0$ [83]) – see [84] for a possible explanation of this non-observation within the tetraquark approach assuming specific diquark correlations and the more recent discussion in Ref. [85]. The claim that $\chi_{c1}(3872)$ must be a compact (tetraquark) state, since it is also produced at very high $p_T$ in $\bar{p}\bar{p}$ collisions [86] and in high multiplicity final states [87], was challenged in [88] and [89], respectively, which in particular stress the importance of rescattering, see also Refs. [90, 91].

The high statistics measurement of the $\chi_{c1}(3872)$ lineshape of Ref. [69] also allowed for an extraction of the effective range for the $D^0\bar{D}^{*0}$ interaction. At face value, the values provided in this publication indicate a large and negative effective range [92], which could be interpreted as a signature of a prominent compact component in the $\chi_{c1}(3872)$ wave function — this follows from the famous Weinberg compositeness criterion [93]. The effect of the one-pion exchange on the effective range turns out to

be small [94–97] and should therefore not invalidate the Weinberg-type analysis. However, it was argued in Ref. [98] that once the effect of the large isospin violation as well as the strong correlations between the effective parameters appearing in Ref. [71] are subtracted, the effective range is consistent with a purely molecular $\chi_{c1}(3872)$. There is no consensus yet how to interpret the line shapes of $\chi_{c1}(3872)$.

### 79.2.2 The $\psi(4230)$ and $\psi(4360)$

A broad structure, originally called $Y(4260)$, decaying into $J/\psi(1S)\pi^+\pi^-$ was reported by BaBar in initial state radiation $e^+e^- \to (\gamma_{ISR})Y(4260)$ [99]. A subsequent measurement with significantly improved precision was reported by BESIII [100], and revealed that the original $Y(4260)$ cannot be described with a simple resonant lineshape. Fitting the BESIII data with two Breit-Wigner distributions leads to a narrower and lighter structure (referred to in the *Listings* as the $\psi(4230)$), but also requires a second state at 4320 MeV. However, note that the $D_1(2420)\bar{D}$ molecular model for the $\psi(4230)$ can describe the same data with just one single pole [101]. How many vector states are in the mass range between 4220 MeV and 4400 MeV is not yet settled, but knowing this is crucial to allow for further progress.

There are no charmonium states expected in this mass region with quantum numbers $1^{--}$ from quark models using the Cornell type of interaction, although this might not be true for some screened versions thereof – for a recent discussion we refer to Ref. [102]. In addition, a charmonium state at this mass is expected to have significant couplings to one or more of the $\bar{D}^{(*)}D^{(*)}$ channels [103, 104], a feature that is not observed for the $\psi(4230)$. This state could be a hybrid charmonium with a spin-1 [105, 106] or a spin-0 [107, 108] $\bar{c}c$ core. However, provided that the observation of $\psi(4230)$ decay into $h_c(1P)\pi\pi$ by BESIII [109] is confirmed, the hybrid hypothesis would be under pressure, since the spin of the heavy quarks (coupled to zero in the $h_c(1P)$) should be conserved in leading order in the expansion in $(\Lambda_{QCD}/m_c)$. (The individual conservation of the heavy quark spin and the total angular momentum of the light quark cloud is a consequence of the heavy-quark spin symmetry, see ‘Heavy-Quark and Soft-Collinear Effective Theory’ in this issue of the *Review*.) The same criticism applies to the hadrocharmonium interpretation of the $\psi(4230)$, which describes this state as spin-1 quarkonium surrounded by a light quark cloud [110]. To circumvent the spin-symmetry argument, Ref. [111] argues that $\psi(4230)$ and $\psi(4360)$ could be mixtures of two hadrocharmonia with spin-triplet and spin-singlet heavy quark pairs. The same kind of mixing could also be at work for hybrid structures. Hybrids can also treated employing the Born-Oppenheimer EFT originally suggested in Ref. [112, 113]. Those studies revealed that while the $\psi(4230)$ is in principle a good candidate for a hybrid state [114], its decays are incompatible with this assignment [115].

A dominant $D_1(2420)\bar{D}$ component in the $\psi(4230)$ [116] explains naturally why the $T_{c\bar{c}1}(3900)^{\pm}$ (interpreted by the authors as a $\bar{D}D^{*}$ bound state) is seen in $\psi(4230) \to \pi^{\mp}T_{c\bar{c}1}(3900)^{\pm}$. A similar mechanism is also found in Ref. [117] linking in addition the $J/\psi(1S)\pi^+\pi^-$ and $\psi(2S)\pi^+\pi^-$ decays of the $\psi(4230)$. Furthermore, a copious production of $\chi_{c1}(3872)$ in $\psi(4230)$ radiative decays was predicted from the prominent $D_1\bar{D}$ component of the $\psi(4230)$ [118], which was confirmed by BESIII [119]. Possible charmonia components both in $\psi(4230)$ and $\chi_{c1}(3872)$ can influence the radiative transition but they are shown to be of subleading order [120]. It was proposed that $\psi(4360)$ as a $D_1\bar{D}^*$ bound state could be the spin partner of the $\psi(4230)$ [121, 122]. In Ref. [123] a dynamical model is presented that identifies the vector states $\psi(4230)$, $\psi(4360)$ and $\psi(4415)$ as $D\bar{D}_1$, $D^*\bar{D}_1$ and $D^*\bar{D}_2$ molecular states, respectively. Moreover, an exotic $0^{-}$ state is predicted as $D^*\bar{D}_1$ molecule with a mass of 4366 $\pm$ 18 MeV and a width of less than 10 MeV.

The tetraquark picture calls for four ground state vector states — once the parameters of the model are fixed from some candidate states in the negative and positive parity sector, states with other quantum numbers can be predicted. Possible scenarios are for instance discussed in Ref. [124] which builds on a tailor-made spin-spin interaction [125] to describe the $\chi_{c1}(3872)$, both $T_{c\bar{c}1}(3900)^{\pm,0}$ and $T_{c\bar{c}}(4020)^{\pm}$ and even the $T_{c\bar{c}1}(4430)^{\pm}$

<footnote n="2">² The $\psi(4230)$, $\psi(4360)$, and $\psi(4660)$ are also known as the $Y(4230)$, $Y(4360)$, and $Y(4660)$, respectively. Before improved mass measurements, the $\psi(4230)$ was originally called the $\psi(4260)$ or $Y(4260)$.</footnote>
confirmed by Belle [126] and LHCb [127]. This model also explains the copious production of $\chi_{c1}(3872)$ in $\psi(4230)$ radiative decays mentioned above [125, 128]. However, tetraquark models (in most cases based on diquark-antidiquark configurations) tend to predict many additional charged and neutral states which have not yet been discovered. In particular, as for the conventional $\bar{q}q$ structures one should expect nearly degenerate isoscalar and isovector states in analogy to the near degeneracy of $\rho(770)$ and $\omega(782)$. The problem and possible explanations are discussed in some detail in Refs. [84, 85].

Ref. [129] found a sizeable SU(3) flavor octet contribution when analysing the $\pi\pi$ final state from $\psi(4230) \rightarrow J/\psi(1S)\pi^+\pi^-$, which is consistent with both a molecular and a tetraquark interpretation of $\psi(4230)$, but at odds with a hybrid or a $c\bar{c}$ interpretation.

### 79.2.3 The $\psi(4660)$

In the mass range above 4600 MeV, the number of poles is also not yet settled. Experimental signals are seen in the $\psi(2S)\pi\pi$, $\Lambda_c^+\bar{\Lambda}_c^-$, $D_s\bar{D}_{s1}$, and $D_s\bar{D}_{s2}^*$ final states. In the *Listings* these structures go into one node, $\psi(4660)$, due to their proximity in parameter values. Moreover, various theoretical works describe these states in a combined analysis [130–132]. The signal in the hidden strangeness mode around 4630 MeV still calls for confirmation and might be yet another realisation of the same state, but there are already speculations about its nature. While Ref. [133] argues in favor of a $D_s^{(*)}\bar{D}_{s1}(2536)$ or $D_s^{(*)}\bar{D}_{s2}(2573)$ [134] molecular nature, Ref. [135] does not confirm these claims. Ref. [136] identifies the structure as P-wave [cs][c$\bar{c}$] tetraquark state. Other explanations of the $\psi(4660)$ include a $\psi(2S)f_0(980)$ molecule [137] and a $\Lambda_c^+\bar{\Lambda}_c^-$ baryonium [130]. Also in this mass range studies of the partner states, driven either by spin or flavor symmetry will be very valuable — see e.g. the predictions in Ref. [138].

### 79.2.4 The $T_{c\bar{c}1}(3900)$, $T_{c\bar{c}}(4020)$, $T_{b\bar{b}1}(10610)$, and $T_{b\bar{b}1}(10650)$

The isovector states $T_{c\bar{c}1}(3900)$ and $T_{c\bar{c}}(4020)$, first observed by BESIII [139, 140], decay predominantly into $\bar{D}D^*$ and $\bar{D}^*D^*$, respectively, while the $T_{b\bar{b}1}(10610)$ and $T_{b\bar{b}1}(10650)$, first observed by Belle [141, 142], decay predominantly into $\bar{B}B^*$ and $\bar{B}^*B^*$ [143], respectively, although all four were discovered in the decay mode heavy quarkonium plus pion. This suggests that these states are close relatives and their interactions are connected via heavy quark flavor symmetry. A molecular interpretation for the bottomonium states was proposed shortly after the discovery of the two $T_{b\bar{b}}$ states [144] and also shortly after that of the $T_{c\bar{c}1}(3900)$ [116]. This picture is confirmed within the meson exchange model of Ref. [145]. Decay patterns of $T_{c\bar{c}1}(3900)$ and the two $T_{b\bar{b}}$ were also shown to be consistent with a molecular interpretation [146–148]. However, some of their properties also appear to be consistent with tetraquark structures [149]. If the molecular picture were correct for the $T_{b\bar{b}}$ states, spin symmetry would lead to the existence of spin partner states [150–152], which are still to be found. In Ref. [153] it was shown that the actual pole locations of these partner states would be good probes of one-pion exchange in the molecular potential, which makes the experimental search for those states even more interesting.

### 79.2.5 The $T_{c\bar{c}1}(4200)$ and $T_{c\bar{c}1}(4430)$

The heaviest confirmed charged state in the charmonium sector is the $T_{c\bar{c}1}(4430)^{\pm}$ observed by Belle [126]. It is interpreted as hadrocharmonium [110], $\bar{D}_1D^*$ molecule [154], as well as tetraquark [125]. Alternatively, in [155, 156] the $T_{c\bar{c}1}(4430)^{\pm}$ is explained as a cross-channel effect enhanced by a triangle singularity from open charm states. These works were criticised in Ref. [157] where an alternative triangle consisting of a $K^*$, a $\pi$, and the $\psi(4230)$, is proposed to generate the $T_{c\bar{c}1}(4430)$. The Argand diagram shows an anticlockwise circle, in line with the experimental analysis [127], while the one of Ref. [156] shows a clockwise motion. By replacing the $\psi(4230)$ with the $\psi(3770)$, and changing the $K^*$, one can also interpret the $T_{c\bar{c}1}(4200)$ as a kinematic effect [157]. A possible interpretation of $T_{c\bar{c}1}(3900)$ and $T_{c\bar{c}}(4020)$ as crossed channel effects is put forward in Ref. [158]. It remains to be seen, however, if this kind of explanation is also

capable of explaining the observations of these lowest $T_{c\bar{c}}$ states, also at other total energies.

### 79.2.6 The $T_{c\bar{c}s1}(4000)$

There is recent evidence for a charged charmonium-like state with strangeness, $T_{c\bar{c}s1}(4000)$, from BESIII [159] and LHCb [160]. The possible existence of a strange partner to the $T_{c\bar{c}}$ near the $D_s\bar{D}^*$ thresholds has been predicted in molecular models [161, 162], for tetraquarks [163, 164], for hadrocharmonium structures [164, 165], and as a coupled-channel effect [166]. Later on this state is interpreted in Refs. [167, 168] as a member of the same multiquark octet and in Refs. [169–171] as a member of the same molecular one as the $T_{c\bar{c}1}(3900)$. Ref. [168] is also able to describe the recent LHCb data in the $J/\psi(1S)K^-$ system [160], although the extracted total width for their lowest $T_{c\bar{c}s}$ state is an order of magnitude larger than that found by BESIII. Refs. [172, 173] claim that both the molecular components and the compact tetraquark core are relevant to describe the $T_{c\bar{c}1}(3900)$ and $T_{c\bar{c}s1}(4000)$ resonances. Again, in [174] the $T_{c\bar{c}s1}(4000)$ can be explained as a coupled-channel effect producing the enhancement close to threshold.

Ref. [169] comes to the conclusion that an understanding of the experimental data for the $T_{c\bar{c}s1}(4000)$ calls for the inclusion of both triangle diagrams as well as a pole term. The authors conclude from their calculations that $T_{c\bar{c}s1}(4000)$ and $T_{c\bar{c}1}(3900)$ are members of the same SU(3) multiplet. This conclusion was confirmed in Ref. [175] and the study extended to higher energies. The fit of the experimental data allowed for two solutions, one where the $T_{c\bar{c}}(4020)$ is a spin partner of the mentioned isovector states and one where it is not. Data e.g. for $e^+e^- \rightarrow K^+D^*-D^*$ is claimed to be decisive which scenario is indeed realised.

It should be stressed that the various scenarios, while describing much of the available data, also make decisive predictions, e.g. yet unobserved quantum numbers [124, 176]. The forthcoming data on heavy meson spectroscopy from various facilities should provide a much deeper understanding of how QCD forms matter out of quarks and gluons.

### 79.2.7 The $T_{cc}(3875)$

There is a recent report and follow-up by LHCb [62, 63] on the observation of a narrow peak in the $D^0D^0\pi^+$ mass spectrum just below the $D^{*+}D^0$ threshold. The observed state is a candidate for an isoscalar double-charm $T_{cc}^+$ tetraquark configuration with a quark content of $cc\bar{u}\bar{d}$. Unsuccessful searches in $D^+D^0\pi^+$ and $D^+D^+$ mass spectra disfavour the isovector assignment. The $D^0\pi^+$ mass distribution indicates that the $T_{cc}^+$ proceeds through the formation of a $D^*$ in the intermediate state. The additional proximity to the $D^*D^0$ mass threshold favors the $J^P = 1^+$ assignment. The $T_{cc}^+$ has a remarkable narrow width and lies very close to the $D^{*+}D^0$ threshold [63], comparable to the properties of the $\chi_{c1}(3872)$ [177].

A first evaluation in the context of QCD sum rules [178] results in mass and width values consistent with the assumption of a compact diquark-antidiquark structure of the $T_{cc}^+$. A pole analysis performed with $D^0D^0\pi^+ - D^{*+}D^0$ coupled-channels [179] suggests that the $T_{cc}^+$ has its origin in a $D^{*+}D^0$ virtual state resulting from the attraction in this channel and the coupling to $D^0D^0\pi^+$. The interpretation of the observed $T_{cc}^+$ as a $D^*D$ molecular structure was pursued in quark models [180–183] and the Bethe-Salpeter framework [184]. Predictions depend on fine model details and are usually accompanied by additional partner states, like an isovector molecule or higher radial excitations. In analogy to the case of the $\chi_{c1}(3872)$ in Ref. [177] the production of $T_{cc}^+$ in high-energy proton-proton collisions at the LHC is studied. Effects of a charm-meson triangle singularity on the inclusive production of $T_{cc}^+$ from the rescattering of DD leads to a narrow peak in the $T_{cc}^+\pi^+$ invariant mass distribution. A spin 2 partner of the $T_{cc}^+$ is predicted in Refs. [182, 183], who's hadronic decays are studied in Ref. [185].

As for the $\chi_{c1}(3872)$ one can employ the effective range deduced from experiment for the effective range in the $DD^*$-channel as a measure of the nature of the $T_{cc}(3875)$. Also here isospin violations are significant. It turns out that in a calculation where the one-pion exchange is kept fully non-perturbative, its effect on the
effective range is small and positive [183], while it is small and negative when being treated perturbatively [97, 186].

First lattice studies are also available for the $T_{cc}(3875)$ [187-189], all performed at pion masses slightly larger than physical such that the $D^*$ and with this also the $T_{cc}(3875)$ are stable. They all find that at these pion masses the $T_{cc}$ is a virtual state. In Ref. [190] it was stressed, however, that all those analyses should be redone, since the proximity of a left-hand cut invalidates the method used.

## 79.3 Systems containing four heavy quarks

### 79.3.1 The $T_{cc\bar{c}\bar{c}}(6900)$

LHCb reported the observation of pronounced structures in a double-$J/\psi(1S)$ invariant mass distribution [191] thus pointing at states with $cc\bar{c}\bar{c}$ quarks contents. First results on di-$J/\psi(1S)$ and $J/\psi(1S)\psi(2S)$ invariant mass distributions in proton-proton collision data by the CMS [192] and ATLAS [193] Collaborations confirm and extend the resonance structures seen by LHCb. Belle reported on a first search for a double-charmonium state with $e^+e^- \to \eta_c J/\psi(1S)$ near threshold via the ISR process. No significant signal of the double charmonium state is found in several bins of the invariant mass of $\eta_c J/\psi(1S)$ [194]. Recently the COMPASS [195] Collaboration also studied double $J/\psi(1S)$ production in pion scattering off nuclear targets. No significant signatures that could be associated with tetraquarks are found in the double $J/\psi(1S)$ mass spectrum, although the COMPASS data sample is limited.

From quark models the possible existence of bound states like this was discussed already long ago [196,197]. There are now also many model calculations available in the literature. For discussions of those data from the compact tetraquark perspective see Refs. [198-204]. Most of these quark model calculations assign the reported structure to a $cc\bar{c}\bar{c}$ state in the $2S$ multiplet with near-degenerate $J^P = 0^+$, $1^+$ and $2^+$ configurations. The dominance of a compact tetraquark component in the apparently exotic structure is also supported by the coupled multichannel study of Ref. [205]. The full coupled-channel study of Ref. [206] leads to a rich spectrum of $c\bar{c}-c\bar{c}$ states where parts are candidates for the observed double $J/\psi(1S)$ structures. For direct analyses of the data within the tetraquark approach, see, e.g., Ref. [207]. Also QCD sum rule studies of the system were published, but do not give consistent results: For example Ref. [208] interprets the structure at 6900 MeV as a hybrid state, Ref. [209] as a tetraquark, and Ref. [210] states that both molecular and tetraquark interpretations are possible. An alternative view on the LHCb double-$J/\psi(1S)$ data is provided in Refs. [211-213] where the analyses are performed using coupled channel $T$-matrices. In these works the structures in the data emerge from the interplay of thresholds and resonances with the non-trivial prediction that there should exist, if this dynamical picture were correct, a state located very near the double-$J/\psi(1S)$ threshold which can be searched for experimentally. In Ref. [214] it was shown that a $J/\psi(1S)J/\psi(1S)$ scattering potential sufficiently strong to bind is in fact consistent with our current understanding of $\pi J/\psi(1S)$ scattering as well as the strength of the $\pi\pi$ interactions.

In contrast, Ref. [215,216] explains the double-$J/\psi(1S)$ data as cusps without nearby poles. This claim can be tested experimentally: if this explanation were correct, there should not be any narrow near threshold structure in the channel that generates the cusp, for that would call for a non-perturbative interaction in that channel, as pointed out in Ref. [217].

### References

[1] Y. S. Kalashnikova and A. V. Nefediev, Phys. Rev. D 80, 074004 (2009), [arXiv:0907.4901].

[2] M. Takizawa and S. Takeuchi, PTEP 2013, 093D01 (2013), [arXiv:1206.4877].

[3] P. G. Ortega, D. R. Entem and F. Fernandez, J. Phys. G 40, 065107 (2013), [arXiv:1205.1699].

[4] S. Coito, G. Rupp and E. van Beveren, Eur. Phys. J. C 73, 3, 2351 (2013), [arXiv:1212.0648].

[5] I. K. Hammer, C. Hanhart and A. V. Nefediev, Eur. Phys. J. A 52, 11, 330 (2016), [arXiv:1607.06971].

[6] C. Hanhart and A. Nefediev, Phys. Rev. D 106, 11, 114003 (2022), [arXiv:2209.10165].

[7] N. Brambilla et al., Eur. Phys. J. C71, 1534 (2011), [arXiv:1010.5827].

[8] H.-X. Chen et al., Phys. Rept. 639, 1 (2016), [arXiv:1601.02092].

[9] A. Ali, J. S. Lange and S. Stone, Prog. Part. Nucl. Phys. 97, 123 (2017), [arXiv:1706.00610].

[10] R. F. Lebed, R. E. Mitchell and E. S. Swanson, Prog. Part. Nucl. Phys. 93, 143 (2017), [arXiv:1610.04528].

[11] S. L. Olsen, T. Skwarnicki and D. Zieminska, Rev. Mod. Phys. 90, 1, 015003 (2018), [arXiv:1708.04012].

[12] Y.-R. Liu et al., Prog. Part. Nucl. Phys. 107, 237 (2019), [arXiv:1903.11976].

[13] N. Brambilla et al., Phys. Rept. 873, 1 (2020), [arXiv:1907.07583].

[14] A. Esposito, A. Pilloni and A. D. Polosa, Phys. Rept. 668, 1 (2017), [arXiv:1611.07920].

[15] F.-K. Guo et al., Rev. Mod. Phys. 90, 1, 015004 (2018), [arXiv:1705.00141].

[16] B. Aubert et al. (BaBar), Phys. Rev. Lett. 90, 242001 (2003), [hep-ex/0304021].

[17] D. Besson et al. (CLEO), Phys. Rev. D68, 032002 (2003), [Erratum: Phys. Rev.D75,119908(2007)], [hep-ex/0305100].

[18] S. Godfrey, Phys. Lett. B568, 254 (2003), [hep-ph/0305122].

[19] H.-Y. Cheng and W.-S. Hou, Phys. Lett. B566, 193 (2003), [hep-ph/0305038].

[20] K. Terasaki, Phys. Rev. D68, 011501 (2003), [hep-ph/0305213].

[21] L. Maiani et al., Phys. Rev. D71, 014028 (2005), [hep-ph/0412098].

[22] V. Dmitrasinovic, Phys. Rev. Lett. 94, 162002 (2005).

[23] T. Barnes, F. E. Close and H. J. Lipkin, Phys. Rev. D68, 054006 (2003), [hep-ph/0305025].

[24] E. E. Kolomeitsev and M. F. M. Lutz, Phys. Lett. B582, 39 (2004), [hep-ph/0307133].

[25] A. Faessler et al., Phys. Rev. D76, 014005 (2007), [arXiv:0705.0254].

[26] A. Faessler et al., Phys. Rev. D 76, 114008 (2007), [arXiv:0709.3946].

[27] M. F. M. Lutz and M. Soyeur, Nucl. Phys. A813, 14 (2008), [arXiv:0710.1545].

[28] L. Liu et al., Phys. Rev. D87, 1, 014508 (2013), [arXiv:1208.4535].

[29] P. Colangelo and F. De Fazio, Phys. Lett. B570, 180 (2003), [hep-ph/0305140].

[30] X.-Y. Guo, Y. Heo and M. F. M. Lutz, Phys. Rev. D98, 1, 014510 (2018), [arXiv:1801.10122].

[31] H.-L. Fu et al., Eur. Phys. J. A 58, 4, 70 (2022), [arXiv:2111.09481].

[32] V. Dmitrasinovic, Phys. Rev. D 86, 016006 (2012).

[33] M.-L. Du et al., Phys. Rev. Lett. 126, 19, 192001 (2021), [arXiv:2012.04599].

[34] R. Aaij et al. (LHCb), Phys. Rev. D94, 7, 072001 (2016), [arXiv:1608.01289].

[35] R. Aaij et al. (LHCb), Phys. Rev. D 94, 7, 072001 (2016), [arXiv:1608.01289].

[36] L. Gayer et al. (Hadron Spectrum), JHEP 07, 123 (2021), [arXiv:2102.04973].

[37] M. Albaladejo et al., Phys. Lett. B767, 465 (2017), [arXiv:1610.06727].

[38] M.-L. Du et al., Phys. Rev. D98, 9, 094018 (2018), [arXiv:1712.07957].
[39] G. Moir et al., JHEP 10, 011 (2016), [arXiv:1607.07093].

[40] A. Asokan et al., Eur. Phys. J. C 83, 9, 850 (2023), [arXiv:2212.07856].

[41] R. J. Hudspith and D. Mohler, Phys. Rev. D 107, 11, 114510 (2023), [arXiv:2303.17295].

[42] R. Aaij et al. (LHCb), Phys. Rev. D 102, 112003 (2020), [arXiv:2009.00026].

[43] R. Aaij et al. (LHCb), Phys. Rev. Lett. 125, 242001 (2020), [arXiv:2009.00025].

[44] M. Karliner and J. L. Rosner, Phys. Rev. D 102, 9, 094016 (2020), [arXiv:2008.05993].

[45] X.-G. He, W. Wang and R. Zhu, Eur. Phys. J. C 80, 11, 1026 (2020), [arXiv:2008.07145].

[46] Z.-G. Wang, Int. J. Mod. Phys. A 35, 30, 2050187 (2020), [arXiv:2008.07833].

[47] R. Molina and E. Oset, Phys. Lett. B 811, 135870 (2020), [arXiv:2008.11171].

[48] Y. Huang et al., Eur. Phys. J. C 80, 10, 973 (2020), [arXiv:2008.07959].

[49] M.-Z. Liu, J.-J. Xie and L.-S. Geng, Phys. Rev. D 102, 9, 091502 (2020), [arXiv:2008.07389].

[50] B. Wang and S.-L. Zhu, Eur. Phys. J. C 82, 5, 419 (2022), [arXiv:2107.09275].

[51] Z.-L. Yue, C.-J. Xiao and D.-Y. Chen, Phys. Rev. D 107, 3, 034018 (2023), [arXiv:2212.03018].

[52] T. J. Burns and E. S. Swanson, Phys. Lett. B 813, 136057 (2021), [arXiv:2008.12838].

[53] X.-H. Liu et al., Eur. Phys. J. C 80, 12, 1178 (2020), [arXiv:2008.07190].

[54] T. J. Burns and E. S. Swanson, Phys. Rev. D 103, 1, 014004 (2021), [arXiv:2009.05352].

[55] R. Aaij et al. (LHCb), Phys. Rev. D 108, 1, 012017 (2023), [arXiv:2212.02717].

[56] R. Molina and E. Oset, Phys. Rev. D 107, 5, 056015 (2023), [arXiv:2211.01302].

[57] H.-W. Ke et al., Phys. Rev. D 106, 11, 114032 (2022), [arXiv:2210.06215].

[58] S. K. Choi et al. (Belle), Phys. Rev. Lett. 91, 262001 (2003), [hep-ex/0309032].

[59] B. Aubert et al. (BaBar), Phys. Rev. D71, 071103 (2005), [hep-ex/0406022].

[60] R. Aaij et al. (LHCb), Phys. Rev. Lett. 115, 072001 (2015), [arXiv:1507.03414].

[61] R. Aaij et al. (LHCb), Phys. Rev. Lett. 122, 22, 222001 (2019), [arXiv:1904.03947].

[62] R. Aaij et al. (LHCb), Nature Phys. 18, 7, 751 (2022), [arXiv:2109.01038].

[63] R. Aaij et al. (LHCb), Nature Commun. 13, 1, 3351 (2022), [arXiv:2109.01056].

[64] R. Aaij et al. (LHCb), Phys. Rev. Lett. 110, 222001 (2013), [arXiv:1302.6269].

[65] R. Aaij et al. (LHCb), Phys. Rev. D92, 1, 011102 (2015), [arXiv:1504.06339].

[66] T. Barnes and S. Godfrey, Phys. Rev. D69, 054008 (2004), [hep-ph/0311162].

[67] K. Abe et al. (Belle), Phys. Rev. Lett. 98, 082001 (2007), [hep-ex/0507019].

[68] S. Uehara et al. (Belle), Phys. Rev. Lett. 96, 082003 (2006), [hep-ex/0512035].

[69] R. Aaij et al. (LHCb), Phys. Rev. D 102, 9, 092005 (2020), [arXiv:2005.13419].

[70] R. Aaij et al. (LHCb), JHEP 08, 123 (2020), [arXiv:2005.13422].

[71] C. Hanhart et al., Phys. Rev. D 76, 034007 (2007), [arXiv:0704.0605].

[72] G. Barucca et al. (PANDA), Eur. Phys. J. A 55, 3, 42 (2019), [arXiv:1812.05132].

[73] G. Barucca et al. (PANDA), Eur. Phys. J. A 57, 6, 184 (2021), [arXiv:2101.11877].

[74] F.-K. Guo, Phys. Rev. Lett. 122, 20, 202002 (2019), [arXiv:1902.11221].

[75] E. Braaten, L.-P. He and K. Ingles, Phys. Rev. D 100, 3, 031501 (2019), [arXiv:1904.12915].

[76] S. Sakai, H.-J. Jing and F.-K. Guo, Phys. Rev. D 102, 11, 114041 (2020), [arXiv:2008.10829].

[77] N. A. Tornqvist, Phys. Lett. B590, 209 (2004), [hep-ph/0402237].

[78] E. S. Swanson, Phys. Lett. B588, 189 (2004), [hep-ph/0311229].

[79] D. Gamermann and E. Oset, Phys. Rev. D80, 014003 (2009), [arXiv:0905.0402].

[80] Y.-b. Dong et al., Phys. Rev. D 77, 094013 (2008), [arXiv:0802.3610].

[81] Y. Dong et al., Phys. Rev. D 79, 094013 (2009), [arXiv:0903.5416].

[82] F.-K. Guo et al., Phys. Lett. B 742, 394 (2015), [arXiv:1410.6712].

[83] B. Aubert et al. (BaBar), Phys. Rev. D71, 031501 (2005), [hep-ex/0412051].

[84] L. Maiani, A. D. Polosa and V. Riquer, Phys. Lett. B778, 247 (2018), [arXiv:1712.05296].

[85] L. Maiani, A. D. Polosa and V. Riquer, Phys. Rev. D 102, 3, 034017 (2020), [arXiv:2005.08764].

[86] C. Bignamini et al., Phys. Rev. Lett. 103, 162001 (2009), [arXiv:0906.0882].

[87] A. Esposito et al., Eur. Phys. J. C 81, 669 (2021), [arXiv:2006.15044].

[88] P. Artoisenet and E. Braaten, Phys. Rev. D81, 114018 (2010), [arXiv:0911.2016].

[89] E. Braaten et al., Phys. Rev. D 103, 7, L071901 (2021), [arXiv:2012.13499].

[90] F.-K. Guo et al., JHEP 05, 138 (2014), [arXiv:1403.4032].

[91] M. Albaladejo et al., Chin. Phys. C41, 12, 121001 (2017), [arXiv:1709.09101].

[92] A. Esposito et al., Phys. Rev. D 105, 3, L031503 (2022), [arXiv:2108.11413].

[93] S. Weinberg, Phys. Rev. 137, B672 (1965).

[94] V. Baru et al., Phys. Rev. D 84, 074029 (2011), [arXiv:1108.5644].

[95] M. Jansen, H. W. Hammer and Y. Jia, Phys. Rev. D 89, 1, 014033 (2014), [arXiv:1310.6937].

[96] E. Braaten, Phys. Rev. D 91, 11, 114007 (2015), [arXiv:1503.04791].

[97] A. Esposito et al., Phys. Lett. B 847, 138285 (2023), [arXiv:2307.11400].

[98] V. Baru et al., Phys. Lett. B 833, 137290 (2022), [arXiv:2110.07484].

[99] B. Aubert et al. (BaBar), Phys. Rev. Lett. 95, 142001 (2005), [hep-ex/0506081].

[100] M. Ablikim et al. (BESIII), Phys. Rev. Lett. 118, 9, 092001 (2017), [arXiv:1611.01317].

[101] M. Cleven et al., Phys. Rev. D90, 7, 074039 (2014), [arXiv:1310.2190].

[102] C. Hanhart and E. Klempt, Int. J. Mod. Phys. A 35, 05, 2050019 (2020), [arXiv:1906.11971].

[103] T. Barnes, S. Godfrey and E. S. Swanson, Phys. Rev. D 72, 054026 (2005), [hep-ph/0505002].
[104] L.-C. Gui et al., Phys. Rev. D **98**, 1, 016010 (2018), [arXiv:1801.08791].

[105] F. E. Close and P. R. Page, Phys. Lett. **B628**, 215 (2005), [hep-ph/0507199].

[106] M. Berwein et al., Phys. Rev. **D92**, 11, 114019 (2015), [arXiv:1510.04299].

[107] E. Kou and O. Pene, Phys. Lett. **B631**, 164 (2005), [hep-ph/0507119].

[108] Yu. S. Kalashnikova and A. V. Nefediev, Phys. Rev. **D77**, 054025 (2008), [arXiv:0801.2036].

[109] M. Ablikim et al. (BESIII), Phys. Rev. Lett. **111**, 24, 242001 (2013), [arXiv:1309.1896].

[110] M. B. Voloshin, Prog. Part. Nucl. Phys. **61**, 455 (2008), [arXiv:0711.4556].

[111] X. Li and M. B. Voloshin, Mod. Phys. Lett. **A29**, 12, 1450060 (2014), [arXiv:1309.1681].

[112] E. Braaten, C. Langmack and D. H. Smith, Phys. Rev. D **90**, 1, 014044 (2014), [arXiv:1402.0438].

[113] N. Brambilla et al., Phys. Rev. D **97**, 1, 016016 (2018), [arXiv:1707.09647].

[114] N. Brambilla et al., Phys. Rev. D **101**, 5, 054040 (2020), [arXiv:1908.11699].

[115] N. Brambilla et al., Phys. Rev. D **107**, 5, 054034 (2023), [arXiv:2212.09187].

[116] Q. Wang, C. Hanhart and Q. Zhao, Phys. Rev. Lett. **111**, 13, 132003 (2013), [arXiv:1303.6355].

[117] Y. Dong et al., Phys. Rev. D **89**, 3, 034018 (2014), [arXiv:1310.4373].

[118] F.-K. Guo et al., Phys. Lett. **B725**, 127 (2013), [arXiv:1306.3096].

[119] M. Ablikim et al. (BESIII), Phys. Rev. Lett. **112**, 9, 092001 (2014), [arXiv:1310.4101].

[120] Y. Dong et al., Phys. Rev. D **90**, 7, 074032 (2014), [arXiv:1404.6161].

[121] Q. Wang et al., Phys. Rev. **D89**, 3, 034001 (2014), [arXiv:1309.4303].

[122] V. Baru et al., Phys. Rev. **D91**, 3, 034002 (2015), [arXiv:1501.02924].

[123] T. Ji et al., Phys. Rev. Lett. **129**, 10, 102002 (2022), [arXiv:2205.10994].

[124] A. Ali et al., Eur. Phys. J. **C78**, 1, 29 (2018), [arXiv:1708.04650].

[125] L. Maiani et al., Phys. Rev. **D89**, 114010 (2014), [arXiv:1405.1551].

[126] K. Chilikin et al. (Belle), Phys. Rev. **D88**, 7, 074026 (2013), [arXiv:1306.4894].

[127] R. Aaij et al. (LHCb), Phys. Rev. Lett. **112**, 22, 222002 (2014), [arXiv:1404.1903].

[128] L. Maiani, A. D. Polosa and V. Riquer, Symmetry **13**, 5, 751 (2021), [arXiv:2103.14356].

[129] Y.-H. Chen et al., Phys. Rev. **D99**, 7, 074016 (2019), [arXiv:1902.10957].

[130] G. Cotugno et al., Phys. Rev. Lett. **104**, 132005 (2010), [arXiv:0911.2178].

[131] F.-K. Guo et al., Phys. Rev. **D82**, 094008 (2010), [arXiv:1005.2055].

[132] L.-Y. Dai, J. Haidenbauer and U. G. Meissner, Phys. Rev. **D96**, 11, 116001 (2017), [arXiv:1710.03142].

[133] J. He et al., Eur. Phys. J. C **80**, 3, 246 (2020), [arXiv:1912.08420].

[134] F.-L. Wang and X. Liu, Phys. Rev. D **102**, 9, 094006 (2020), [arXiv:2008.13484].

[135] H.-W. Ke, X.-H. Liu and X.-Q. Li, Chin. Phys. C **44**, 9, 093104 (2020), [arXiv:2004.03167].

[136] J.-R. Zhang, Phys. Rev. D **102**, 5, 054006 (2020), [arXiv:2004.10985].

[137] F.-K. Guo, C. Hanhart and U.-G. Meissner, Phys. Lett. B **665**, 26 (2008), [arXiv:0803.1392].

[138] F.-K. Guo, C. Hanhart and U.-G. Meissner, Phys. Rev. Lett. **102**, 242004 (2009), [arXiv:0904.3338].

[139] M. Ablikim et al. (BESIII), Phys. Rev. Lett. **110**, 252001 (2013), [arXiv:1303.5949].

[140] M. Ablikim et al. (BESIII), Phys. Rev. Lett. **112**, 2, 022001 (2014), [arXiv:1310.1163].

[141] P. Krokovny et al. (Belle), Phys. Rev. **D88**, 5, 052016 (2013), [arXiv:1308.2646].

[142] A. Bondar et al. (Belle), Phys. Rev. Lett. **108**, 122001 (2012), [arXiv:1110.2251].

[143] A. Garmash et al. (Belle), Phys. Rev. Lett. **116**, 21, 212001 (2016), [arXiv:1512.07419].

[144] A. E. Bondar et al., Phys. Rev. **D84**, 054010 (2011), [arXiv:1105.4473].

[145] Z.-M. Ding, H.-Y. Jiang and J. He, Eur. Phys. J. C **80**, 12, 1179 (2020), [arXiv:2011.04980].

[146] Y. Dong et al., Phys. Rev. D **88**, 1, 014030 (2013), [arXiv:1306.0824].

[147] F. Goerke et al., Phys. Rev. D **94**, 9, 094017 (2016), [arXiv:1608.04656].

[148] F. Goerke et al., Phys. Rev. D **96**, 5, 054028 (2017), [arXiv:1707.00539].

[149] A. Ali et al., Phys. Rev. **D91**, 1, 017502 (2015), [arXiv:1412.2049].

[150] M. B. Voloshin, Phys. Rev. **D84**, 031502 (2011), [arXiv:1105.5829].

[151] T. Mehen and J. W. Powell, Phys. Rev. **D84**, 114013 (2011), [arXiv:1109.3479].

[152] V. Baru et al., JHEP **06**, 158 (2017), [arXiv:1704.07332].

[153] V. Baru et al., Phys. Rev. **D99**, 9, 094013 (2019), [arXiv:1901.10319].

[154] T. Branz, T. Gutsche and V. E. Lyubovitskij, Phys. Rev. **D82**, 054025 (2010), [arXiv:1005.3168].

[155] P. Pakhlov, Phys. Lett. **B702**, 139 (2011), [arXiv:1105.2945].

[156] P. Pakhlov and T. Uglov, Phys. Lett. **B748**, 183 (2015), [arXiv:1408.5295].

[157] S. X. Nakamura and K. Tsushima, Phys. Rev. **D100**, 5, 051502 (2019), [arXiv:1901.07385].

[158] J.-Z. Wang et al., Eur. Phys. J. C **80**, 11, 1040 (2020), [arXiv:2007.02263].

[159] M. Ablikim et al. (BESIII), Phys. Rev. Lett. **126**, 10, 102001 (2021), [arXiv:2011.07855].

[160] R. Aaij et al. (LHCb), Phys. Rev. Lett. **127**, 8, 082001 (2021), [arXiv:2103.01803].

[161] S. H. Lee, M. Nielsen and U. Wiedner, J. Korean Phys. Soc. **55**, 424 (2009), [arXiv:0803.1168].

[162] J. M. Dias, X. Liu and M. Nielsen, Phys. Rev. D **88**, 9, 096014 (2013), [arXiv:1307.7100].

[163] D. Ebert, R. N. Faustov and V. O. Galkin, Eur. Phys. J. C **58**, 399 (2008), [arXiv:0808.3912].

[164] J. Ferretti and E. Santopinto, JHEP **04**, 119 (2020), [arXiv:2001.01067].

[165] M. B. Voloshin, Phys. Lett. B **798**, 135022 (2019), [arXiv:1901.01936].

[166] D.-Y. Chen, X. Liu and T. Matsuki, Phys. Rev. Lett. **110**, 23, 232001 (2013), [arXiv:1303.6842].

[167] L. Meng, B. Wang and S.-L. Zhu, Phys. Rev. D **102**, 11, 111502 (2020), [arXiv:2011.08656].
[168] P. G. Ortega, D. R. Entem and F. Fernandez, Phys. Lett. B 818, 136382 (2021), [arXiv:2103.07871].

[169] Z. Yang et al., Phys. Rev. D 103, 7, 074029 (2021), [arXiv:2011.08725].

[170] B. Wang, L. Meng and S.-L. Zhu, Phys. Rev. D 103, 2, L021501 (2021), [arXiv:2011.10922].

[171] M.-L. Du et al., Phys. Rev. D 105, 7, 074018 (2022), [arXiv:2201.08253].

[172] R. M. Albuquerque, S. Narison and D. Rabetiarivony, Phys. Rev. D 103, 7, 074015 (2021), [arXiv:2101.07281].

[173] Z.-H. Guo and J. A. Oller, Phys. Rev. D 103, 5, 054021 (2021), [arXiv:2012.11904].

[174] N. Ikeno, R. Molina and E. Oset, Phys. Lett. B 814, 136120 (2021), [arXiv:2011.13425].

[175] V. Baru et al., Phys. Rev. D 105, 3, 034014 (2022), [arXiv:2110.00398].

[176] M. Cleven et al., Phys. Rev. D92, 1, 014005 (2015), [arXiv:1505.01771].

[177] E. Braaten et al., Phys. Rev. D 106, 3, 034033 (2022), [arXiv:2202.03900].

[178] S. S. Agaev, K. Azizi and H. Sundu, Nucl. Phys. B 975, 115650 (2022), [arXiv:2108.00188].

[179] L.-Y. Dai et al., Phys. Rev. D 105, 5, L051507 (2022), [arXiv:2108.06002].

[180] C. Deng and S.-L. Zhu, Phys. Rev. D 105, 5, 054015 (2022), [arXiv:2112.12472].

[181] P. G. Ortega et al., Phys. Lett. B 841, 137918 (2023), [arXiv:2211.06118].

[182] M. Albaladejo, Phys. Lett. B 829, 137052 (2022), [arXiv:2110.02944].

[183] M.-L. Du et al., Phys. Rev. D 105, 1, 014024 (2022), [arXiv:2110.13765].

[184] H.-W. Ke, X.-H. Liu and X.-Q. Li, Eur. Phys. J. C 82, 2, 144 (2022), [arXiv:2112.14142].

[185] Z.-S. Jia et al., Phys. Rev. D 107, 7, 074029 (2023), [arXiv:2211.02479].

[186] M. Mikhasenko (2022), [arXiv:2203.04622].

[187] M. Padmanath and S. Prelovsek, Phys. Rev. Lett. 129, 3, 032002 (2022), [arXiv:2202.10110].

[188] S. Chen et al., Phys. Lett. B 833, 137391 (2022), [arXiv:2206.06185].

[189] Y. Lyu et al., Phys. Rev. Lett. 131, 16, 161901 (2023), [arXiv:2302.04505].

[190] M.-L. Du et al., Phys. Rev. Lett. 131, 13, 131903 (2023), [arXiv:2303.09441].

[191] R. Aaij et al. (LHCb), Sci. Bull. 65, 23, 1983 (2020), [arXiv:2006.16957].

[192] A. Hayrapetyan et al. (CMS) (2023), [arXiv:2306.07164].

[193] Y. Xu (ATLAS), Acta Phys. Polon. Supp. 16, 3, 21 (2023), [arXiv:2209.12173].

[194] J. H. Yin et al. (Belle) (2023), [arXiv:2305.17947].

[195] G. D. Alexeev et al. (COMPASS), Phys. Lett. B 838, 137702 (2023), [arXiv:2204.01817].

[196] J. P. Ader, J. M. Richard and P. Taxil, Phys. Rev. D 25, 2370 (1982).

[197] Y. Iwasaki, Prog. Theor. Phys. 54, 492 (1975).

[198] L. Maiani, Sci. Bull. 65, 1949 (2020), [arXiv:2008.01637].

[199] K.-T. Chao and S.-L. Zhu, Sci. Bull. 65, 23, 1952 (2020), [arXiv:2008.07670].

[200] J.-M. Richard, Sci. Bull. 65, 1954 (2020), [arXiv:2008.01962].

[201] R. N. Faustov, V. O. Galkin and E. M. Savchenko, Phys. Rev. D 102, 11, 114030 (2020), [arXiv:2009.13237].

[202] J. F. Giron and R. F. Lebed, Phys. Rev. D 102, 7, 074003 (2020), [arXiv:2008.01631].

[203] X. Jin et al., Eur. Phys. J. C 80, 11, 1083 (2020), [arXiv:2006.13745].

[204] Q.-F. Lü, D.-Y. Chen and Y.-B. Dong, Eur. Phys. J. C 80, 9, 871 (2020), [arXiv:2006.14445].

[205] Z.-H. Guo and J. A. Oller, Phys. Rev. D 103, 3, 034024 (2021), [arXiv:2011.00978].

[206] P. G. Ortega, D. R. Entem and F. Fernandez, Phys. Rev. D 108, 9, 094023 (2023), [arXiv:2307.00532].

[207] M. Karliner and J. L. Rosner, Phys. Rev. D 102, 11, 114039 (2020), [arXiv:2009.04429].

[208] B.-D. Wan and C.-F. Qiao, Phys. Lett. B 817, 136339 (2021), [arXiv:2012.00454].

[209] B.-C. Yang, L. Tang and C.-F. Qiao, Eur. Phys. J. C 81, 4, 324 (2021), [arXiv:2012.04463].

[210] R. M. Albuquerque et al., Phys. Rev. D 102, 9, 094001 (2020), [arXiv:2008.01569].

[211] X.-K. Dong et al., Phys. Rev. Lett. 126, 13, 132001 (2021), [arXiv:2009.07795].

[212] C. Gong et al., Phys. Lett. B 824, 136794 (2022), [arXiv:2011.11374].

[213] Z.-R. Liang, X.-Y. Wu and D.-L. Yao, Phys. Rev. D 104, 3, 034034 (2021), [arXiv:2104.08589].

[214] X.-K. Dong et al., Sci. Bull. 66, 24, 2462 (2021), [arXiv:2107.03946].

[215] J.-Z. Wang et al., Phys. Rev. D 103, 7, 071503 (2021), [arXiv:2008.07430].

[216] J.-Z. Wang, X. Liu and T. Matsuki, Phys. Lett. B 816, 136209 (2021), [arXiv:2012.03281].

[217] F.-K. Guo et al., Phys. Rev. D 91, 5, 051504 (2015), [arXiv:1411.5584].
# 80. Baryon Decay Parameters

Revised August 2019 by E.D. Commins (UC Berkeley).

## 80.1 Baryon semileptonic decays

The typical spin-1/2 baryon semileptonic decay is described by a matrix element, the hadronic part of which may be written as:

$$ \bar{B}_f [ f_1(q^2)\gamma_\lambda + i f_2(q^2)\sigma_{\lambda\mu}q^\mu + g_1(q^2)\gamma_\lambda\gamma_5 + g_3(q^2)\gamma_5q_\lambda ] B_i . \quad (80.1) $$

Here $B_i$ and $\bar{B}_f$ are spinors describing the initial and final baryons, and $q = p_i - p_f$, while the terms in $f_1, f_2, g_1$, and $g_3$ account for vector, induced tensor (“weak magnetism”), axial vector, and induced pseudoscalar contributions [1]. Second-class current contributions are ignored here. In the limit of zero momentum transfer, $f_1$ reduces to the vector coupling constant $g_V$, and $g_1$ reduces to the axial-vector coupling constant $g_A$. The latter coefficients are related by Cabibbo’s theory [2], generalized to six quarks (and three mixing angles) by Kobayashi and Maskawa [3]. The $g_3$ term is negligible for transitions in which an $e^\pm$ is emitted, and gives a very small correction, which can be estimated by PCAC [4], for $\mu^\pm$ modes. Recoil effects include weak magnetism, and are taken into account adequately by considering terms of first order in

$$ \delta = \frac{m_i - m_f}{m_i + m_f}, \quad (80.2) $$

where $m_i$ and $m_f$ are the masses of the initial and final baryons.

The experimental quantities of interest are the total decay rate, the lepton-neutrino angular correlation, the asymmetry coefficients in the decay of a polarized initial baryon, and the polarization of the decay baryon in its own rest frame for an unpolarized initial baryon. Formulae for these quantities are derived by standard means [5] and are analogous to formulae for nuclear beta decay [6]. We use the notation of Ref. [6] in the Listings for neutron beta decay. For comparison with experiments at higher $q^2$, it is necessary to modify the form factors at $q^2 = 0$ by a “dipole” $q^2$ dependence, and for high-precision comparisons to apply appropriate radiative corrections [7].

The ratio $g_A/g_V$ may be written as

$$ g_A/g_V = |g_A/g_V| e^{i\phi_{AV}}. \quad (80.3) $$

The presence of a “triple correlation” term in the transition probability, proportional to $\text{Im}(g_A/g_V)$ and of the form

$$ \sigma_i \cdot (p_\ell \times p_\nu) \quad (80.4) $$

for initial baryon polarization or

$$ \sigma_f \cdot (p_\ell \times p_\nu) \quad (80.5) $$

for final baryon polarization, would indicate failure of time-reversal invariance. The phase angle $\phi$ has been measured precisely only in neutron decay (and in $^{19}$Ne nuclear beta decay), and the results are consistent with T invariance.

## 80.2 Hyperon nonleptonic decays

The amplitude for a spin-1/2 hyperon decaying into a spin-1/2 baryon and a spin-0 meson may be written in the form

$$ M = G_F m_\pi^2 \cdot \bar{B}_f (A - B\gamma_5) B_i, \quad (80.6) $$

where A and B are constants [1]. The transition rate is proportional to

$$ R = 1 + \gamma \hat{\omega}_f \cdot \hat{\omega}_i + (1 - \gamma)(\hat{\omega}_f \cdot \hat{n})(\hat{\omega}_i \cdot \hat{n}) \\ + \alpha(\hat{\omega}_f \cdot \hat{n} + \hat{\omega}_i \cdot \hat{n}) + \beta \hat{n} \cdot (\hat{\omega}_f \times \hat{\omega}_i), \quad (80.7) $$

where $\hat{n}$ is a unit vector in the direction of the final baryon momentum, and $\hat{\omega}_i$ and $\hat{\omega}_f$ are unit vectors in the directions of the

initial and final baryon spins. (The sign of the last term in the above equation was incorrect in our 1988 and 1990 editions.) The parameters $\alpha, \beta$, and $\gamma$ are defined as

$$ \begin{aligned} \alpha &= 2 \text{Re}(s^*p)/(|s|^2 + |p|^2), \\ \beta &= 2 \text{Im}(s^*p)/(|s|^2 + |p|^2), \\ \gamma &= (|s|^2 - |p|^2)/(|s|^2 + |p|^2), \end{aligned} \quad (80.8) $$

where $s = A$ and $p = |\mathbf{p}_f| B/(E_f + m_f)$; here $E_f$ and $\mathbf{p}_f$ are the energy and momentum of the final baryon. The parameters $\alpha, \beta$, and $\gamma$ satisfy

$$ \alpha^2 + \beta^2 + \gamma^2 = 1 \quad (80.9) $$

If the hyperon polarization is $\mathbf{P}_Y$, the polarization $\mathbf{P}_B$ of the decay baryons is

$$ \mathbf{P}_B = \frac{(\alpha + \mathbf{P}_Y \cdot \hat{\mathbf{n}})\hat{\mathbf{n}} + \beta(\mathbf{P}_Y \times \hat{\mathbf{n}}) + \gamma\hat{\mathbf{n}} \times (\mathbf{P}_Y \times \hat{\mathbf{n}})}{1 + \alpha\mathbf{P}_Y \cdot \hat{\mathbf{n}}}. \quad (80.10) $$

Here $\mathbf{P}_B$ is defined in the rest system of the baryon, obtained by a Lorentz transformation along $\hat{\mathbf{n}}$ from the hyperon rest frame, in which $\hat{\mathbf{n}}$ and $\mathbf{P}_Y$ are defined.

An additional useful parameter $\phi$ is defined by

$$ \beta = (1 - \alpha^2)^{1/2} \sin\phi. \quad (80.11) $$

In the Listings, we compile $\alpha$ and $\phi$ for each decay, since these quantities are most closely related to experiment and are essentially uncorrelated. When necessary, we have changed the signs of reported values to agree with our sign conventions. In the Baryon Summary Table, we give $\alpha, \phi$, and $\Delta$ (defined below) with errors, and also give the value of $\gamma$ without error.

Time-reversal invariance requires, in the absence of final-state interactions, that $s$ and $p$ be relatively real, and therefore that $\beta = 0$. However, for the decays discussed here, the final-state interaction is strong. Thus

$$ s = |s| e^{i\delta_s} \quad \text{and} \quad p = |p| e^{i\delta_p}, \quad (80.12) $$

where $\delta_s$ and $\delta_p$ are the pion-baryon s- and p-wave strong interaction phase shifts. We then have

$$ \beta = \frac{-2|s| |p|}{|s|^2 + |p|^2} \sin(\delta_s - \delta_p). \quad (80.13) $$

One also defines $\Delta = -\tan^{-1}(\beta/\alpha)$. If $T$ invariance holds, $\Delta = \delta_s - \delta_p$. For $\Lambda \to p\pi^-$ decay, the value of $\Delta$ may be compared with the s- and p-wave phase shifts in low-energy $\pi^-p$ scattering, and the results are consistent with $T$ invariance.

See also the note on “Radiative Hyperon Decays” in this Review.

## References

[1] E. Commins and P. Bucksbaum, *Weak Interactions of Leptons and Quarks* (1983), ISBN 978-0-521-27370-1, (Cambridge, USA: University Press, 473p.).

[2] N. Cabibbo, Phys. Rev. Lett. **10**, 531 (1963).

[3] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. **49**, 652 (1973).

[4] M. Goldberger and S. Treiman, Phys. Rev. **111**, 354 (1958).

[5] P. Frampton and W.-K. Tung, Phys. Rev. D **3**, 1114 (1971).

[6] J. Jackson, S. Treiman and H. Wyld, Nucl. Phys. **4**, 206 (1957).

[7] Y. Yokoo, S. Suzuki and M. Morita, Prog. Theor. Phys. **50**, 1894 (1973).
# 81. N and Δ Resonances

Revised August 2019 by V.D. Burkert (Jefferson Lab), V. Crede (Florida State U.), U. Thoma (Bonn U.), L. Tiator (KPH, JGU Mainz) and R.L. Workman (George Washington U.).

## 81.1 Introduction

The excited states of the nucleon have been studied in a large number of formation and production experiments. Until recently, the Breit-Wigner masses and widths, the pole positions, and the elasticities of the N and Δ resonances in the Baryon Summary Table came largely from partial-wave analyses of $\pi N$ total, elastic, and charge-exchange scattering data. The most comprehensive analyses were carried out by the Karlsruhe-Helsinki (KH80) [1], Carnegie Mellon-Berkeley (CMB80) [2], and George Washington U (GWU) [3] groups. Partial-wave analyses have also been performed on much smaller $\pi N$ reaction data sets to get $\eta N$, $K\Lambda$, and $K\Sigma$ branching fractions (see the Listings for references). Other branching fractions come from analyses of $\pi N \to \pi\pi N$ data.

In recent years, a large amount of data on photoproduction of many final states has been accumulated, and these data are beginning to tell us much about the properties of baryon resonances. A survey of data on photoproduction can be found in the proceedings of recent conferences [4] and workshops [5], and in recent reviews [6, 7].

## 81.2 Naming scheme for baryon resonances

In the past, when nearly all resonance information came from elastic $\pi N$ scattering, it was common to label resonances with the incoming partial wave $L_{21,2J}$, as in $\Delta(1232)P_{33}$ and $N(1680)F_{15}$. However, most recent information has come from $\gamma N$ experiments. Therefore, we have replaced $L_{21,2J}$ with the spin-parity $J^P$ of the state, as in $\Delta(1232)3/2^+$ and $N(1680)5/2^+$; this name gives intrinsic properties of the resonance that are independent of the specific particles and reactions used to study them. This applies equally to all baryons, including $\Xi$ resonances and charm baryons that are not produced in formation experiments. We do not, however, attach the mass or spin-parity to the names of the ground-state ("stable") baryons $N, \Lambda, \Sigma, \Xi, \Omega, \Lambda_c, \dots$.

## 81.3 Using the N and Δ listings

Tables 81.1 and 81.2 list all the N and Δ entries in the Baryon Listings and give our evaluation of the overall status and the status channel by channel. Only the established resonances (overall status 3 or 4 stars) are promoted to the Baryon Summary Table. We long ago omitted from the Listings information from old analyses, prior to KH80 and CMB80, which can be found in earlier editions. A rather complete survey of older results was given in our 1982 edition [8].

As a rule, we award an overall status **** or *** only to those resonances which are derived from analyses of data sets that include precision differential cross sections and polarization observables, and are confirmed by independent analyses. All other signals are given ** or * status. New results that are not accompanied by proper error evaluation are less valuable for evaluating star ratings. The following criteria are guidelines for future error analysis.

1. Uncertainties in resonance parameters: The publication should have a detailed discussion on how the uncertainties of parameters were estimated. This requires that the error estimates go beyond the simple fit error as e.g. given by MINUIT, and the robustness of the results should be demonstrated.

2. Fit quality: Concrete measures for the fit quality should be provided. The reduced global $\chi^2$ value of the fit, while useful, is insufficient. Other possibilities include quoting variations of local $\chi^2$ values in kinematic regions where evidence for new resonances, or significantly improved information on resonance parameters, is claimed.

3. Weight factors in observables: Analyses sometimes use weight factors for certain data sets to either increase or reduce their impact on the results. This has been particularly important when polarization observables are involved, which often are sensitive to resonance amplitudes through interferences, but usually have much poorer statistics than differential cross section data. To evaluate sensitivities, the resulting resonance parameters should

be checked against variations of the specific weight factors.

<table><caption>**Table 81.1:** The status of the N resonances and their decays. Sub-threshold decay modes are omitted. Only resonances with an overall status of *** or **** are included in the main Baryon Summary Table.</caption><thead><tr><th rowspan="2">Particle<br>N</th><th rowspan="2">J<sup>P</sup><br>1/2<sup>+</sup></th><th colspan="10">Status as seen in</th></tr><tr><th>overall</th><th>N<sub>γ</sub></th><th>N<sub>π</sub></th><th>Δ<sub>π</sub></th><th>N<sub>σ</sub></th><th>N<sub>η</sub></th><th>Λ<sub>K</sub></th><th>Σ<sub>K</sub></th><th>N<sub>ρ</sub></th><th>N<sub>ω</sub></th><th>N<sub>η</sub><sup>†</sup></th></tr></thead><tbody><tr><td>N(1440)</td><td>1/2<sup>+</sup></td><td>****</td><td>****</td><td>****</td><td>****</td><td>***</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>N(1520)</td><td>3/2<sup>-</sup></td><td>****</td><td>****</td><td>****</td><td>***</td><td>**</td><td>****</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>N(1535)</td><td>1/2<sup>-</sup></td><td>****</td><td>****</td><td>****</td><td>***</td><td>*</td><td>****</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>N(1650)</td><td>1/2<sup>-</sup></td><td>****</td><td>****</td><td>****</td><td>***</td><td>*</td><td>****</td><td>*</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>N(1675)</td><td>5/2<sup>-</sup></td><td>****</td><td>****</td><td>****</td><td>****</td><td>***</td><td>*</td><td>*</td><td>*</td><td></td><td></td><td></td><td></td></tr><tr><td>N(1680)</td><td>5/2<sup>+</sup></td><td>****</td><td>****</td><td>****</td><td>****</td><td>***</td><td>*</td><td>*</td><td>*</td><td></td><td></td><td></td><td></td></tr><tr><td>N(1700)</td><td>3/2<sup>-</sup></td><td>***</td><td>**</td><td>***</td><td>***</td><td>*</td><td>*</td><td>*</td><td>*</td><td></td><td></td><td></td><td></td></tr><tr><td>N(1710)</td><td>1/2<sup>+</sup></td><td>****</td><td>****</td><td>****</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>N(1720)</td><td>3/2<sup>+</sup></td><td>****</td><td>****</td><td>****</td><td>***</td><td>*</td><td>*</td><td>****</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>N(1860)</td><td>5/2<sup>+</sup></td><td>**</td><td>*</td><td>**</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>N(1875)</td><td>3/2<sup>-</sup></td><td>***</td><td>**</td><td>**</td><td>*</td><td>**</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>N(1880)</td><td>1/2<sup>+</sup></td><td>***</td><td>**</td><td>*</td><td>**</td><td>*</td><td>*</td><td>**</td><td>**</td><td>**</td><td>*</td><td>*</td><td>**</td></tr><tr><td>N(1895)</td><td>1/2<sup>-</sup></td><td>****</td><td>****</td><td>*</td><td>*</td><td>**</td><td>****</td><td>**</td><td>**</td><td>**</td><td>*</td><td>*</td><td>****</td></tr><tr><td>N(1900)</td><td>3/2<sup>+</sup></td><td>****</td><td>****</td><td>**</td><td>**</td><td>*</td><td>*</td><td>**</td><td>**</td><td>**</td><td>*</td><td>*</td><td>**</td></tr><tr><td>N(1990)</td><td>7/2<sup>+</sup></td><td>**</td><td>**</td><td>**</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>N(2000)</td><td>5/2<sup>+</sup></td><td>**</td><td>**</td><td>*</td><td>**</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>N(2040)</td><td>3/2<sup>-</sup></td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>N(2060)</td><td>5/2<sup>-</sup></td><td>***</td><td>***</td><td>**</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>N(2100)</td><td>1/2<sup>+</sup></td><td>***</td><td>**</td><td>***</td><td>**</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>N(2120)</td><td>3/2<sup>-</sup></td><td>***</td><td>***</td><td>**</td><td>**</td><td>**</td><td>**</td><td>**</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>N(2190)</td><td>7/2<sup>-</sup></td><td>****</td><td>****</td><td>****</td><td>****</td><td>**</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>N(2220)</td><td>9/2<sup>+</sup></td><td>****</td><td>**</td><td>****</td><td>***</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>N(2250)</td><td>9/2<sup>-</sup></td><td>****</td><td>**</td><td>****</td><td>***</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>N(2300)</td><td>1/2<sup>+</sup></td><td>**</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>N(2570)</td><td>5/2<sup>-</sup></td><td>**</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>N(2600)</td><td>11/2<sup>+</sup></td><td>***</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>N(2700)</td><td>13/2<sup>+</sup></td><td>**</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr></tbody></table>

<table><caption>**Table 81.2:** The status of the Δ resonances and their decays. Sub-threshold decay modes are omitted. Only resonances with an overall status of *** or **** are included in the main Baryon Summary Table.</caption><thead><tr><th rowspan="2">Particle</th><th rowspan="2">J<sup>P</sup></th><th colspan="10">Status as seen in</th></tr><tr><th>overall</th><th>N<sub>γ</sub></th><th>N<sub>π</sub></th><th>Δ<sub>π</sub></th><th>Σ<sub>K</sub></th><th>N<sub>ρ</sub></th><th>N<sub>ω</sub></th><th>N<sub>η</sub><sup>†</sup></th></tr></thead><tbody><tr><td>Δ(1232)</td><td>3/2<sup>+</sup></td><td>****</td><td>****</td><td>****</td><td>****</td><td>***</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Δ(1600)</td><td>3/2<sup>+</sup></td><td>****</td><td>****</td><td>****</td><td>****</td><td>***</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Δ(1620)</td><td>1/2<sup>-</sup></td><td>****</td><td>****</td><td>****</td><td>****</td><td>***</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Δ(1700)</td><td>3/2<sup>-</sup></td><td>****</td><td>****</td><td>****</td><td>****</td><td>***</td><td>*</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Δ(1750)</td><td>1/2<sup>+</sup></td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Δ(1900)</td><td>1/2<sup>-</sup></td><td>***</td><td>***</td><td>***</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>Δ(1905)</td><td>5/2<sup>+</sup></td><td>****</td><td>****</td><td>****</td><td>**</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>Δ(1910)</td><td>1/2<sup>+</sup></td><td>****</td><td>***</td><td>****</td><td>**</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>Δ(1920)</td><td>3/2<sup>+</sup></td><td>***</td><td>***</td><td>***</td><td>***</td><td>**</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>Δ(1930)</td><td>5/2<sup>-</sup></td><td>*</td><td>*</td><td>**</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>Δ(1940)</td><td>3/2<sup>-</sup></td><td>**</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>Δ(1950)</td><td>7/2<sup>+</sup></td><td>****</td><td>****</td><td>****</td><td>**</td><td>*</td><td>**</td><td>**</td><td>**</td><td>**</td><td>*</td><td>*</td><td>*</td></tr><tr><td>Δ(2000)</td><td>5/2<sup>+</sup></td><td>**</td><td>*</td><td>**</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>Δ(2150)</td><td>1/2<sup>-</sup></td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>Δ(2200)</td><td>7/2<sup>-</sup></td><td>***</td><td>**</td><td>**</td><td>**</td><td>***</td><td>**</td><td>**</td><td>**</td><td>**</td><td>*</td><td>*</td><td>*</td></tr><tr><td>Δ(2300)</td><td>9/2<sup>+</sup></td><td>**</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>Δ(2350)</td><td>5/2<sup>-</sup></td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>Δ(2390)</td><td>7/2<sup>+</sup></td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>Δ(2400)</td><td>9/2<sup>-</sup></td><td>**</td><td>**</td><td>**</td><td>**</td><td>**</td><td>**</td><td>**</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>Δ(2420)</td><td>11/2<sup>+</sup></td><td>****</td><td>*</td><td>****</td><td>***</td><td>*</td><td>**</td><td>**</td><td>**</td><td>**</td><td>*</td><td>*</td><td>*</td></tr><tr><td>Δ(2750)</td><td>13/2<sup>-</sup></td><td>**</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr><tr><td>Δ(2950)</td><td>15/2<sup>+</sup></td><td>**</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td><td>*</td></tr></tbody></table>

**** Existence is certain.

*** Existence is very likely.

** Evidence of existence is fair.

* Evidence of existence is poor.

Claims of evidence for new baryon states must be based on a sufficiently complete set of partial waves in the fit. The robustness of signals must be demonstrated, e.g. by examining the effect of higher partial waves in the fit.

## 81.4 Properties of resonances

Resonances are defined by poles of the S-matrix, whether in scattering, production or decay matrix elements. These are poles
in the complex plane in s, as discussed in the new review on Resonances. As is traditional, we quote here the pole positions in the complex energy $w = \sqrt{s}$ plane. Crucially, the position of the pole of the S-matrix is independent of the process, and the production and decay properties factorize. This is the rationale for listing the pole position first for each resonance. These key properties of the S-matrix pole are in contrast to other quantities related to resonance phenomena, such as Breit-Wigner parameters or any K-matrix pole. Breit-Wigner parameters depend on the formalism used, such as angular-momentum barrier factors, or cut-off parameters, and the assumed or modeled background. However, the accurate determination of pole parameters from the analysis of data on the real energy axis is not necessarily simple, or even straightforward. It requires the implementation of the correct analytic structure of the relevant (often coupled) channels.

In principle, there are two ways to extract pole parameters from experimental data: (i) analytic continuation of theoretical single- or multi-channel models into the complex energy plane or (ii) local expansions of the partial-wave T-matrix amplitudes in the complex energy plane in the vicinity of a pole.

At present, poles are usually extracted using the first method [9–14], but considerable effort has been put into the development of alternate approaches, such as the speed plot [15], time delay [16], N/D method [17], regularization procedure [18], or Padé approximation [19].

Methods of the second type are based on the idea to use first or higher-order derivatives in energy to reduce the importance of, or totally eliminate, the background contribution. One either has to model the background contribution and introduce model dependence, or one is faced with numerical derivatives of single-energy data. In both cases, one reaches almost unsurmountable difficulties.

An alternate way to extract pole parameters from partial waves has been proposed by introducing a Laurent+Pietarinen (L+P) expansion [20–22]

$$T(W) = \sum_{i=1}^{N} \frac{Res_i}{W - W_i} + \sum_{j=1}^{M} \sum_{n=0}^{n_{max}} c_n^j \left( \frac{\alpha_j - \sqrt{x_j - W}}{\alpha_j + \sqrt{x_j - W}} \right)^n, \quad (81.1)$$

where $T(W)$ is a given partial wave amplitude, $W_i$ and $Res_i$ are the N complex pole positions and residues. The background is parameterized with M Pietarinen functions, where $\alpha_j$ are positive range parameters and $x_j$ are real or complex branch points; $c_n^j$ are real expansion coefficients.

The main idea of this procedure is to find the simplest analytic function, with well-defined poles and cuts, regardless of whether they are generated by a theoretical model or some energy-independent procedure. Instead of searching for the function which reproduces the input amplitudes over the complete complex energy plane, on all Riemann sheets, a representation is searched only in a limited complex energy range, near the real axis, which is defined by the radius of convergence of the Laurent decomposition, and which contains the input amplitudes. All details are found in Ref. [21]. Applications of the method can be found in [20–27].

## 81.5 Photoproduction

A new approach to the nucleon excitation spectrum is provided by dedicated facilities at the Universities of Bonn, Grenoble, and Mainz, and at the national laboratories Jefferson Lab in the US and SPring-8 in Japan. High-precision cross sections and polarization observables for the photoproduction of pseudoscalar mesons provide a data set that is approaching a “complete experiment,” one that fully constrains the four complex amplitudes describing the spin-structure of the reaction [28]. A large number of photoproduction reactions has been studied.

In pseudoscalar meson photoproduction, the four independent helicity amplitudes can be expressed in terms of the four CGLN [29] amplitudes allowed by Lorentz and gauge invariance. These amplitudes can be expanded in a series of electric and magnetic multipoles. Except for $J = 1/2$, one electric and one magnetic multipole contributes to each $J^P$ combination.

For a given state, these two amplitudes determine the resonance photo-decay helicity amplitudes $A_{1/2}$ and $A_{3/2}$. As described below, this resonance extraction has been carried out either assuming a Breit-Wigner resonance or at the pole.

If a Breit-Wigner parametrization is used, the $N\gamma$ partial width, $\Gamma_\gamma$, is given in terms of the helicity amplitudes $A_{1/2}$ and $A_{3/2}$ by

$$\Gamma_{\gamma} = \frac{k_{\text{BW}}^2}{\pi} \frac{2m_N}{(2J+1)m_{\text{BW}}} \left( |A_{1/2}|^2 + |A_{3/2}|^2 \right). \quad (81.2)$$

Here $m_N$ and $m_{BW}$ are the nucleon and resonance masses, $J$ is the resonance spin, and $k_{BW}$ is the photon c.m. decay momentum. Most earlier analyses have provided these real quantities $A_{1/2}$ and $A_{3/2}$.

More recent studies have quoted related complex quantities, evaluated at the T-matrix pole. These complex helicity amplitudes, $\tilde{A}_{1/2}$ and $\tilde{A}_{3/2}$, can be cast onto the form

$$\tilde{A}_h = \sqrt{\frac{\pi(2J+1)w_{\text{pole}}}{m_N k_{\text{pole}}^2}} \frac{\text{Res}(T_h(\gamma N \rightarrow N b))}{\sqrt{\text{Res}(T(N b \rightarrow N b))}} \quad (81.3)$$

where the residues ($\text{Res}$) are evaluated at the pole position, $w_{\text{pole}}$, and $k_{\text{pole}}^2 = (w_{\text{pole}}^2 - m_N^2)^2 / 4w_{\text{pole}}^2$ [30]. For Breit-Wigner amplitudes, $w_{\text{pole}} = m_{BW}$ and $\tilde{A}_h = A_h$. Similar relations for the photo and electro couplings at the pole position can be found in [31, 32].

The determination of eight real numbers from four complex amplitudes (with one overall phase undetermined) requires at least seven independent measurements. At least one further measurement is required to resolve discrete ambiguities that result from the fact that data are proportional to squared amplitudes. Photon beams and nucleon targets can be polarized (with linear or circular polarization $P_\perp$, $P_\odot$ and $\vec{T}$, respectively); the recoil polarization of the outgoing baryon $\vec{R}$ can be measured. The experiments can be divided into three classes: (1) the beam and target are polarized (BT); (2) the beam is polarized and the recoil baryon polarization is measured (BR); (3) the target is polarized and the recoil polarization is measured (TR). Different sign conventions are used in the literature, as summarized in [33].

One of the best studied reactions is $\gamma p \rightarrow \Lambda K^+$. Published data include differential cross sections, the beam asymmetry $\Sigma$, the target asymmetry $T$, the recoil polarization $P$, and the BR double-polarization variables $C_{x'}$, $C_{z'}$, $O_{x'}$, and $O_{z'}$. For the photoproduction of pions and etas, off proton and neutron targets, differential cross sections, single- and double-polarization asymmetries have been measured, mainly for pions.

## 81.6 Electroproduction

Electroproduction of mesons provides information on the internal structure of resonances. The helicity amplitudes are functions of the (squared) momentum transfer $Q^2 = -(e^-e')^2$, where $e$ and $e'$ are the 4-momenta of the incident and scattered electron, and a third amplitude, $S_{1/2}$, measures the resonance response to the longitudinal component of the virtual photon. Most data stem from the reactions $e^-p \rightarrow e^- n\pi^+$ and $e^-p \rightarrow e^- p\pi^0$ but also the reactions $e^-p \rightarrow e^- p\eta$, $e^-p \rightarrow e^- p\pi^+\pi^-$, and $e^-p \rightarrow e^- \Lambda(\Sigma^0)K^+$ have been studied. The data and their interpretation are reviewed in Refs. [34, 35].

The transition to the $\Delta(1232)3/2^+$ is often quantified in terms of the magnetic dipole transition moment $M_{1+}$ (or the magnetic transition form factor $G^*_{M,Ash}(Q^2)$) [36], and the electric and scalar quadrupole transition moments $E_{1+}$ and $S_{1+}$. Figure 81.1 shows the strength of the $p \rightarrow \Delta^+$ transition plotted versus the photon virtuality $Q^2$. At $Q^2 = 0$, $M_{1+}$ dominates the resonance transition strength. The two amplitudes $E_{1+}$ and $S_{1+}$ imply a quadrupole deformation of the transition to the lowest excited state. The magnitude of $R_{EM} = E_{1+}/M_{1+}$ remains nearly constant, while the magnitude of $R_{SM} = S_{1+}/M_{1+}$ increases rapidly up to 25% at the highest $Q^2$ value.

Figure 81.2 shows the transverse and scalar helicity amplitudes for the $N(1440)1/2^+$, $N(1520)3/2^-$, and $N(1535)1/2^-$ resonances from JLab [34]. Similar results have been achieved at
<figure><img src="image_350.png" /><figcaption>**Figure 81.1:** Left: The magnetic transition form factor for the $\gamma^*p \rightarrow \Delta^+(1232)$ transition versus the photon virtuality $Q^2$. Right: The electric and scalar quadrupole ratios $R_{EM}$ and $R_{SM}$. The different symbols are results from different experiments at JLab (squares, diamonds, circle) and MAMI (triangle, cross). The boxes near the horizontal axis indicate model uncertainties of the squares. Curves to guide the eyes.</figcaption></figure>

Mainz [35]. For the states $N(1440)1/2^+$ and $N(1520)3/2^-$, helicity amplitudes and $\pi\Delta$ and $\rho p$ decays were determined at JLab in an analysis of $\pi^+\pi^-p$ electroproduction [37]. The data show distinctly different $Q^2$ dependencies that indicate different internal structures.

The $N(1520)3/2^-$ helicity amplitudes reveal the dominance of its three-quark nature: the $A_{3/2}$ amplitude is large at the photon point and decreases rapidly $\sim Q^{-5}$ with increasing $Q^2$; $A_{1/2}$ is small at the photon point, increases rapidly with $Q^2$ and then falls off with $\sim Q^{-3}$. Quantitative agreement with the data is, however, achieved only when meson cloud effects are included.

At high $Q^2$, both amplitudes for $N(1440)1/2^+$ are qualitatively described by light front quark models [38]: at short distances the resonance behaves as expected from a radial excitation of the nucleon. On the other hand, $A_{1/2}$ changes sign at about 0.6 GeV$^2$. This remarkable behavior has not been observed before for any nucleon form factor or transition amplitude. Obviously, an important change in the structure occurs when the resonance is probed as a function of $Q^2$.

The $Q^2$ dependence of $A_{1/2}$ of the $N(1535)1/2^-$ resonance exhibits the expected $Q^{-3}$ dependence, except for small $Q^2$ values where meson cloud effects set in.

Figure 81.3 shows the transverse and scalar amplitudes for three states in the 3rd nucleon resonance region, the $\Delta(1620)1/2^-$, the $N(1675)5/2^-$ and $N(1680)5/2^+$. The latter two states have nearly degenerate masses and are parity partners. In the quark model picture, the transverse amplitudes for $N(1675)5/2^-$ on the proton are suppressed due to the Moorhouse selection rule, allowing for a quantitative evaluation of the meson-baryon contributions. The data show significant meson-baryon strength in the $A_{1/2}$ amplitude even at quite high $Q^2$, while $A_{3/2}$ drops much faster with $Q^2$. $N(1680)5/2^+$ shows qualitatively the features predicted in constituent quark models, a dominant $A_{3/2}$ at the real photon point that drops rapidly with increasing $Q^2$, while $A_{1/2}$ becomes the dominant contribution at high $Q^2$, indicating a switch of the helicity structure in the resonance transition at short distances.

<figure><img src="image_351.png" /><figcaption>**Figure 81.2:** Transverse and scalar (longitudinal) helicity amplitudes for $\gamma p \rightarrow N(1440)1/2^+$ (top), $\gamma p \rightarrow N(1520)3/2^-$ (center), and $\gamma p \rightarrow N(1535)1/2^-$ (bottom) as extracted from the JLab-/CLAS data in $n\pi^+$ production (full circles), MAMI/A1 data in $p\pi^0$ production (full down triangle), in $p\pi^+\pi^-$ (open triangles), and combined single and double pion production (open squares). The solid triangle is the PDG 2014 value at $Q^2 = 0$. The open boxes are the model uncertainties of the full circles.</figcaption></figure>
<figure><img src="image_352.png" /><figcaption>Figure 81.3: Transverse and scalar helicity amplitudes for $\gamma p \to \Delta(1620)1/2^-$ (top), $\gamma p \to N(1675)5/2^-$ (center), and $\gamma p \to N(1680)5/2^+$ (bottom) as extracted from the JLab/CLAS data in $n\pi^+$ production (full circles), $p\pi^+\pi^-$ (open triangles), combined single and double pion production (open square). The solid triangle is the 2014 PDG value at $Q^2 = 0$. The open boxes are the model uncertainties of the full circles. The curves are to guide the eye.</figcaption></figure>

## 81.7 Partial wave analyses

Several PWA groups are now actively involved in the analysis of the new data. The GWU group maintains a nearly complete database covering reactions from $\pi N$ and $KN$ elastic scattering to $\gamma N \to N\pi$, $N\eta$, and $N\eta'$. It is presently the only group determining $\pi N$ elastic amplitudes from scattering data in sliced energy bins. Given the high-precision of photoproduction data already or soon to be collected, the spectrum of $N$ and $\Delta$ resonances will in the near future be better known.

Fits to the data are performed by various groups with the aim to understand the reaction dynamics and to identify $N$ and $\Delta$ resonances. For practical reasons, approximations have to be made. We mention several analyses here: (1) The Mainz unitary isobar model [39] focuses on the correct treatment of the low-energy domain. Resonances are added to the unitary amplitude as a sum of Breit-Wigner amplitudes. This model also obtains resonance transition form factors and helicity amplitudes from electroproduction [35]. (2) For $N\pi$ electroproduction, the Yerevan/JLab group uses both the unitary isobar model and the dispersion relation approach developed in [38]. A phenomenological model was developed to extract resonance couplings and partial decay widths from exclusive $\pi^+\pi^-p$ electroproduction [37]. (3) Multichannel analyses using K-matrix parameterizations derive background terms from a chiral Lagrangian - providing a microscopical description of the background - (Giessen [40, 41]) or from phenomenology (KSU [42, 43], Bonn-Gatchina [44]). (4.) Several groups (EBAC-Jlab [45, 46], ANL-Osaka [47], Dubna-Mainz-Taipeh [48], Bonn-Jülich [49-51], Valencia [52]) use dynamical reaction models, driven by chiral Lagrangians, which take dispersive parts of intermediate states into account. Several other groups have made important contributions. The Giessen group pioneered multichannel analyses of large data sets on pion- and photo-induced reactions [40, 41]. The Bonn-Gatchina group included recent high-statistics data and reported systematic searches for new baryon resonances in all relevant partial waves. A summary of their results can be found in [44].

## References

[1] G. Höhler, Pion-Nucleon Scattering, Landolt-Börnstein Vol. I/9b2 (1983), ed. H. Schopper, Springer Verlag.

[2] R. E. Cutkosky et al., in "VI International Conference on Baryon Resonances," 19 (1980), Toronto, ed. N. Isgur.

[3] R. Arndt et al., Phys. Rev. C 74, 045205 (2006), [arXiv:nucl-th/0605082].

[4] "Hadron 2011: 14th International Conference on Hadron Spectroscopy", München, Germany, June, 13 - 17, 2011, published in eConf.

[5] "NSTAR 2013: 9th International Workshop on the Physics of Excited Nucleons", 27-30 May 2013, Peñíscola, Spain.

[6] E. Klempt and J.-M. Richard, Rev. Mod. Phys. 82, 1095 (2010), [arXiv:0901.2055].

[7] V. Crede and W. Roberts, Rept. Prog. Phys. 76, 076301 (2013), [arXiv:1302.7299].

[8] M. Roos et al. (Particle Data Group), Phys. Lett. B 111, 1 (1982).

[9] M. Doring et al., Nucl. Phys. A 829, 170 (2009), [arXiv:0903.4337].

[10] N. Suzuki et al., Phys. Rev. Lett. 104, 042302 (2010), [arXiv:0909.1356].

[11] R. Cutkosky et al., Phys. Rev. D 20, 2839 (1979).

[12] R. Arndt et al., Phys. Rev. C 69, 035213 (2004), [arXiv:nucl-th/0311089].

[13] M. Batinic et al., Phys. Rev. C 51, 2310 (1995), [Erratum: Phys. Rev. C 57, 1004-1005 (1998)], [arXiv:nucl-th/9501011]; M. Batinić et al., Phys. Scr. 58, 15 (1998).

[14] A. Anisovich et al., Eur. Phys. J. A 48, 15 (2012), and references therein., [arXiv:1112.4937].

[15] G. Höhler, PiN Newslett. 1993, 9, 1 (1993).

[16] N. Kelkar and M. Nowakowski, Phys. Rev. A 78, 012709 (2008), and references therein., [arXiv:0805.0608].

[17] G. F. Chew and S. Mandelstam, Phys. Rev. 119, 467 (1960).

[18] S. Ceci et al., Phys. Rev. D 77, 116007 (2008), [hep-ph/0609236].

[19] P. Masjuan and J. J. Sanz-Cillero, Eur. Phys. J. C 73, 2594 (2013), [arXiv:1306.6308].

[20] A. Švarc et al., Phys. Rev. C 88, 3, 035206 (2013), [arXiv:1307.4613].

[21] A. Švarc et al., Phys. Rev. C 89, 4, 045205 (2014), [arXiv:1401.1947].

[22] A. Švarc et al., Phys. Lett. B 755, 452 (2016), [arXiv:1512.07403].

[23] A. Švarc et al., Phys. Rev. C 89, 6, 065208 (2014), [arXiv:1404.1544].

[24] A. Švarc et al., Phys. Rev. C 91, 1, 015207 (2015), [arXiv:1405.6474].

[25] L. Tiator et al., Phys. Rev. C 94, 6, 065204 (2016), [arXiv:1606.00371].

[26] A. Anisovich et al., Eur. Phys. J. A 53, 12, 242 (2017), [arXiv:1712.07537].

[27] A. Anisovich et al., Phys. Rev. Lett. 119, 6, 062004 (2017), [arXiv:1712.07549].

[28] C. Fasano, F. Tabakin and B. Saghai, Phys. Rev. C 46, 2430 (1992).

[29] G. Chew et al., Phys. Rev. 106, 1345 (1957).

[30] R. Workman, L. Tiator and A. Sarantsev, Phys. Rev. C 87, 6, 068201 (2013), [arXiv:1304.4029].

[31] N. Suzuki, T. Sato and T.-S. Lee, Phys. Rev. C 82, 045206 (2010), [arXiv:1006.2196].

[32] H. Kamano, Phys. Rev. C 88, 045203 (2013), [arXiv:1305.6678].
[33] A. Sandorfi et al., AIP Conf. Proc. **1432**, 1, 219 (2012), [arXiv:1108.5411].

[34] I. Aznauryan and V. Burkert, Prog. Part. Nucl. Phys. **67**, 1 (2012), [arXiv:1109.1720].

[35] L. Tiator et al., Eur. Phys. J. ST **198**, 141 (2011), [arXiv:1109.6745]; S. Štajner et al., Phys. Rev. Lett. **119**, 2, 022001 (2017).

[36] W. Ash et al., Phys. Lett. B **24**, 165 (1967).

[37] V. Mokeev et al. (CLAS), Phys. Rev. C **86**, 035203 (2012), [arXiv:1205.3948].

[38] I. Aznauryan, Phys. Rev. C **67**, 015209 (2003), [arXiv:nucl-th/0206033].

[39] D. Drechsel, S. Kamalov and L. Tiator, Eur. Phys. J. A **34**, 69 (2007), [arXiv:0710.0306].

[40] G. Penner and U. Mosel, Phys. Rev. C **66**, 055211 (2002), [arXiv:nucl-th/0207066].

[41] G. Penner and U. Mosel, Phys. Rev. C **66**, 055212 (2002), [arXiv:nucl-th/0207069].

[42] D. Manley and E. Saleski, Phys. Rev. D **45**, 4002 (1992).

[43] M. Shrestha and D. Manley, Phys. Rev. C **86**, 055203 (2012), [arXiv:1208.2710].

[44] A. Anisovich et al., Eur. Phys. J. A **48**, 15 (2012), [arXiv:1112.4937].

[45] A. Matsuyama, T. Sato and T.-S. Lee, Phys. Rept. **439**, 193 (2007), [arXiv:nucl-th/0608051].

[46] T. Sato and T.-S. Lee, J. Phys. G **36**, 073001 (2009), [arXiv:0902.3653].

[47] H. Kamano et al., Phys. Rev. C **88**, 3, 035209 (2013), [arXiv:1305.4351].

[48] G. Y. Chen et al., Phys. Rev. C **76**, 035206 (2007), [arXiv:nucl-th/0703096].

[49] M. Doring et al., Phys. Lett. B **681**, 26 (2009), [arXiv:0903.1781].

[50] M. Doring et al., Nucl. Phys. A **829**, 170 (2009), [arXiv:0903.4337].

[51] D. Ronchen et al., Eur. Phys. J. A **49**, 44 (2013), [arXiv:1211.6998].

[52] S. Sarkar, E. Oset and M. Vicente Vacas, Nucl. Phys. A **750**, 294 (2005), [Erratum: Nucl.Phys.A 780, 90-90 (2006)], [arXiv:nucl-th/0407025].
## 82. Λ and Σ Resonances

Revised August 2021 by V.D. Burkert (Jefferson Lab), V. Crede (Florida State U.), E. Klempt (Bonn U.), U. Thoma (Bonn U.), L. Tiator (KPH, JGU Mainz) and R.L. Workman (George Washington U.).

### 82.1 Introduction

For several decades, there has been very little new experimental data bearing on the properties of Λ and Σ resonances. An exception was the study at JLab of the reactions $γp → K^{+}Σ^{±}π^{∓}$ and $γp → K^{+}Σ^{0}π^{0}$ [1], which established the spin and parity of the Λ(1405) [2]. There was also from BNL new data on the very low energy region of $K^{-}p$ scattering [3-7]. Otherwise, the field is starved for data. Recent analyses (see below) have improved what we know about the properties of the known Λ and Σ resonances, but the established resonances are the same ones that were listed in our 1984 edition [8] except for the $Σ(2250)$ which we consider 2-star only due to its unknown spin-parity. The 1990 Review [9] gave a full report of the status of Lambda and Sigma hyperons, and included Argand plots from the partial-wave analyses. The 2018 Review [10] has a short survey of the $Σ(1670)$-region.

In the last few years, four groups have re-analyzed $K^{-}p$ reactions using more extensive collections of the old data. These analyses justify an update of the status of the Λ and Σ resonances. Although they have not established any new resonances, they have provided at least some evidence for new states and have given a better understanding of the old ones.

Tables 82.1 and 82.2 give our evaluation of the status, both overall and channel by channel, of each Λ and Σ resonance in the Particle Listings. In making these evaluations, we considered, in addition to the four analyses [11-14], the ratings that predated them. The ratings use a 1- to 4-star system. For more details on the evaluation of the overall star ratings, see [15]. The main Summary Table includes only established states with an overall status of 3 or 4 stars; as has already been noted, these are the same fourteen Λ resonances (including Λ(1116)) and nine Σ resonances instead of the former ten (including $Σ(1193)3/2^{+}$, and $Σ(1385)3/2^{+}$) that had long been in the Table. In addition, there are seven 1-star and two 2-star Λ's, and fourteen 1-star and three 2-star Σ's in the Particle Listings.

### 82.2 New analyses

The new analysis progress was pioneered by the Kent group which collected a large fraction of the available data and performed a comprehensive partial wave analysis [11, 16]. $K^{-}p$ scattering into a pseudoscalar meson and an octet baryon is governed by two complex amplitudes; hence four quantities need to be measured to fully construct the amplitudes (up to an arbitrary phase per energy and angular bin). Discussions of complete experiments also generally assume perfect data (no experimental uncertainties); realistic uncertainties further complicate the task of amplitude extraction. Here, the available data are limited to the differential cross section and the target or hyperon recoil polarization $P$; data on the polarization transfer do not exist. The authors of Ref. [16] overcame this difficulty by using start values for the partial wave amplitudes determined in [17] and/or from an energy-dependent fit and by freezing or releasing sets of amplitudes. The resulting amplitudes were fitted with a unitary multichannel parameterization [11].

The JPAC group presented a coupled-channel fit to the $\bar{K}N$ partial waves derived by the Kent group [13]. The JPAC approach was based on the K-matrix formalism. Special attention was paid to the analytical properties of the amplitudes determined by the square-root unitary branch points and the continuation to the complex angular momentum plane. The fit described the Kent partial waves reasonably well. However, when observables were calculated from their partial-wave amplitudes, significant discrepancies became apparent. The results were therefore not included in the RPP.

The ANL-Osaka group derived the energy-dependent amplitudes in fits to a large subset of the data collected in Ref. [16] and further data sets described in Ref. [18]. Their fits were based on

a phenomenological SU(3) Lagrangian [18]. The two ANL-Osaka models agree on the leading contributions but differ significantly in cases with weaker candidates [12].

<table><caption>**Table 82.1:** The status of the Λ resonances. Only those with an overall status of *** or **** are included in the main Baryon Summary Table. Decay channels other than $N\bar{K}$ and $Σπ$ are only given for *** and **** resonances.</caption><thead><tr><th rowspan="2">Particle</th><th rowspan="2">J<sup>P</sup></th><th colspan="3">Status as seen in —</th><th rowspan="2">Other channels</th></tr><tr><th>Overall status</th><th>N<sup>3</sup>K</th><th>Σπ</th></tr></thead><tbody><tr><td>Λ(1116)</td><td>1/2<sup>+</sup></td><td>****</td><td></td><td></td><td>Nπ (weak decay)</td></tr><tr><td>Λ(1380)</td><td>1/2<sup>-</sup></td><td>**</td><td>**</td><td>**</td><td></td></tr><tr><td>Λ(1405)</td><td>1/2<sup>-</sup></td><td>****</td><td>****</td><td>****</td><td></td></tr><tr><td>Λ(1520)</td><td>3/2<sup>-</sup></td><td>****</td><td>****</td><td>****</td><td>Λππ, Λγ, Σππ</td></tr><tr><td>Λ(1600)</td><td>1/2<sup>+</sup></td><td>****</td><td>***</td><td>****</td><td>Λππ, Σ(1385)π</td></tr><tr><td>Λ(1670)</td><td>1/2<sup>-</sup></td><td>****</td><td>****</td><td>****</td><td>Λη</td></tr><tr><td>Λ(1690)</td><td>3/2<sup>-</sup></td><td>****</td><td>****</td><td>***</td><td>Λππ, Σ(1385)π</td></tr><tr><td>Λ(1710)</td><td>1/2<sup>+</sup></td><td>*</td><td>*</td><td>*</td><td></td></tr><tr><td>Λ(1800)</td><td>1/2<sup>-</sup></td><td>***</td><td>***</td><td>**</td><td>Λππ, N<sup>3</sup>K*</td></tr><tr><td>Λ(1810)</td><td>1/2<sup>+</sup></td><td>***</td><td>**</td><td>**</td><td>N<sup>3</sup>K*</td></tr><tr><td>Λ(1820)</td><td>5/2<sup>+</sup></td><td>****</td><td>****</td><td>****</td><td>Σ(1385)π</td></tr><tr><td>Λ(1830)</td><td>5/2<sup>-</sup></td><td>****</td><td>****</td><td>****</td><td>Σ(1385)π</td></tr><tr><td>Λ(1890)</td><td>3/2<sup>+</sup></td><td>****</td><td>****</td><td>**</td><td>Σ(1385)π, N<sup>3</sup>K*</td></tr><tr><td>Λ(2000)</td><td>1/2<sup>-</sup></td><td>*</td><td>*</td><td>*</td><td></td></tr><tr><td>Λ(2050)</td><td>3/2<sup>-</sup></td><td>*</td><td>*</td><td>*</td><td></td></tr><tr><td>Λ(2070)</td><td>3/2<sup>+</sup></td><td>*</td><td>*</td><td>*</td><td></td></tr><tr><td>Λ(2080)</td><td>5/2<sup>-</sup></td><td>*</td><td>*</td><td>*</td><td></td></tr><tr><td>Λ(2085)</td><td>7/2<sup>+</sup></td><td>**</td><td>**</td><td>*</td><td></td></tr><tr><td>Λ(2100)</td><td>7/2<sup>-</sup></td><td>****</td><td>****</td><td>**</td><td>N<sup>3</sup>K*</td></tr><tr><td>Λ(2110)</td><td>5/2<sup>+</sup></td><td>***</td><td>**</td><td>**</td><td>N<sup>3</sup>K*</td></tr><tr><td>Λ(2325)</td><td>3/2<sup>-</sup></td><td>*</td><td>*</td><td></td><td></td></tr><tr><td>Λ(2350)</td><td>9/2<sup>+</sup></td><td>***</td><td>***</td><td>*</td><td></td></tr><tr><td>Λ(2585)</td><td></td><td>*</td><td>*</td><td></td><td></td></tr></tbody></table>

The Bonn-Gatchina (BnGa) group added further (old) data to those analyzed in Ref. [16]. The data set was fitted in a modified K-matrix approach and the resulting amplitudes were compared with those from Refs. [16, 18]. New resonances were found, other states, mostly one and two-star states could not be confirmed; all resonances were tested for their statistical significance. Additional states with any set of quantum numbers were tested and were found to produce only small improvements in the fit [14]. In Ref. [19], properties of the full set of contributing hyperons were reported.

The star ratings of Λ and Σ resonances given in our earlier editions, and the new results from the Kent, ANL-Osaka and BnGa groups were used to update the star rating of the hyperon resonances. In [19], the overall star rating is directly estimated, for [11] we estimate the star rating from the branching-ratio uncertainties. In [12], two solutions are given but no uncertainties for branching ratios. The overall star ratings are based on the evidence for the resonances in the new analyses as well as their consistency including also the results of earlier analyses given in [20]. For further details see also [15].

We decided to remove the three so-called bumps $Σ(1480)$, $Σ(1560)$, $Σ(1670)$, and $Σ(1690)$ as well as $Σ(1620)$ from production experiments. The entries from $Σ(1770)1/2^{+}$ are now listed under $Σ(1660)1/2^{+}$ and $Σ(1880)1/2^{+}$, the entries from $Σ(1730)3/2^{+}$ and $Σ(1840)3/2^{+}$ are now combined to one $Σ(1780)3/2^{+}$. The one-star $Σ(2000)1/2^{-}$ is combined with a new one-star $Σ(2160)1/2^{-}$ to a single one-star $Σ(2110)1/2^{-}$. Apart from $Λ(1380)1/2^{-}$, four further new resonances were included in the Listings: $Λ(2070)3/2^{+}$, $Λ(2080)5/2^{-}$, $Σ(2010)3/2^{-}$, and $Σ(2230)3/2^{+}$, all with one star.
<table><caption>**Table 82.2:** The status of the Σ resonances. Only those with an overall status of *** or **** are included in the main Baryon Summary Table. Decay channels other than $N\bar{K}$, $\Lambda\pi$ and $\Sigma\pi$ are only given for *** and **** resonances.</caption><thead><tr><th rowspan="2">Particle</th><th rowspan="2">J<sup>P</sup></th><th colspan="4">Status as seen in —</th><th rowspan="2">Other channels</th></tr><tr><th>Overall status</th><th>N<span style="text-decoration: overline">K</span></th><th>Λπ</th><th>Σπ</th></tr></thead><tbody><tr><td>Σ(1193)</td><td>1/2<sup>+</sup></td><td>****</td><td></td><td>****</td><td>****</td><td>Nπ (weak decay)</td></tr><tr><td>Σ(1385)</td><td>3/2<sup>+</sup></td><td>****</td><td></td><td>****</td><td>****</td><td>Λγ</td></tr><tr><td>Σ(1580)</td><td>3/2<sup>-</sup></td><td>*</td><td>*</td><td>*</td><td>*</td><td></td></tr><tr><td>Σ(1620)</td><td>1/2<sup>-</sup></td><td>*</td><td>*</td><td>*</td><td>*</td><td></td></tr><tr><td>Σ(1660)</td><td>1/2<sup>+</sup></td><td>***</td><td>***</td><td>***</td><td>***</td><td></td></tr><tr><td>Σ(1670)</td><td>3/2<sup>-</sup></td><td>****</td><td>****</td><td>****</td><td>****</td><td></td></tr><tr><td>Σ(1750)</td><td>1/2<sup>-</sup></td><td>***</td><td>***</td><td>**</td><td>***</td><td>Ση</td></tr><tr><td>Σ(1775)</td><td>5/2<sup>-</sup></td><td>****</td><td>****</td><td>****</td><td>**</td><td></td></tr><tr><td>Σ(1780)</td><td>3/2<sup>+</sup></td><td>*</td><td>*</td><td>*</td><td>*</td><td></td></tr><tr><td>Σ(1880)</td><td>1/2<sup>+</sup></td><td>**</td><td>**</td><td>*</td><td></td><td></td></tr><tr><td>Σ(1900)</td><td>1/2<sup>-</sup></td><td>**</td><td>**</td><td>*</td><td>**</td><td></td></tr><tr><td>Σ(1910)</td><td>3/2<sup>-</sup></td><td>***</td><td>*</td><td>*</td><td>**</td><td></td></tr><tr><td>Σ(1915)</td><td>5/2<sup>+</sup></td><td>****</td><td>***</td><td>***</td><td>***</td><td></td></tr><tr><td>Σ(1940)</td><td>3/2<sup>+</sup></td><td>*</td><td>*</td><td></td><td>*</td><td></td></tr><tr><td>Σ(2010)</td><td>3/2<sup>-</sup></td><td>*</td><td>*</td><td>*</td><td></td><td></td></tr><tr><td>Σ(2030)</td><td>7/2<sup>+</sup></td><td>****</td><td>****</td><td>****</td><td>**</td><td>Δ(1232)<span style="text-decoration: overline">K</span>, N<span style="text-decoration: overline">K</span>*,<br>Σ(1385)π</td></tr><tr><td>Σ(2070)</td><td>5/2<sup>+</sup></td><td>*</td><td>*</td><td></td><td>*</td><td></td></tr><tr><td>Σ(2080)</td><td>3/2<sup>+</sup></td><td>*</td><td>*</td><td>*</td><td></td><td></td></tr><tr><td>Σ(2100)</td><td>7/2<sup>-</sup></td><td>*</td><td>*</td><td>*</td><td>*</td><td></td></tr><tr><td>Σ(2110)</td><td>1/2<sup>-</sup></td><td>*</td><td>*</td><td>*</td><td>*</td><td></td></tr><tr><td>Σ(2230)</td><td>3/2<sup>+</sup></td><td>*</td><td>*</td><td>*</td><td>*</td><td></td></tr><tr><td>Σ(2250)</td><td></td><td>**</td><td>**</td><td>*</td><td>*</td><td></td></tr><tr><td>Σ(2455)</td><td></td><td>*</td><td>*</td><td></td><td></td><td></td></tr><tr><td>Σ(2620)</td><td></td><td>*</td><td>*</td><td></td><td></td><td></td></tr><tr><td>Σ(3000)</td><td></td><td>*</td><td>*</td><td>*</td><td></td><td></td></tr><tr><td>Σ(3170)</td><td></td><td>*</td><td></td><td></td><td></td><td></td></tr></tbody></table>

## 82.3 Sign conventions for resonance couplings

In terms of the isospin-0 and isospin-1 elastic scattering amplitudes $A_0$ and $A_1$, the amplitude for $K^-p \to \bar{K}^0n$ scattering is $\pm(A_1 - A_0)/2$, where the sign depends on conventions used in conjunction with the Clebsch-Gordan coefficients (such as, is the baryon or the meson the “first” particle). If this reaction is partial-wave analyzed and if the overall phase is chosen so that, say, the $\Sigma(1775)5/2^-(D_{15})$ amplitude at resonance points along the positive imaginary axis (points “up”), then any $\Sigma$ at resonance will point “up” and any $\Lambda$ at resonance will point “down” (along the negative imaginary axis). Thus the phase at resonance determines the isospin. The above ignores background amplitudes in the resonating partial waves.

That is the basic idea. In a similar but somewhat more complicated way, the phases of the $\bar{K}N \to \Lambda\pi$ and $\bar{K}N \to \Sigma\pi$ amplitudes for a resonating wave help determine the SU(3) multiplet to which the resonance belongs. Again, a convention has to be adopted for some overall arbitrary phases: which way is “up”? Our convention is that of Levi-Setti [21] and is shown in Fig. 82.1, which also compares experimental results with theoretical predictions for the signs of several resonances. In the Listings, a + or – sign in front of a measurement of an inelastic resonance coupling indicates the sign (the absence of a sign means that the sign is not determined, not that it is positive). Also other decay modes can be used to assign a hyperon to a SU(3) multiplet [22, 23]. Modern analyses determine properties of resonances at the pole position. In these analyses, the + or – sign is replaced by a phase. Background amplitudes can lead to significant phase shifts, and an additional phase shift due to rescattering is admitted in some analyses. In comparison to quark model predictions [19, 24], three $\Lambda$ spin doublets can be identified as being mainly SU(3) singlets: the well-known ($\Lambda(1405)1/2^-$, $\Lambda(1520)3/2^-$), the ($\Lambda(2080)5/2^-$, $\Lambda(2100)7/2^-$), and ($\Lambda(2070)3/2^+$, $\Lambda(2110)5/2^+$).

<figure><img src="image_353.png" /><figcaption>**Figure 82.1:** The signs of the imaginary parts of resonating amplitudes in the $\bar{K}N \to \Lambda\pi$ and $\Sigma\pi$ channels. The signs of the $\Sigma(1385)$ and $\Lambda(1405)$, marked with a ●, are set by convention, and then the others are determined relative to them. The signs required by the SU(3) assignments of the resonances are shown with an arrow, and the experimentally determined signs are shown with an ×.</figcaption></figure>

## 82.4 The $\Lambda(1405)$

In coupled-channels calculations based on the chiral SU(3) effective field theory, the strongly attractive forces between $N\bar{K}$

and $\Sigma\pi$ generate five poles, one SU(3) singlet pole, two $\Lambda$ octet poles and two (or one) octet $\Sigma$ poles (see Section 83). In quark models, these five states are $\Lambda(1405)$, $\Lambda(1670)$, $\Lambda(1800)$, $\Sigma(1620)$,
and Σ(1750). The octet states are found 100 to 150 MeV above the corresponding nucleon resonances. In chiral SU(3) effective field theories, at least three of these states are seen in the 1300 to 1600 MeV mass range. The appearance of two Λ poles in this mass range, a narrow SU(3) octet at ~1420 MeV and a wider SU(3) singlet at ~1380 MeV, was unexpected. This approach has been pursued by a number of groups; for a summary of the results see our Review 83, "Pole Structure of the Λ(1405) Region". In the Listings, we have introduced the Λ(1380) as a new candidate resonance (with two stars), named in accordance with its approximate pole position. The second SU(3) octet Λ state is the well-known Λ(1670). The masses of the two associated Σ states are uncertain so far, and no new entries are introduced in the Listings.

In traditional approaches only one resonance was seen in this mass region, the narrow state at 1405 MeV. It was reported to be the SU(3) singlet state a long time ago in Ref. [25], in agreement with the quark-model expectations but in contrast to the findings based on coupled-channels calculations within chiral SU(3) effective field theories. In the Listings, the Λ(1405) has been retained with its traditional name. In quark models, this state is identified with the SU(3) singlet state, the two Λ octet states with Λ(1670) and Λ(1800), and the two Σ states with Σ(1620) and Σ(1750).

## 82.5 Uncertainties on masses and widths

The uncertainties quoted on resonance parameters from partial-wave analyses are often only statistical, and the parameters can change beyond these uncertainties when a different parametrization of the waves is used. Furthermore, the different analyses use more or less the same data, so it is not really appropriate to treat the different determinations of the resonance parameters as independent or to average them. In any case, the spread of the masses, widths, and branching fractions from the different analyses is certainly a better indication of the uncertainties than are the quoted uncertainties. In the Listings, we usually give a range reflecting the spread of the values rather than a particular value with error.

## 82.6 Production experiments

Partial-wave analyses of course separate partial waves, whereas a peak in a cross section or an invariant mass distribution usually cannot be disentangled from background and analyzed for its quantum numbers; and more than one resonance may be contributing to the peak. The Σ(1385) and Λ(1405) lie below the $\bar{K}N$ threshold and nearly everything about Σ(1385) is learned from production experiments. Our knowledge on Λ(1405) benefits greatly from photoproduction of the three Σπ charge states [1,2] and from the precise measurement of the energy shift and width of the kaonic hydrogen atom [26].

Production and formation experiments agree quite well in the case of Λ(1520) and results have been combined. Above this mass, no new results on peak hunting have been reported for about 40 years. For these early results, we refer the reader to our earlier editions. In photoproduction with energetic photons [27, 28] or at LHCb [29], hyperons are produced abundantly. So far, no attempt has been made to extract hyperon properties from these data. New data on hyperon spectroscopy can be expected from J-PARC [30], JLAB [31], and the forthcoming PANDA experiment [32].

### References

[1] K. Moriya et al. (CLAS), Phys. Rev. C87, 3, 035206 (2013), [arXiv:1301.5000].

[2] K. Moriya et al. (CLAS), Phys. Rev. Lett. 112, 8, 082004 (2014), [arXiv:1402.2296].

[3] A. Starostin et al. (Crystal Ball), Phys. Rev. C64, 055205 (2001).

[4] S. Prakhov et al., Phys. Rev. C69, 042202 (2004).

[5] S. Prakhov et al. (Crystall Ball), Phys. Rev. C70, 034605 (2004).

[6] R. Manweiler et al., Phys. Rev. C77, 015205 (2008).

[7] S. Prakhov et al., Phys. Rev. C80, 025204 (2009), [arXiv:0812.1888].

[8] C. G. Wohl et al. (Particle Data Group), Rev. Mod. Phys. 56, S1 (1984).

[9] J. J. Hernandez et al. (Particle Data Group), Phys. Lett. B239, 1 (1990), [Erratum: Phys. Lett. B253, 524 (1991)].

[10] M. Tanabashi et al. (Particle Data Group), Phys. Rev. D98, 3, 030001 (2018).

[11] H. Zhang et al., Phys. Rev. C88, 3, 035205 (2013), [arXiv:1305.4575].

[12] H. Kamano et al., Phys. Rev. C92, 2, 025205 (2015), [Erratum: Phys. Rev. C95, no.4, 049903 (2017)], [arXiv:1506.01768].

[13] C. Fernandez-Ramirez et al., Phys. Rev. D93, 3, 034029 (2016), [arXiv:1510.07065].

[14] M. Matveev et al., Eur. Phys. J. A55, 10, 179 (2019), [arXiv:1907.03645].

[15] E. Klempt et al. (Baryon@PDG Group), Eur. Phys. J. A 56, 10, 261 (2020), [arXiv:2007.04232].

[16] H. Zhang et al., Phys. Rev. C88, 3, 035204 (2013), [arXiv:1305.3598].

[17] J. Tulpan, PhD Dissertation, Kent State University (2007).

[18] H. Kamano et al., Phys. Rev. C90, 6, 065204 (2014), [arXiv:1407.6839].

[19] A. V. Sarantsev et al., Eur. Phys. J. A55, 10, 180 (2019), [arXiv:1907.13387].

[20] J. Beringer et al. (Particle Data Group), Phys. Rev. D86, 010001 (2012).

[21] R. Levi Setti, Conf. Proc. C690625, 339 (1969).

[22] V. Guzey and M. V. Polyakov, Annalen Phys. 13, 673 (2004).

[23] V. Guzey and M. V. Polyakov (2005), [hep-ph/0512355].

[24] U. Loring, B. C. Metsch and H. R. Petry, Eur. Phys. J. A10, 447 (2001), [hep-ph/0103290].

[25] R. D. Tripp et al., Phys. Rev. Lett. 21, 1721 (1968).

[26] M. Bazzi et al. (SIDDHARTA), Phys. Lett. B704, 113 (2011), [arXiv:1105.3090].

[27] K. Moriya et al. (CLAS), Phys. Rev. C88, 045201 (2013), [Addendum: Phys. Rev. C88, no.4, 049902 (2013)], [arXiv:1305.6776].

[28] B. Dey et al. (CLAS), Phys. Rev. C89, 5, 055208 (2014), [Addendum: Phys. Rev. C90, no.1, 019901 (2014)], [arXiv:1403.2110].

[29] R. Aaij et al. (LHCb), Phys. Rev. Lett. 115, 072001 (2015), [arXiv:1507.03414].

[30] K. Hicks and H. Sako, Proposal for J-PARC E45.

[31] M. Amaryan et al. (KLF) (2020), [arXiv:2008.08215].

[32] F. Iazzi (PANDA), AIP Conf. Proc. 1743, 1, 050006 (2016).
## 83. Pole Structure of the $\Lambda(1405)$ Region

Revised June 2021 by T. Hyodo (Tokyo Metropolitan U.) and U.-G. Meißner (Bonn U.; FZ Jülich).

The $\Lambda(1405)$ resonance emerges in the meson-baryon scattering amplitude with the strangeness $S = -1$ and isospin $I = 0$. It is the archetyp of what is called a dynamically generated resonance, as pioneered by Dalitz and Tuan [1]. The most powerful and systematic approach for the low-energy regime of the strong interactions is chiral perturbation theory (ChPT), see e.g. Ref. [2]. A perturbative calculation is, however, not applicable to this sector because of the existence of the $\Lambda(1405)$ just below the $KN$ threshold. In this case, ChPT has to be combined with a non-perturbative resummation technique, just as in the case of the nuclear forces. By solving the Lippmann-Schwinger equation with the interaction kernel determined by ChPT and using a particular regularization, in Ref. [3] a successful description of the low-energy $K^-p$ scattering data as well as the mass distribution of the $\Lambda(1405)$ was achieved (for further developments, see Ref. [4-7] and references therein).

The study of the pole structure was initiated by Ref. [8], which finds two poles of the scattering amplitude in the complex energy plane between the $KN$ and $\pi\Sigma$ thresholds. The spectrum in experiments exhibits one effective resonance shape, while the existence of two poles results in the reaction-dependent lineshape [9]. The origin of this two-pole structure is attributed to the two attractive channels of the leading order interaction in the $SU(3)$ basis (singlet and octet) [9] and in the isospin basis ($KN$ and $\pi\Sigma$) [10]. It is remarkable that the sign and the strength of the leading order interaction is determined by a low-energy theorem of chiral symmetry, i.e. the so-called Weinberg-Tomozawa term. The two-pole nature of the $\Lambda(1405)$ is qualitatively different from the case of the $N(1440)$ resonance. Two poles of the $N(1440)$ appear on different Riemann sheets of the complex energy plane separated by the $\pi\Delta$ branch point. These poles reflect a single state, with a nearby pole and a more distant shadow pole. In contrast, the two poles in the $\Lambda(1405)$ region on the same Riemann sheet (where $\pi\Sigma$ channels are unphysical and all other channels physical, correspondingly to the one, connected to the real axis between the $\pi\Sigma$ and $KN$ thresholds) are generated from two attractive forces mentioned above [9, 10].

Recently, various new experimental results on the $\Lambda(1405)$ have become available [4]. Among these, the most striking measurement is the precise determination of the energy shift and width of kaonic hydrogen by the SIDDHARTA collaboration [11, 12], which provides a quantitative and stringent constraint on the $K^-p$ amplitude at threshold through the improved Deser formula [13]. Systematic studies with error analyses based on the next-to-leading order ChPT interaction including the SIDDHARTA constraint have been performed by various groups [14-18] All these studies confirm that the new kaonic hydrogen data are compatible with the scattering data above threshold.

The results of the pole positions of $\Lambda(1405)$ in the various approaches are summarized in Table 83.1. We may regard the difference among the calculations as a systematic error, which stems from the various approximations of the Bethe-Salpeter equation, the fitting procedure, and also the inclusion of $SU(3)$ breaking effects such as the choice of the various meson decay constants, and so on. A detailed comparison of the various approaches that enter the table is given in Ref. [19]. A recent analysis including also the $JP = 1/2^+ P$-wave contribution (and also an explicit $\Sigma(1385) 3/2^+$ state) gives results consistent with the findings reported above, with the pole positions at $(1364 - i43)$ MeV and $(1430 - i15)$ MeV, respectively [20].

The main component for the $\Lambda(1405)$ is the pole 1, whose position converges within a relatively small region near the $KN$ threshold. On the other hand, the position of the pole 2 shows a sizeable scatter. Detailed studies of the $\pi\Sigma$ spectrum in various reaction processes, together with the precise experimental lineshape (see e.g. the recent precise photoproduction data from the LEPS collaboration [21] and from the CLAS collaboration [22, 23], electroproduction data from the CLAS collaboration [24], and proton-proton collision data from COSY [25] and the HADES collaboration [26]), will shed light on the position of the second pole.

The $\pi\Sigma$ spectra from the CLAS data are analyzed in Ref. [27] and Ref. [18]. It was shown in Ref. [18] that several solutions, which agree with the scattering data, are ruled out if confronted with the recent CLAS data. The remaining solutions are collected as solution #2 and solution #4 in Table 83.1. The HADES data are analyzed in Ref. [28] and Ref [29]. Although the result of the pole found in Ref. [28] is not compatible with other results, the authors of Ref. [29] invoke the anomalous triangle singularity mechanism to argue that the invariant mass distribution of the $\pi\Sigma$ system is found at lower masses than in other reactions. It is thus desirable to perform more comprehensive analyses of $\pi\Sigma$ spectra together with the systematic error analysis of the scattering data.

<table><caption>**Table 83.1:** Comparison of the pole positions of $\Lambda(1405)$ in the complex energy plane from next-to-leading order chiral unitary coupled-channel approaches including the SIDDHARTA constraint. The lower two results also include the CLAS photoproduction data.</caption><thead><tr><th>approach</th><th>pole 1 [MeV]</th><th>pole 2 [MeV]</th></tr></thead><tbody><tr><td>Refs. [14, 15], NLO</td><td>1424<sup>+7</sup><sub>-23</sub> - i 26<sup>+3</sup><sub>-14</sub></td><td>1381<sup>+18</sup><sub>-6</sub> - i 81<sup>+19</sup><sub>-8</sub></td></tr><tr><td>Ref. [17], Fit II</td><td>1421<sup>+3</sup><sub>-2</sub> - i 19<sup>+8</sup><sub>-5</sub></td><td>1388<sup>+9</sup><sub>-9</sub> - i 114<sup>+24</sup><sub>-25</sub></td></tr><tr><td>Ref. [18], solution #2</td><td>1434<sup>+2</sup><sub>-2</sub> - i 10<sup>+2</sup><sub>-1</sub></td><td>1330<sup>+4</sup><sub>-5</sub> - i 56<sup>+17</sup><sub>-11</sub></td></tr><tr><td>Ref. [18], solution #4</td><td>1429<sup>+8</sup><sub>-7</sub> - i 12<sup>+2</sup><sub>-3</sub></td><td>1325<sup>+15</sup><sub>-15</sub> - i 90<sup>+12</sup><sub>-18</sub></td></tr></tbody></table>

### References

[1] R. Dalitz and S. Tuan, Phys. Rev. Lett. **2**, 425 (1959).

[2] V. Bernard, N. Kaiser and U.-G. Meißner, Int. J. Mod. Phys. E **4**, 193 (1995), [hep-ph/9501384].

[3] N. Kaiser, P. Siegel and W. Weise, Nucl. Phys. A **594**, 325 (1995), [arXiv:nucl-th/9505043].

[4] T. Hyodo and D. Jido, Prog. Part. Nucl. Phys. **67**, 55 (2012), [arXiv:1104.4474].

[5] U.-G. Meißner, Symmetry **12**, 6, 981 (2020), [arXiv:2005.06909].

[6] M. Mai, Eur. Phys. J. ST **230**, 6, 1593 (2021), [arXiv:2010.00056].

[7] T. Hyodo and M. Niiyama, Prog. Part. Nucl. Phys. **120**, 103868 (2021), [arXiv:2010.07592].

[8] J. Oller and U.-G. Meißner, Phys. Lett. B **500**, 263 (2001), [hep-ph/0011146].

[9] D. Jido et al., Nucl. Phys. A **725**, 181 (2003), [arXiv:nucl-th/0303062].

[10] T. Hyodo and W. Weise, Phys. Rev. C **77**, 035204 (2008), [arXiv:0712.1613].

[11] M. Bazzi et al. (SIDDHARTA), Phys. Lett. B **704**, 113 (2011), [arXiv:1105.3090].

[12] M. Bazzi et al. (SIDDHARTA), Nucl. Phys. A **881**, 88 (2012), [arXiv:1201.4635].

[13] U.-G. Meißner, U. Raha and A. Rusetsky, Eur. Phys. J. C **35**, 349 (2004), [hep-ph/0402261].

[14] Y. Ikeda, T. Hyodo and W. Weise, Phys. Lett. B **706**, 63 (2011), [arXiv:1109.3005].

[15] Y. Ikeda, T. Hyodo and W. Weise, Nucl. Phys. A **881**, 98 (2012), [arXiv:1201.6549].

[16] M. Mai and U.-G. Meißner, Nucl. Phys. A **900**, 51 (2013), [arXiv:1202.2030].

[17] Z.-H. Guo and J. Oller, Phys. Rev. C **87**, 3, 035202 (2013), [arXiv:1210.3485].

[18] M. Mai and U.-G. Meißner, Eur. Phys. J. A **51**, 3, 30 (2015), [arXiv:1411.7884].

[19] A. Cieplý et al., Nucl. Phys. A **954**, 17 (2016), [arXiv:1603.02531].

[20] D. Sadasivan, M. Mai and M. Döring, Phys. Lett. B **789**, 329 (2019), [arXiv:1805.04534].
[21] M. Niiyama et al., Phys. Rev. C **78**, 035202 (2008), [arXiv:0805.4051].

[22] K. Moriya et al. (CLAS), Phys. Rev. C **87**, 035206 (2013), [arXiv:1301.5000].

[23] K. Moriya et al. (CLAS), Phys. Rev. Lett. **112**, 082004 (2014), [arXiv:1402.2296].

[24] H. Lu et al. (CLAS), Phys. Rev. C **88**, 045202 (2013), [arXiv:1307.4411].

[25] I. Zychor et al., Phys. Lett. B **660**, 167 (2008), [arXiv:0705.1039].

[26] G. Agakishiev et al. (HADES), Phys. Rev. C **87**, 025201 (2013), [arXiv:1208.0205].

[27] L. Roca and E. Oset, Phys. Rev. C **87**, 055201 (2013), [arXiv:1301.5741].

[28] M. Hassanvand et al., Phys. Rev. C **87**, 055202 (2013), [Addendum: Phys.Rev.C **88**, 019905 (2013)], [arXiv:1210.7725].

[29] M. Bayar et al., Phys. Rev. C **97**, 035203 (2018), [arXiv:1710.03964].