$\mathbb{CP}^{1}$ model, at the same model parameters of $\theta/\pi=0.06$ and
$\phi/\pi=0.305$. The top row shows the spectrum along the path in momentum
space as indicated by green lines in the left panel of Fig. 9(d). The bottom
row depicts frequency cuts at the explicit momentum points $\Gamma$ and K′.

For better comparison we replot in Fig. 16(a) the result from the
$\mathbb{CP}^{2}$ model, as previously shown in Fig. 10(b). The spectrum
exhibits three energy regimes: a zero-energy flat band, a slightly gapped
diffuse band, and a largely gapped and diffuse continuum with broad bandwidth
at higher energies. Additionally, a flat, weak intensity band around
$\omega/J\approx 2$ is observed, arising from the mixing between dipole and
quadrupole excitations due to spin anisotropy. These energy regimes are
quantified in the bottom row.

In Fig. 16(b), qualitatively similar features are visible in the dispersion
obtained from $\mathbb{CP}^{1}$ model simulations, performed at nearly the
same temperature. Here, the diffuse band at intermediate energies exhibits a
larger bandwidth while clearly showing an energy gap to the high-intensity
zero-energy mode. The broad continuum is slightly shifted to higher energies
compared to the $\mathbb{CP}^{2}$ case but demonstrates almost the same
intensity distribution. The nearly flat band around $\omega/J\approx 2$ is
absent in the $\mathbb{CP}^{1}$ case, as quadrupoles are strictly excluded in
these simulations.

In Fig. 16(c), we additionally present the spectrum for the $\mathbb{CP}^{1}$
model at $T=0.001$, a temperature that is not accessible in the
$\mathbb{CP}^{2}$ model, as the FM CSL transitions into the NC-1 ordered state
before reaching such low temperatures (see thermodynamics in Sec. III.4.2).
Since thermal fluctuations are reduced, the spectrum becomes slightly sharper,
especially the gap of the diffuse intermediate band to the zero-energy mode
becomes more pronounced. However, importantly, qualitative features between
$\mathbb{CP}^{1}$ and $\mathbb{CP}^{2}$ model simulations remain the same.

This comparison confirms our intuition that dominant properties of the CSL are
still captured in the $\mathbb{CP}^{1}$ model, which includes only dipole spin
degree of freedom. Additionally, simulation results for the dynamics in both
the $\mathbb{CP}^{1}$ and $\mathbb{CP}^{2}$ models reveal a very diffuse and
gapped continuum, associated with the excitations of the FM CSL, as discussed
in the eight-color model in Sec. III.2.

## References

  * Chen  _et al._ [2010] G. Chen, R. Pereira, and L. Balents, Exotic phases induced by strong spin-orbit coupling in ordered double perovskites, Phys. Rev. B 82, 174440 (2010).
  * Fu [2015] L. Fu, Parity-Breaking Phases of Spin-Orbit-Coupled Metals with Gyrotropic, Ferroelectric, and Multipolar Orders, Phys. Rev. Lett. 115, 026401 (2015).
  * Hirai  _et al._ [2020] D. Hirai, H. Sagayama, S. Gao, H. Ohsumi, G. Chen, T.-h. Arima, and Z. Hiroi, Detection of multipolar orders in the spin-orbit-coupled $5d$ Mott insulator $\mathrm{B}{\mathrm{a}}_{2}\mathrm{MgRe}{\mathrm{O}}_{6}$, Phys. Rev. Res. 2, 022063 (2020).
  * Maharaj  _et al._ [2020] D. D. Maharaj, G. Sala, M. B. Stone, E. Kermarrec, C. Ritter, F. Fauth, C. A. Marjerrison, J. E. Greedan, A. Paramekanti, and B. D. Gaulin, Octupolar versus Néel Order in Cubic $5{d}^{2}$ Double Perovskites, Phys. Rev. Lett. 124, 087206 (2020).
  * Paramekanti  _et al._ [2020] A. Paramekanti, D. D. Maharaj, and B. D. Gaulin, Octupolar order in $d$-orbital Mott insulators, Phys. Rev. B 101, 054439 (2020).
  * Thalmeier  _et al._ [2008] P. Thalmeier, T. Takimoto, J. Chang, and I. Eremin, Multipolar Order and Superconductivity in f-Electron Compounds, Journal of the Physical Society of Japan 77, 43 (2008).
  * Matsubayashi  _et al._ [2012] K. Matsubayashi, T. Tanaka, A. Sakai, S. Nakatsuji, Y. Kubo, and Y. Uwatoko, Pressure-Induced Heavy Fermion Superconductivity in the Nonmagnetic Quadrupolar System ${\mathrm{PrTi}}_{2}{\mathrm{Al}}_{20}$, Phys. Rev. Lett. 109, 187004 (2012).
  * Freyer  _et al._ [2018] F. Freyer, J. Attig, S. Lee, A. Paramekanti, S. Trebst, and Y. B. Kim, Two-stage multipolar ordering in $\mathrm{Pr}{T}_{2}{\mathrm{Al}}_{20}$ Kondo materials, Phys. Rev. B 97, 115111 (2018).
  * Lee  _et al._ [2018] S. Lee, S. Trebst, Y. B. Kim, and A. Paramekanti, Landau theory of multipolar orders in $\mathrm{Pr}{(Y)}_{2}{X}_{20}$ Kondo materials ($Y=\mathrm{Ti}$, V, Rh, Ir; $X=\mathrm{Al}$, Zn), Phys. Rev. B 98, 134447 (2018).
  * Sim  _et al._ [2020] G. Sim, A. Mishra, M. J. Park, Y. B. Kim, G. Y. Cho, and S. Lee, Multipolar superconductivity in Luttinger semimetals, Phys. Rev. Res. 2, 023416 (2020).
  * Patri and Kim [2022] A. S. Patri and Y. B. Kim, Unconventional Superconductivity arising from Multipolar Kondo Interactions, SciPost Phys. 12, 057 (2022).
  * Wladimir A. Benalcazar and B. Andrei Bernevig and Taylor L. Hughes [2017] Wladimir A. Benalcazar and B. Andrei Bernevig and Taylor L. Hughes , Quantized electric multipole insulators, Science 357, 61 (2017).
  * Romhányi [2019] J. Romhányi, Multipolar edge states in the anisotropic kagome antiferromagnet, Phys. Rev. B 99, 014408 (2019).
  * Sibille  _et al._ [2015] R. Sibille, E. Lhotel, V. Pomjakushin, C. Baines, T. Fennell, and M. Kenzelmann, Candidate Quantum Spin Liquid in the ${\mathrm{Ce}}^{3+}$ Pyrochlore Stannate ${\mathrm{Ce}}_{2}{\mathrm{Sn}}_{2}{\mathrm{O}}_{7}$, Phys. Rev. Lett. 115, 097202 (2015).
  * Gaudet  _et al._ [2019] J. Gaudet, E. M. Smith, J. Dudemaine, J. Beare, C. R. C. Buhariwalla, N. P. Butch, M. B. Stone, A. I. Kolesnikov, G. Xu, D. R. Yahne, K. A. Ross, C. A. Marjerrison, J. D. Garrett, G. M. Luke, A. D. Bianchi, and B. D. Gaulin, Quantum Spin Ice Dynamics in the Dipole-Octupole Pyrochlore Magnet ${\mathrm{Ce}}_{2}{\mathrm{Zr}}_{2}{\mathrm{O}}_{7}$, Phys. Rev. Lett. 122, 187201 (2019).
  * Sibille  _et al._ [2020] R. Sibille, N. Gauthier, E. Lhotel, V. Porée, V. Pomjakushin, R. A. Ewings, T. G. Perring, J. Ollivier, A. Wildes, C. Ritter, T. C. Hansen, D. A. Keen, G. J. Nilsen, L. Keller, S. Petit, and T. Fennell, A quantum liquid of magnetic octupoles on the pyrochlore lattice, Nature Physics 16, 546 (2020).
  * Akagi  _et al._ [2021] Y. Akagi, Y. Amari, N. Sawado, and Y. Shnir, Isolated skyrmions in the $C{P}^{2}$ nonlinear sigma model with a Dzyaloshinskii-Moriya type interaction, Phys. Rev. D 103, 065008 (2021).
  * Amari  _et al._ [2022] Y. Amari, Y. Akagi, S. B. Gudnason, M. Nitta, and Y. Shnir, $\mathbb{C}{P}^{2}$ skyrmion crystals in an SU(3) magnet with a generalized Dzyaloshinskii-Moriya interaction, Phys. Rev. B 106, L100406 (2022).
  * Zhang  _et al._ [2023] H. Zhang, Z. Wang, D. Dahlbom, K. Barros, and C. D. Batista, CP2 skyrmions and skyrmion crystals in realistic quantum magnets, Nature Communications 14, 3626 (2023).
  * Lai  _et al._ [2018] H.-H. Lai, E. M. Nica, W.-J. Hu, S.-S. Gong, S. Paschen, and Q. Si, Kondo Destruction and Multipolar Order– Implications for Heavy Fermion Quantum Criticality (2018), arXiv:1807.09258 [cond-mat.str-el] .
  * Masui and Totsuka [2022] R. Masui and K. Totsuka, Electric and magnetic properties of higher-spin Kondo-Heisenberg models at strong coupling, Phys. Rev. B 106, 014411 (2022).
  * Papanicolaou [1988] N. Papanicolaou, Unusual phases in quantum spin-1 systems, Nuclear Physics B 305, 367 (1988).
  * Tsunetsugu and Arikawa [2006] H. Tsunetsugu and M. Arikawa, Spin Nematic Phase in $S$=1 Triangular Antiferromagnets, Journal of the Physical Society of Japan 75, 083701 (2006).
  * Läuchli  _et al._ [2006a] A. Läuchli, F. Mila, and K. Penc, Quadrupolar Phases of the $S=1$ Bilinear-Biquadratic Heisenberg Model on the Triangular Lattice, Phys. Rev. Lett. 97, 087205 (2006a).
  * Penc and Läuchli [2011] K. Penc and A. M. Läuchli, Spin Nematic Phases in Quantum Spin Systems, in  _Introduction to Frustrated Magnetism: Materials, Experiments, Theory_, edited by C. Lacroix, P. Mendels, and F. Mila (Springer Berlin Heidelberg, Berlin, Heidelberg, 2011) pp. 331–362.
  * Kelker [1973] H. Kelker, History of Liquid Crystals, Molecular Crystals and Liquid Crystals 21, 1 (1973).
  * Kelker [1988] H. Kelker, Survey of the Early History of Liquid Crystals, Molecular Crystals and Liquid Crystals Incorporating Nonlinear Optics 165, 1 (1988).
  * de Gennes and Prost [1993] P. de Gennes and J. Prost,  _The Physics of Liquid Crystals_, International Series of Monographs on Physics (Clarendon Press, 1993).
  * Mermin and Wagner [1966] N. D. Mermin and H. Wagner, Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models, Phys. Rev. Lett. 17, 1133 (1966).
  * Alexander  _et al._ [2012] G. P. Alexander, B. G.-g. Chen, E. A. Matsumoto, and R. D. Kamien, Colloquium: Disclination loops, point defects, and all that in nematic liquid crystals, Rev. Mod. Phys. 84, 497 (2012).
  * Chuang  _et al._ [1991] I. Chuang, R. Durrer, N. Turok, and B. Yurke, Cosmology in the Laboratory: Defect Dynamics in Liquid Crystals, Science 251, 1336 (1991).
  * Bowick  _et al._ [1994] M. J. Bowick, L. Chandar, E. A. Schiff, and A. M. Srivastava, The Cosmological Kibble Mechanism in the Laboratory: String Formation in Liquid Crystals, Science 263, 943 (1994).
  * Volovik [1990] G. Volovik, Superfluid 3He-B and gravity, Physica B: Condensed Matter 162, 222 (1990).
  * Volovik [2009] G. E. Volovik,  _The universe in a helium droplet_ , Repr ed., International series of monographs on physics No. 117 (Oxford University Press, Oxford, 2009).
  * Chojnacki  _et al._ [2023] L. Chojnacki, R. Pohle, H. Yan, Y. Akagi, and N. Shannon, Gravitational wave analogues in spin nematics and cold atoms (2023), arXiv:2310.10078 [cond-mat.str-el] .
  * Blume and Hsieh [1969] M. Blume and Y. Y. Hsieh, Biquadratic Exchange and Quadrupolar Ordering, Journal of Applied Physics 40, 1249 (1969).
  * Matveev [1974] V. M. Matveev, Quantum quadrupolar magnetism and phase transitions in the presence of biquadratic exchange, JETP 38, 813 (1974).
  * Andreev and Grishchuk [1984] A. F. Andreev and I. A. Grishchuk, Spin nematics, JETP 60, 267 (1984).
  * Smerald and Shannon [2013] A. Smerald and N. Shannon, Theory of spin excitations in a quantum spin-nematic state, Phys. Rev. B 88, 184430 (2013).
  * Michaud  _et al._ [2011] F. Michaud, F. Vernay, and F. Mila, Theory of inelastic light scattering in spin-1 systems: Resonant regimes and detection of quadrupolar order, Phys. Rev. B 84, 184424 (2011).
  * Valentine  _et al._ [2020] M. E. Valentine, T. Higo, Y. Nambu, D. Chaudhuri, J. Wen, C. Broholm, S. Nakatsuji, and N. Drichko, mpact of the Lattice on Magnetic Properties and Possible Spin Nematicity in the $S=1$ Triangular Antiferromagnet ${\mathrm{NiGa}}_{2}{\mathrm{S}}_{4}$, Phys. Rev. Lett. 125, 197201 (2020).
  * Savary and Senthil [2015] L. Savary and T. Senthil, Probing Hidden Orders with Resonant Inelastic X-Ray Scattering (2015), arXiv:1506.04752 [cond-mat.str-el] .
  * Takahashi  _et al._ [2021] H. Takahashi, H. Suzuki, J. Bertinshaw, S. Bette, C. Mühle, J. Nuss, R. Dinnebier, A. Yaresko, G. Khaliullin, H. Gretarsson, T. Takayama, H. Takagi, and B. Keimer, Nonmagnetic $J=0$ State and Spin-Orbit Excitations in ${\mathrm{K}}_{2}{\mathrm{RuCl}}_{6}$, Phys. Rev. Lett. 127, 227201 (2021).
  * Nag  _et al._ [2022] A. Nag, A. Nocera, S. Agrestini, M. Garcia-Fernandez, A. C. Walters, S.-W. Cheong, S. Johnston, and K.-J. Zhou, Quadrupolar magnetic excitations in an isotropic spin-1 antiferromagnet, Nature Communications 13, 2327 (2022).
  * Kohama  _et al._ [2019] Y. Kohama, H. Ishikawa, A. Matsuo, K. Kindo, N. Shannon, and Z. Hiroi, Possible observation of quantum spin-nematic phase in a frustrated magnet, Proceedings of the National Academy of Sciences 116, 10686 (2019).
  * Zhang and Batista [2021] H. Zhang and C. D. Batista, Classical spin dynamics based on $\mathrm{SU}(N)$ coherent states, Phys. Rev. B 104, 104409 (2021).
  * Remund  _et al._ [2022] K. Remund, R. Pohle, Y. Akagi, J. Romhányi, and N. Shannon, Semi-classical simulation of spin-1 magnets, Phys. Rev. Research 4, 033106 (2022).
  * Dahlbom  _et al._ [2022a] D. Dahlbom, H. Zhang, C. Miles, X. Bai, C. D. Batista, and K. Barros, Geometric integration of classical spin dynamics via a mean-field Schrödinger equation, Phys. Rev. B 106, 054423 (2022a).
  * Dahlbom  _et al._ [2022b] D. Dahlbom, C. Miles, H. Zhang, C. D. Batista, and K. Barros, Langevin dynamics of generalized spins as SU($N$) coherent states, Phys. Rev. B 106, 235154 (2022b).
  * Iwazaki  _et al._ [2023] R. Iwazaki, H. Shinaoka, and S. Hoshino, Material-based analysis of spin-orbital Mott insulators, Phys. Rev. B 108, L241108 (2023).
  * [51] https://github.com/SunnySuite/Sunny.jl.
  * Nakatsuji  _et al._ [2005] S. Nakatsuji, Y. Nambu, H. Tonomura, O. Sakai, S. Jonas, C. Broholm, H. Tsunetsugu, Y. Qiu, and Y. Maeno, Spin Disorder on a Triangular Lattice, Science 309, 1697 (2005), https://www.science.org/doi/pdf/10.1126/science.1114727 .
  * Stoudenmire  _et al._ [2009] E. M. Stoudenmire, S. Trebst, and L. Balents, Quadrupolar correlations and spin freezing in $S=1$ triangular lattice antiferromagnets, Phys. Rev. B 79, 214436 (2009).
  * Bai  _et al._ [2021] X. Bai, S.-S. Zhang, Z. Dun, H. Zhang, Q. Huang, H. Zhou, M. B. Stone, A. I. Kolesnikov, F. Ye, C. D. Batista, and M. Mourigal, Hybridized quadrupolar excitations in the spin-anisotropic frustrated magnet ${\mathrm{FeI}}_{2}$, Nature Physics 17, 467 (2021).
  * Dahlbom  _et al._ [2024] D. Dahlbom, F. T. Brooks, M. S. Wilson, S. Chi, A. I. Kolesnikov, M. B. Stone, H. Cao, Y.-W. Li, K. Barros, M. Mourigal, C. D. Batista, and X. Bai, Quantum-to-classical crossover in generalized spin systems: Temperature-dependent spin dynamics of ${\mathrm{FeI}}_{2}$, Phys. Rev. B 109, 014427 (2024).
  * Do  _et al._ [2023] S.-H. Do, H. Zhang, D. A. Dahlbom, T. J. Williams, V. O. Garlea, T. Hong, T.-H. Jang, S.-W. Cheong, J.-H. Park, K. Barros, C. D. Batista, and A. D. Christianson, Understanding temperature-dependent SU(3) spin dynamics in the $S=1$ antiferromagnet $\mathrm{Ba_{2}FeSi_{2}O_{7}}$, npj Quantum Materials 8, 5 (2023).
  * Baskaran  _et al._ [2008] G. Baskaran, D. Sen, and R. Shankar, Spin-$S$ Kitaev model: Classical ground states, order from disorder, and exact correlation functions, Phys. Rev. B 78, 115116 (2008).
  * Koga  _et al._ [2018] A. Koga, H. Tomishige, and J. Nasu, Ground-state and Thermodynamic Properties of an $S=1$ Kitaev Model, Journal of the Physical Society of Japan 87, 063703 (2018).
  * Suzuki and Yamaji [2018] T. Suzuki and Y. Yamaji, Thermal properties of spin-S Kitaev-Heisenberg model on a honeycomb lattice, Physica B: Condensed Matter 536, 637 (2018).
  * Oitmaa  _et al._ [2018] J. Oitmaa, A. Koga, and R. R. P. Singh, Incipient and well-developed entropy plateaus in spin-$S$ Kitaev models, Phys. Rev. B 98, 214404 (2018).
  * Dong and Sheng [2020] X.-Y. Dong and D. N. Sheng, Spin-1 Kitaev-Heisenberg model on a honeycomb lattice, Phys. Rev. B 102, 121102 (2020).
  * Hickey  _et al._ [2020] C. Hickey, C. Berke, P. P. Stavropoulos, H.-Y. Kee, and S. Trebst, Field-driven gapless spin liquid in the spin-1 Kitaev honeycomb model, Phys. Rev. Res. 2, 023361 (2020).
  * Lee  _et al._ [2020] H.-Y. Lee, N. Kawashima, and Y. B. Kim, Tensor network wave function of $S=1$ Kitaev spin liquids, Phys. Rev. Res. 2, 033318 (2020).
  * Khait  _et al._ [2021] I. Khait, P. P. Stavropoulos, H.-Y. Kee, and Y. B. Kim, Characterizing spin-one Kitaev quantum spin liquids, Phys. Rev. Res. 3, 013160 (2021).
  * Fukui  _et al._ [2022] K. Fukui, Y. Kato, J. Nasu, and Y. Motome, Ground-state phase diagram of spin-$S$ Kitaev-Heisenberg models, Phys. Rev. B 106, 174416 (2022).
  * Rayyan  _et al._ [2023] A. Rayyan, D. Churchill, and H.-Y. Kee, Field-induced Kitaev multipolar liquid in spin-orbit coupled ${d}^{2}$ honeycomb Mott insulators, Phys. Rev. B 107, L020408 (2023).
  * Ma [2023] H. Ma, ${\mathbb{Z}}_{2}$ Spin Liquids in the Higher Spin-$S$ Kitaev Honeycomb Model: An Exact Deconfined ${\mathbb{Z}}_{2}$ Gauge Structure in a Nonintegrable Model, Phys. Rev. Lett. 130, 156701 (2023).
  * Liu  _et al._ [2023] R. Liu, H. T. Lam, H. Ma, and L. Zou, Symmetries and anomalies of Kitaev spin-$S$ models (2023), arXiv:2310.16839 [cond-mat.str-el] .
  * Jin  _et al._ [2022] H.-K. Jin, W. M. H. Natori, F. Pollmann, and J. Knolle, Unveiling the $S$=3/2 Kitaev honeycomb spin liquids, Nature Communications 13, 3813 (2022).
  * de Carvalho  _et al._ [2023] V. S. de Carvalho, H. Freire, and R. G. Pereira, Multipolar spin liquid in an exactly solvable model for ${j}_{\mathrm{eff}}$ = $\frac{3}{2}$ moments, Phys. Rev. B 108, 094418 (2023).
  * Stavropoulos  _et al._ [2019] P. P. Stavropoulos, D. Pereira, and H.-Y. Kee, Microscopic Mechanism for a Higher-Spin Kitaev Model, Phys. Rev. Lett. 123, 037203 (2019).
  * Scheie  _et al._ [2019] A. Scheie, K. Ross, P. P. Stavropoulos, E. Seibel, J. A. Rodriguez-Rivera, J. A. Tang, Y. Li, H.-Y. Kee, R. J. Cava, and C. Broholm, Counterrotating magnetic order in the honeycomb layers of ${\mathrm{NaNi}}_{2}{\mathrm{BiO}}_{6-\delta}$, Phys. Rev. B 100, 214421 (2019).
  * Taddei  _et al._ [2023] K. M. Taddei, V. O. Garlea, A. M. Samarakoon, L. D. Sanjeewa, J. Xing, T. W. Heitmann, C. dela Cruz, A. S. Sefat, and D. Parker, Zigzag magnetic order and possible Kitaev interactions in the spin-1 honeycomb lattice ${\mathrm{KNiAsO}}_{4}$, Phys. Rev. Res. 5, 013022 (2023).
  * Pohle  _et al._ [2023] R. Pohle, N. Shannon, and Y. Motome, Spin nematics meet spin liquids: Exotic quantum phases in the spin-1 bilinear-biquadratic model with Kitaev interactions, Phys. Rev. B 107, L140403 (2023).
  * Kitaev [2006] A. Kitaev, Anyons in an exactly solved model and beyond, Annals of Physics 321, 2 (2006), January Special Issue.
  * Perelomov [1972] A. M. Perelomov, Coherent states for arbitrary Lie group, Communications in Mathematical Physics 26, 222 (1972).
  * Gnutzmann and Kus [1998] S. Gnutzmann and M. Kus, Coherent states and the classical limit on irreducible representations, Journal of Physics A: Mathematical and General 31, 9871 (1998).
  * Nemoto [2000] K. Nemoto, Generalized coherent states for SU(n) systems, Journal of Physics A: Mathematical and General 33, 3493 (2000).
  * Bradbury  _et al._ [2018] J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, and Q. Zhang, JAX: composable transformations of Python+NumPy programs (2018).
  * Babuschkin  _et al._ [2020] I. Babuschkin, K. Baumli, A. Bell, S. Bhupatiraju, J. Bruce, P. Buchlovsky, D. Budden, T. Cai, A. Clark, I. Danihelka, C. Fantacci, J. Godwin, C. Jones, R. Hemsley, T. Hennigan, M. Hessel, S. Hou, S. Kapturowski, T. Keck, I. Kemaev, M. King, M. Kunesch, L. Martens, H. Merzic, V. Mikulik, T. Norman, J. Quan, G. Papamakarios, R. Ring, F. Ruiz, A. Sanchez, R. Schneider, E. Sezener, S. Spencer, S. Srinivasan, L. Wang, W. Stokowiec, and F. Viola, The DeepMind JAX Ecosystem (2020).
  * Zhao  _et al._ [2012] H. H. Zhao, C. Xu, Q. N. Chen, Z. C. Wei, M. P. Qin, G. M. Zhang, and T. Xiang, Plaquette order and deconfined quantum critical point in the spin-1 bilinear-biquadratic Heisenberg model on the honeycomb lattice, Phys. Rev. B 85, 134416 (2012).
  * Martin and Batista [2008] I. Martin and C. D. Batista, Itinerant Electron-Driven Chiral Magnetic Ordering and Spontaneous Quantum Hall Effect in Triangular Lattice Models, Phys. Rev. Lett. 101, 156402 (2008).
  * Akagi and Motome [2010] Y. Akagi and Y. Motome, Spin Chirality Ordering and Anomalous Hall Effect in the Ferromagnetic Kondo Lattice Model on a Triangular Lattice, Journal of the Physical Society of Japan 79, 083711 (2010).
  * Messio  _et al._ [2011] L. Messio, C. Lhuillier, and G. Misguich, Lattice symmetries and regular magnetic orders in classical frustrated antiferromagnets, Phys. Rev. B 83, 184401 (2011).
  * Niesen and Corboz [2017] I. Niesen and P. Corboz, A tensor network study of the complete ground state phase diagram of the spin-1 bilinear-biquadratic Heisenberg model on the square lattice, SciPost Phys. 3, 030 (2017).
  * Note [1] We notice several complicated phases around the two SU(3) points at $(\theta/\pi,\phi/\pi)=(0.5,0.25)$ and $(0.5,0.5)$, which will be discussed elsewhere.
  * Corboz  _et al._ [2013] P. Corboz, M. Lajkó, K. Penc, F. Mila, and A. M. Läuchli, Competing states in the SU(3) Heisenberg model on the honeycomb lattice: Plaquette valence-bond crystal versus dimerized color-ordered state, Phys. Rev. B 87, 195113 (2013).
  * Note [2] Values in parentheses represent the statistical error in the last digit, determined by the standard deviation among five independent simulation runs.
  * Samarakoon  _et al._ [2017] A. M. Samarakoon, A. Banerjee, S.-S. Zhang, Y. Kamiya, S. E. Nagler, D. A. Tennant, S.-H. Lee, and C. D. Batista, Comprehensive study of the dynamics of a classical Kitaev spin liquid, Phys. Rev. B 96, 134408 (2017).
  * Krause [1986] E. Krause,  _Taxicab Geometry: An Adventure in Non-Euclidean Geometry_, Dover Books on Mathematics Series (Dover Publications, 1986).
  * Note [3] We observed several unconventional phases near the two SU(3) points at $(\theta/\pi,\phi/\pi)=(0.5,0.25)$ and $(0.5,0.5)$, which we reserve for a more in-depth discussion in the future.
  * Kalmeyer and Laughlin [1987] V. Kalmeyer and R. B. Laughlin, Equivalence of the resonating-valence-bond and fractional quantum Hall states, Phys. Rev. Lett. 59, 2095 (1987).
  * Wen  _et al._ [1989] X. G. Wen, F. Wilczek, and A. Zee, Chiral spin states and superconductivity, Phys. Rev. B 39, 11413 (1989).
  * Schroeter  _et al._ [2007] D. F. Schroeter, E. Kapit, R. Thomale, and M. Greiter, Spin Hamiltonian for which the Chiral Spin Liquid is the Exact Ground State, Phys. Rev. Lett. 99, 097202 (2007).
  * Yao and Kivelson [2007] H. Yao and S. A. Kivelson, Exact Chiral Spin Liquid with Non-Abelian Anyons, Phys. Rev. Lett. 99, 247203 (2007).
  * Momoi  _et al._ [1997] T. Momoi, K. Kubo, and K. Niki, Possible Chiral Phase Transition in Two-Dimensional Solid ${}^{3}\mathrm{He}$, Phys. Rev. Lett. 79, 2081 (1997).
  * Kato  _et al._ [2010] Y. Kato, I. Martin, and C. D. Batista, Stability of the Spontaneous Quantum Hall State in the Triangular Kondo-Lattice Model, Phys. Rev. Lett. 105, 266405 (2010).
  * Nasu and Motome [2015] J. Nasu and Y. Motome, Thermodynamics of Chiral Spin Liquids with Abelian and Non-Abelian Anyons, Phys. Rev. Lett. 115, 087203 (2015).
  * Kato  _et al._ [2017] Y. Kato, Y. Kamiya, J. Nasu, and Y. Motome, Chiral spin liquids at finite temperature in a three-dimensional Kitaev model, Phys. Rev. B 96, 174409 (2017).
  * Mishchenko  _et al._ [2020] P. A. Mishchenko, Y. Kato, K. O’Brien, T. A. Bojesen, T. Eschmann, M. Hermanns, S. Trebst, and Y. Motome, Chiral spin liquids with crystalline ${\mathbb{Z}}_{2}$ gauge order in a three-dimensional Kitaev model, Phys. Rev. B 101, 045118 (2020).
  * Lozano-Gómez  _et al._ [2024] D. Lozano-Gómez, Y. Iqbal, and M. Vojta, A Classical Chiral Spin-Liquid from Chiral Interactions on the Pyrochlore Lattice (2024), arXiv:2403.10601 [cond-mat.str-el] .
  * Yan  _et al._ [2023a] H. Yan, O. Benton, R. Moessner, and A. H. Nevidomskyy, Classification of Classical Spin Liquids: Typology and Resulting Landscape (2023a), arXiv:2305.00155 [cond-mat.str-el] .
  * Yan  _et al._ [2023b] H. Yan, O. Benton, A. H. Nevidomskyy, and R. Moessner, Classification of Classical Spin Liquids: Detailed Formalism and Suite of Examples (2023b), arXiv:2305.19189 [cond-mat.str-el] .
  * Dahlbom  _et al._ [2023] D. Dahlbom, H. Zhang, Z. Laraib, D. M. Pajerowski, K. Barros, and C. Batista, Renormalized Classical Theory of Quantum Magnets (2023), arXiv:2304.03874 [cond-mat.str-el] .
  * Kawamura and Yamamoto [2007] H. Kawamura and A. Yamamoto, Vortex-Induced Topological Transition of the Bilinear–Biquadratic Heisenberg Antiferromagnet on the Triangular Lattice, Journal of the Physical Society of Japan 76, 073704 (2007).
  * Korshunov  _et al._ [2012] S. E. Korshunov, F. Mila, and K. Penc, Degeneracy and ordering of the noncoplanar phase of the classical bilinear-biquadratic Heisenberg model on the triangular lattice, Phys. Rev. B 85, 174420 (2012).
  * Mellado [2023] P. Mellado, Spin model for the honeycomb NiPS3, Applied Physics Letters 123, 242403 (2023).
  * Mashiko and Okubo [2024] T. Mashiko and T. Okubo, Quantum phase transition between spin liquid and spin nematics in spin-1 Kitaev honeycomb model (2024), arXiv:2403.11490 [cond-mat.stat-mech] .
  * Luo  _et al._ [2024] Q. Luo, J. Zhao, J. Li, and X. Wang, Chiral spin state and nematic ferromagnet in the spin-1 Kitaev-$\Gamma$ model (2024), arXiv:2403.08382 [cond-mat.str-el] .
  * Singhania  _et al._ [2023] A. Singhania, J. van den Brink, and S. Nishimoto, Emergence of vortex state in the $S=1$ Kitaev-Heisenberg model with single-ion anisotropy (2023), arXiv:2311.14161 [cond-mat.str-el] .
  * Läuchli  _et al._ [2006b] A. Läuchli, G. Schmid, and S. Trebst, Spin nematics correlations in bilinear-biquadratic $S=1$ spin chains, Phys. Rev. B 74, 144426 (2006b).
  * Savary [2021] L. Savary, Quantum loop states in spin-orbital models on the honeycomb lattice, Nature Communications 12, 3004 (2021).
  * Gohlke  _et al._ [2022] M. Gohlke, J. C. Pelayo, and T. Suzuki, Extended Quantum Spin Liquid with Spinon-like Excitations in an Anisotropic Kitaev-Gamma Model (2022), arXiv:2212.11000 [cond-mat.str-el] .
  * Feng  _et al._ [2023] S. Feng, A. Agarwala, and N. Trivedi, Dimensional reduction of Kitaev spin liquid at quantum criticality (2023), arXiv:2308.08116 [cond-mat.str-el] .
  * Tchernyshyov and Chern [2011] O. Tchernyshyov and G.-W. Chern, Spin-Lattice Coupling in Frustrated Antiferromagnets, in  _Introduction to Frustrated Magnetism: Materials, Experiments, Theory_, edited by C. Lacroix, P. Mendels, and F. Mila (Springer Berlin Heidelberg, Berlin, Heidelberg, 2011) pp. 269–291.
  * Yoshimori and Inagaki [1981] A. Yoshimori and S. Inagaki, Fourth Order Interaction Effects on the Antiferromagnetic Structures. II. A Phenomenological Model for ${\mathrm{NiS_{2}}}$, Journal of the Physical Society of Japan 50, 769 (1981).
  * Hoffmann and Blügel [2020] M. Hoffmann and S. Blügel, Systematic derivation of realistic spin models for beyond-Heisenberg solids, Phys. Rev. B 101, 024418 (2020).
  * Soni  _et al._ [2022] R. Soni, N. Kaushal, C. Şen, F. A. Reboredo, A. Moreo, and E. Dagotto, Estimation of biquadratic and bicubic Heisenberg effective couplings from multiorbital Hubbard models, New Journal of Physics 24, 073014 (2022).
  * Katsura  _et al._ [2010] H. Katsura, N. Nagaosa, and P. A. Lee, Theory of the Thermal Hall Effect in Quantum Magnets, Phys. Rev. Lett. 104, 066403 (2010).
  * Onose  _et al._ [2010] Y. Onose, T. Ideue, H. Katsura, Y. Shiomi, N. Nagaosa, and Y. Tokura, Observation of the Magnon Hall Effect, Science 329, 297 (2010).
  * Matsumoto and Murakami [2011] R. Matsumoto and S. Murakami, Theoretical Prediction of a Rotating Magnon Wave Packet in Ferromagnets, Phys. Rev. Lett. 106, 197202 (2011).
  * Janssen  _et al._ [2016] L. Janssen, E. C. Andrade, and M. Vojta, Honeycomb-Lattice Heisenberg-Kitaev Model in a Magnetic Field: Spin Canting, Metamagnetism, and Vortex Crystals, Phys. Rev. Lett. 117, 277202 (2016).
  * Krüger  _et al._ [2023] W. G. F. Krüger, W. Chen, X. Jin, Y. Li, and L. Janssen, Triple-q Order in ${\mathrm{Na}}_{2}{\mathrm{Co}}_{2}{\mathrm{TeO}}_{6}$ from Proximity to Hidden-SU(2)-Symmetric Point, Phys. Rev. Lett. 131, 146702 (2023).
  * Francini and Janssen [2024] N. Francini and L. Janssen, Spin vestigial orders in extended Heisenberg-Kitaev models near hidden SU(2) points: Application to ${\mathrm{Na}}_{2}{\mathrm{Co}}_{2}{\mathrm{TeO}}_{6}$, Phys. Rev. B 109, 075104 (2024).
  * Gu  _et al._ [2024] Y. Gu, X. Li, Y. Chen, K. Iida, A. Nakao, K. Munakata, V. O. Garlea, Y. Li, G. Deng, I. A. Zaliznyak, J. M. Tranquada, and Y. Li, In-plane multi-$\mathrm{q}$ magnetic ground state of ${\mathrm{Na}}_{3}{\mathrm{Co}}_{2}{\mathrm{SbO}}_{6}$, Phys. Rev. B 109, L060410 (2024).
  * Metropolis  _et al._ [1953] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equation of State Calculations by Fast Computing Machines, The Journal of Chemical Physics 21, 1087 (1953).
  * Swendsen and Wang [1986] R. H. Swendsen and J.-S. Wang, Replica Monte Carlo Simulation of Spin-Glasses, Phys. Rev. Lett. 57, 2607 (1986).
  * Hukushima and Nemoto [1996] K. Hukushima and K. Nemoto, Exchange Monte Carlo Method and Application to Spin Glass Simulations, Journal of the Physical Society of Japan 65, 1604 (1996).
  * Earl and Deem [2005] D. J. Earl and M. W. Deem, Parallel tempering: Theory, applications, and new perspectives, Phys. Chem. Chem. Phys. 7, 3910 (2005).
  * Hairer  _et al._ [1993] E. Hairer, G. Wanner, and S. P. Nørsett,  _Solving Ordinary Differential Equations I – Nonstiff Problems_ (Springer Berlin Heidelberg, 1993).
  * Press  _et al._ [2007] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery,  _Numerical Recipes 3rd Edition: The Art of Scientific Computing_ , 3rd ed. (Cambridge University Press, New York, NY, USA, 2007).
  * Frigo and Johnson [2005] M. Frigo and S. G. Johnson, The Design and Implementation of FFTW3, Proceedings of the IEEE 93, 216 (2005), special issue on “Program Generation, Optimization, and Platform Adaptation”.
  * Arfken and Weber [1995] G. B. Arfken and H. J. Weber,  _Mathematical Methods for Physicists – International Edition_ , 4th ed. (Academic Press, INC, 1995).
  * Zhang  _et al._ [2019] S. Zhang, H. J. Changlani, K. W. Plumb, O. Tchernyshyov, and R. Moessner, Dynamical Structure Factor of the Three-Dimensional Quantum Spin Liquid Candidate ${\mathrm{NaCaNi}}_{2}{\mathrm{F}}_{7}$, Phys. Rev. Lett. 122, 167203 (2019).
  * Price and Perkins [2012] C. C. Price and N. B. Perkins, Critical Properties of the Kitaev-Heisenberg Model, Phys. Rev. Lett. 109, 187201 (2012).
  * Price and Perkins [2013] C. Price and N. B. Perkins, Finite-temperature phase diagram of the classical Kitaev-Heisenberg model, Phys. Rev. B 88, 024410 (2013).


# Threshold Upsilon-meson Photoproduction at EIC and EicC

Igor I. Strakovsky Corresponding author<EMAIL_ADDRESS>Institute for Nuclear
Studies, Department of Physics, The George Washington University, Washington,
DC 20052, USA  William J. Briscoe Institute for Nuclear Studies, Department of
Physics, The George Washington University, Washington, DC 20052, USA  Lubomir
Pentchev Thomas Jefferson National Accelerator Facility, Newport News,
Virginia 23606, USA  Axel Schmidt Institute for Nuclear Studies, Department of
Physics, The George Washington University, Washington, DC 20052, USA

###### Abstract

High-accuracy $\Upsilon$-meson photoproduction data from EIC and EicC
experiments will allow the measurement of the near-threshold total cross
section of the reaction $\gamma p\to\Upsilon p$, from which the absolute value
of the $\Upsilon p$ scattering length, $|\alpha_{\Upsilon p}|$, can be
extracted using a Vector-Meson Dominance model. For this evaluation, we used
$\Upsilon$-meson photoproduction quasi-data from the QCD approach (the
production amplitude can be factorized in terms of gluonic generalized parton
distributions and the quarkonium distribution amplitude). A comparative
analysis of $|\alpha_{\Upsilon p}|$ with the recently determined scattering
lengths for $\omega p$, $\phi p$, and $J/\psi p$ using the A2, CLAS, and GlueX
experimental data are performed. The role of the “young” vector-meson effect
is evaluated.

##  I Introduction

In 1977, the observation of an enhancement at 9.5 GeV in the di-muon mass
spectrum produced in 400-GeV proton-nucleus collisions at Fermi Lab resulted
in a new vector-meson ($V$) Herb:1977ek , then called $\Upsilon(1S)(9460)$.

Compared to other mesons, vector-mesons can be measured to very high
precision. This stems from the fact that vector-mesons have the same quantum
numbers as the photon $I^{G}(J^{PC})=0^{-}(1^{-~{}-})$.

The photoproduction of $J/\psi(1S)$ and $\Upsilon(1S)$ are among the key
reactions that will be measured at the electron-ion colliders EIC hosted by
the Brookhaven National Laboratory (BNL) AbdulKhalek:2021gbh  and EicC at the
High Intensity heavy-ion Accelerator Facility (HIAF) in China Anderle:2021wcy
. Charmonium and bottomonium are important not only for understanding the
interaction mechanisms of the photoproduction of the heavy vector-mesons, but
also for probing the gluonic properties of the nucleon. The large statistics
of the exclusive $J/\psi(1S)$ and $\Upsilon(1S)$ production data at the hard
scale are very helpful in extracting the generalized parton distribution (GPD)
of the gluon. These measurements will advance our understanding of QCD which
governs the properties of hadrons and the interactions involving hadrons.

Exclusive vector meson photoproduction can also shed light on the bound meson-
nucleon system, since the generated charmonium and bottomonium interact with
an intact nucleon. We point out that the bottomonium, $\Upsilon(1S)$, can be
measured using real photons at EIC and EicC, where the quality of the expected
data near-threshold will give access to a variety of interesting physics
aspects, e.g., trace anomaly, pentaquarks, cusp effects, vector-meson-nucleon
scattering length, and so on. The main object of this note is to estimate the
magnitude of the absolute value of the $\Upsilon p$ scattering length,
$|\alpha_{\Upsilon p}|$, using quasi data generated from the QCD model of Ref.
Guo:2021ibg , and compare it with the results for the other vector mesons. Our
analysis is based also on the Vector-Meson Dominance (VMD) model
GellMann:1961tg ; Sakurai:1969  relying on the transparent current-field
identities of Kroll, Lee, and Zumino Kroll:1967it . The VMD model can be used
for a variety of qualitative estimates of observables in vector-meson
photoproduction Lutz:2001mi ; Titov:2007xb  at least as the first step towards
their more extended theoretical studies. The use of the VMD model in case of
the $J/\psi$ and $\Upsilon$ requires special attention due to the heavy mass
of these vector-mesons. For the critical review of the VMD model, we refer to
papers of Boreskov and Ioffe Boreskov:1976dj  and Kopeliovich and co-workers
Kopeliovich:2017jpy  and references therein. Recently, in Ref. Xu:2021mju ,
the VMD approach has been examined based on Dyson-Schwinger equations (DSE).

On the basis of recent threshold measurements of the photoproduction of three
vector mesons off the proton by the A2 (MAMI), CLAS (JLab), and GlueX (JLab)
Collaborations, one can determine vector-meson–proton scattering lengths (SLs)
using the VMD model Strakovsky:2014wja ; Strakovsky:2020uqs ;
Strakovsky:2019bev . This results in

$|\alpha_{J/\psi p}|~{}\ll~{}|\alpha_{\phi p}|~{}\ll~{}|\alpha_{\omega p}|,$
(1)

which indicates that the proton is more transparent for the $\phi$-meson
compared to the $\omega$-meson, and at the same time it is much less
transparent for the $\phi$-meson than for the $J/\psi$-meson. Due to the small
size of “young” vector mesons relative to those that have had time to fully
form, scattering lengths determined phenomenologically in the near threshold
photoproduction are smaller. Recall that when the photon produces a vector
meson, $V$, it first creates a $q\bar{q}$ pair in point-like configuration.
Near the threshold, this pair lacks sufficient time to form the complete wave
function of the vector meson; that is, the proton interacts with the “young”
(undressed) vector meson whose size is smaller than that of the “old” one
participating in the elastic $Vp\to Vp$ scattering. Therefore, one observes
stronger suppression for the vector-meson–proton interaction Feinberg:1980yu .

##  II Exclusive Vector-Meson Photoproduction at the EIC

Exclusive vector-meson photoproduction is one of the key physics measurements
for the EIC as discussed in the Yellow Report AbdulKhalek:2021gbh . The
proposed design for EIC is to collide electrons with energy of $E_{e}=5$–$18$
GeV and protons with energy of $T_{p}=41$–$275$ GeV. The proposed EIC detector
has a number of specific features that will enable photo-production
measurements, and to ensure exclusivity. First, the energy of quasi-real
photons will be determined by tagger detectors in the so-called “far-backward”
region, i.e., in the direction of the electron beam. Tagger detectors placed
approximately 24 m and 37 m from the interaction point can cover a low-$Q^{2}$
acceptance better than $10^{-7}$ GeV${}^{2}/c^{2}$. Produced $\Upsilon(1S)$
mesons will be reconstructed from their leptonic decays, e.g., to an
$e^{+}e^{-}$ pair. The proposed EIC detector aims for momentum resolution
sufficient to cleanly separate the $\Upsilon$-states, a $\Delta p/p$ of better
than 1% in the 4–10 GeV/$c$ momentum range. Finally, exclusivity can be
ensured by the detection of the final state proton, which, at threshold, will
largely travel to the so-called “far-forward” region of the detector. The far-
forward region, covering angles within approximately 13 mrad from the proton
beam line, will be instrumented with a series of tracking detectors including
Roman Pots, to track charged particles with slightly different magnetic
rigidities than beam protons. Thus threshold $\Upsilon$ production will
require the combination of the far-backward, far-forward and main EIC
detectors in order to reconstruct the full event and ensure exlusivity.

##  III Scattering Length for Upsilon - Proton

The total cross section of a binary reaction $ab\to cd$ with particle masses
$m_{a}+M_{b}<m_{c}+M_{d}$ can be written as

$\sigma_{t}=\frac{q}{k}\cdot F(q,s),$ (2)

where $s$ is the square of the total center-of-mass energy, $k$ is center-of-
mass momentum of $a$ (and $b$), and $q$ is the center-of-mass momentum of $c$
(and $d$). In photoproduction, i.e., when $m_{a}=0$,
$k=(s-M_{b}^{2})/(2\cdot\sqrt{s})$. Vector-meson kinematical parameters for
the vector-meson photoproduction off the proton, i.e., $\gamma p\to Vp$, are
given in Table 1 and Figure 1, where $m_{V}$ is the vector-meson mass,
$s_{thr}$ is the value of $s$ at threshold, $k_{thr}$ is the value of $k$ at
threshold, and $E_{thr}$ is the photon energy at threshold in the frame where
the proton is initially at rest.

The factor $F(q,s)$ in Eq. 2 is proportional to the square of the invariant
amplitude and does not vanish at threshold, i.e., when $q\to 0$ and $k\to
k_{thr}$, but instead approaches a constant value. Thus, near threshold,
$\sigma_{t}\to 0$ and is at least proportional to $q$.

Table 1: Kinematical parameters for the vector-meson photoproduction off the proton at thresholds Zyla:2020zbs . Vector- | $m_{V}$ | $\sqrt{s_{thr}}$ | $E_{thr}$ | $k_{thr}$
---|---|---|---|---
Meson | (MeV) | (MeV) | (MeV) | (MeV/$c$)
$\omega(782)$ | 782.65 | 1720.9 | 1109.1 | 604.7
$\phi(1020)$ | 1019.461 | 1957.7 | 1573.3 | 754.0
$J/\psi(1S)$ | 3096.900 | 4035.2 | 8207.8 | 1908.5
$\Upsilon(1S)$ | 9460.30 | 10398.6 | 57152.9 | 5156.9
Figure 1: Thresholds of meson photoproduction off the proton. Blue vertical
lines are for vector-mesons including charmonium and b-quarkonium.

Traditionally, the $\sigma_{t}$ behavior of a near-threshold binary inelastic
reaction is described as a series of odd powers in $q$ (see, for instance,
Ref. Strakovsky:2014wja ). In the energy range under study, we use:

$\sigma_{t}(q)=b_{1}\cdot q+b_{3}\cdot q^{3}+b_{5}\cdot q^{5},$ (3)

which assumes contributions from only the lowest $S$-, $P$-, and $D$-waves.
Very close to threshold, the higher-order terms can be neglected and the
linear term is determined by the $S$-wave only with a total spin of $1/2$
and/or $3/2$.

For the evaluation of the absolute value of the vector-meson–proton SL, we
apply the commonly used and effective VMD approach (Fig. 2), which links the
near-threshold cross sections of the vector-meson photoproduction, $\gamma
p\to Vp$, and the elastic scattering, $Vp\to Vp$, processes via Titov:2007xb :

$\displaystyle\frac{d\sigma^{\gamma p\to Vp}}{d\Omega}|_{\rm thr}$
$\displaystyle=$ $\displaystyle\frac{q}{k}\cdot\frac{1}{64\pi}\cdot|T^{\gamma
p\to Vp}|^{2}$ (4) $\displaystyle=$
$\displaystyle\frac{q}{k}\cdot\frac{\pi\alpha}{g_{V}^{2}}\cdot\frac{d\sigma^{Vp\to
Vp}}{d\Omega}|_{\rm thr}$ $\displaystyle=$
$\displaystyle\frac{q}{k}\cdot\frac{\pi\alpha}{g_{V}^{2}}\cdot|\alpha_{Vp}|^{2},$

where $T^{\gamma p\to Vp}$ is the invariant amplitude of the vector-meson
photoproduction, $\alpha$ is the fine-structure constant, and $g_{V}$ is the
VMD coupling constant, related to the vector-meson electromagnetic (EM) decay
width $\Gamma_{V\to e^{+}e^{-}}$

$g_{V}^{2}=\frac{\pi\cdot\alpha^{2}\cdot m_{V}}{3\cdot\Gamma(V\to
e^{+}e^{-})},$ (5)

where $m_{V}$ is the vector-meson mass.

Figure 2: Schematic diagrams of vector-meson photoproduction (left) and the
VMD model (right) in the energy region at threshold experiments.

Combining Eq. (3) (which is also valid for $\omega$\- $\phi$-, and
$J/\psi$-photoproduction Strakovsky:2014wja ; Strakovsky:2019bev ;
Strakovsky:2020uqs ) and Eq. (4) with Eq. (5), one can express the absolute
value of the SL as a product of the pure EM, VMD-motivated kinematic factor

$\displaystyle B_{V}^{2}=\frac{{\alpha\cdot m_{V}\cdot
k}}{{12\pi\cdot\Gamma(V\to e^{+}e^{-})}}$ (6)

and the factor

$\displaystyle h_{Vp}=\sqrt{b_{1}}$ (7)

that is determined by an interplay of strong (hadronic) and EM dynamics as

$\displaystyle|\alpha_{Vp}|=B_{V}\cdot h_{Vp}.$ (8)

Numerical values of both $g_{V}$ and $B_{V}$ are given in Table 2. Let us note
that these EM quantities $B_{V}$ for each vector-meson are close to each other
except for $\Upsilon$.

Table 2: Vector-meson EM properties. The decay $\Gamma(V\to e^{+}e^{-})$ from PDG2020 Zyla:2020zbs  (2nd column), $g_{V}$ was calculated using Eq. (5) (3rd column) and $B_{V}$ using Eq. (6) (4th column). Vector- | $\Gamma(V\to e^{+}e^{-})$ | $g_{V}$ | $B_{V}$
---|---|---|---
Meson | (keV) |  | (MeV1/2)
$\omega(782)$ | 0.60$\pm$0.02 | 8.53$\pm$0.14 | 390.49$\pm$6.35
$\phi(1020)$ | 1.27$\pm$0.04 | 6.69$\pm$0.10 | 342.50$\pm$5.27
$J/\psi(1S)$ | 5.53$\pm$0.10 | 5.59$\pm$0.05 | 454.92$\pm$4.06
$\Upsilon(1S)$ | 1.340$\pm$0.018 | 19.85$\pm$1.21 | 2654.96$\pm$162.15

Figure 3 shows the fit of the QCD model of Ref. Guo:2021ibg  to the GlueX
$J/\psi$ photoproduction data Ali:2019lzf  and the prediction of this model
for the $\Upsilon$ photoproduction total cross section. Our phenomenological
fit of the GlueX data using Eq. (3) is shown, as well. This is a rather
general model assuming factorization in terms of gluonic Generalized Parton
Distributions (GPD) and quarkonium wave function on one side and hard quark-
gluon interaction on the other side. This work extends the validity of the
factorization, as studied previously in Ref. Ivanov:2004vd  for high energies,
down to the threshold region in leading order and in the case of a heavy quark
mass.

Figure 3: The total $\gamma p\to J/\psi p$ and $\gamma p\to\Upsilon p$ cross
sections $\sigma_{t}$. GlueX threshold $J/\psi$ photoproduction data
Ali:2019lzf  shown by blue filled circles and the best-fit results are from
Ref. Strakovsky:2019bev  and shown by red solid curve. Theoretical fit of the
GlueX data from Ref. Guo:2021ibg  is shown by yellow shaded band at 95% C.L.
Theoretical predictions for $\Upsilon$ photoproduction Guo:2021ibg  are shown
by magenta shaded band.

We have generated quasi-data (Fig.4) for the $\Upsilon$ cross sections from
this model Guo:2021ibg  and fit it with Eq. (3) in order to extrapolate the
data to the threshold. The exact $q_{min}$ attainable in an $ep$ collider
experiment will depend heavily on the exact placement of the low-$Q^{2}$
tagging detectors. For the purposes of this note, we assume that the detector
will have comparable coverage to the EIC Yellow Report detector
AbdulKhalek:2021gbh , where the backward calorimeter coverage down to a
pseudo-rapidity of $-4$ would allow $q_{min}$ for $\Upsilon$ photoproduction
to be as low as $\approx 500$ MeV/c. Further optimization of the low-$Q^{2}$
taggers may allow an even smaller $q_{min}$ to be achieved. Gryniuk and co-
workers assumed a total integrated luminosity of 100 $fb^{-1}$ for the
$\Upsilon$ photoproduction at EIC, which corresponds to 116 days of the beam
with the $10^{34}~{}cm^{-2}~{}s^{-1}$, for Monte Carlo calculations
Gryniuk:2020mlh . As Guo and co-workers Guo:2021ibg  pointed out, the large
mass of the $\Upsilon$-meson implies that the calculations in the heavy meson
limit works better for the $\Upsilon$ photoproduction, as there are fewer
theoretical uncertainties from higher-order corrections.

Figure 4: The total $\gamma p\to J/\psi p$ cross sections $\sigma_{t}$. Quasi-data from Ref. Guo:2021ibg  ($q_{min}$ = 521 MeV/$c$) shown by yellow filled stars The phenomenological best fit for quasi cross section Guo:2021ibg  is shown by red solid curve. The uncertainties for quasi-data were taken from Ref. Guo:2021ibg .  Table 3: The 2nd column showed the minimal momentum $q_{min}$ for vector-mesons in photoproduction experiments and the source of data. The linear term (3rd column) of the best-fit of the total cross section data using Eq. (3). The errors represent the total uncertainties (summing statistical and systematic uncertainties in quadrature. The 4th column showed vector-meson–proton SLs. Vector- | $q_{min}$ | $b_{1}$ | $|\alpha_{Vp}|$
---|---|---|---
Meson | (MeV/$c$) | ($\mu b$/MeV/$c$) | (fm)
$\omega(782)$ | 49 Strakovsky:2014wja  | (0.44$\pm$0.01)$\times 10^{-1}$ Strakovsky:2014wja  | 0.82$\pm$0.03 Strakovsky:2014wja
$\phi(1020)$ | 216 Dey:2014tfa  | (0.34$\pm$0.12)$\times 10^{-3}$ Strakovsky:2020uqs  | 0.063$\pm$0.010 Strakovsky:2020uqs
$J/\psi(1S)$ | 230 Ali:2019lzf  | (0.46$\pm$0.16)$\times 10^{-6}$ Strakovsky:2019bev  | (3.08$\pm$0.55)$\times 10^{-3}$ Strakovsky:2019bev
$\Upsilon(1S)$ | 521 Guo:2021ibg  | (0.37$\pm$0.04)$\times 10^{-9}$ | (0.51$\pm$0.03)$\times 10^{-3}$

Figure 5 illustrates the dramatic differences in the hadronic factors
$h_{Vp}=\sqrt{b_{1}}$ (see Table 3), as the slopes ($b_{1}$ from Eq. (3)) of
the total cross sections at threshold as a function of $q$ vary significantly
from $\omega$ to $J/\psi$ and now to $\Upsilon$.

Figure 5: The total $\gamma p\to Vp$ cross section $\sigma_{t}$ derived from
the A2 (magenta open circles) Strakovsky:2014wja , CLAS (blue filled circles)
Strakovsky:2020uqs , and GlueX (red filled triangles) Strakovsky:2019bev ,
data, and EIC/EicC (yellow filled stars) quasi-data is shown as a function of
the center-of-mass momentum $q$ of the final-state particles. The vertical
(horizontal) error bars represent the total uncertainties of the data summing
statistical and systematic uncertainties in quadrature (energy binning). Solid
curves are the fit of the data with Eq. (3).

Therefore, such a big difference in SLs of the vector-meson–proton systems is
determined mainly by the hadronic factor $h_{Vp}$, and reflects a strong
weakening of the interaction in the $\bar{b}b-p$ and $\bar{c}c-p$ systems
compared to that of the light $\bar{q}q-p$ ($q=u,d$) configurations. The
interaction in the $\bar{s}s-p$ has an intermediate strength that is
manifested in an intermediate value of the $\phi p$ SL.

The corresponding results for the scattering lengths are shown in Fig. 6 as a
function of the inverse vector-meson mass. Starting from the $\phi$ meson and
for higher masses, the scattering lengths are significantly smaller than the
typical hadron size of 1 fm, indicating increasing transparency of the proton
for these mesons. Moreover, our analysis shows almost linear exponential
increase $|\alpha_{Vp}|\propto\exp(1/m_{V})$ with increasing $1/m_{V}$.
Actually, $p\to V$ coupling is proportional to the strong coupling
$\alpha_{s}$ and the separation of the corresponding quarks. This separation
(in zero approximation) is proportional to $1/m_{V}$, which may explain the
above relation.

Figure 6: Comparison of the $|\alpha_{Vp}|$ SLs estimated from vector-meson
photoproduction at threshold vs. the inverse mass of the vector mesons. The
magenta open circle shows the analysis of the A2 $\omega$-meson data
Strakovsky:2014wja , the blue filled circle shows the analysis of the CLAS
$\phi$-meson data Strakovsky:2020uqs , the red filled triangle shows the
analysis of the GlueX $J/\psi$-meson data Strakovsky:2019bev , and the yellow
filled star shows the analysis of the EIC and EicC $\Upsilon$-meson quasi-data
Guo:2021ibg . The green dashed line is hypotetical.

##  IV Discussions and Outlook

In summary, one can extend the relationship (1) including the result for
$\Upsilon$ (Table 3):

$|\alpha_{\Upsilon p}|\ll|\alpha_{J/\psi p}|\ll|\alpha_{\phi
p}|\ll|\alpha_{\omega p}|.$ (9)

The values of $|\alpha_{Vp}|$ for the heavy vector-mesons, as determined in
the recent papers Strakovsky:2014wja ; Strakovsky:2020uqs ; Strakovsky:2019bev
using the VMD model, are smaller than most of the theoretical predictions; see
references in Refs. Strakovsky:2014wja ; Strakovsky:2020uqs ;
Strakovsky:2019bev . As for the $\Upsilon$ case, we are not aware of any
theoretical predictions for the $\Upsilon p$ scattering length except, related
to that, a non-relativistic potential result for the radius of the bottominium
which is $r_{\Upsilon}=(0.14\pm 0.0014$) fm Satz:2006kba . The same approach
gives a radius of the charmonium of $r_{J/\psi}=(0.25\pm 0.0025$) fm. Gryniuk
and co-workers reported recently $|\alpha_{\Upsilon p}|=(0.066\pm 0.001$) fm
and $|\alpha_{\Upsilon p}|=(0.016\pm 0.001$) fm Gryniuk:2020mlh  for two
different subtraction constants. The latter results are based on the
dispersive relations, taking some assumptions from their $J/\psi$
photoproduction model Gryniuk2016 , and extrapolating the high-energy
measurements at $W\approx 100$ GeV down to the threshold.

The smallness of the scattering lengths, as extracted with the help of the VMD
model, can be related to the “young age” of the vector mesons participating in
the interaction with the proton as introduced by Feinberg Feinberg:1980yu .
For more quantitative estimate of the theoretical uncertainty related to the
VMD model, we refer to the paper by Boreskov and Ioffe Boreskov:1976dj . They
evaluated the cross section of $J/\psi$ photoproduction in a peripheral model
and found a strong energy dependence because the non-diagonal process $\gamma
p\to J/\psi p$ must have larger transfer momenta versus the elastic scattering
$J/\psi p\to J/\psi p$. This result is in a violation of VMD by a factor of 5
or so. In the case of the $\Upsilon$ meson, we expect the disagreement to be
even worse. Boreskov and co-workers showed that a fluctuation of a photon into
open charm particles is preferable than into charmonium Boreskov:1992ur .
Later on, it was shown by Kopeliovich and co-workers that the open-charm cross
section is larger than the charmonium one by a factor of 10 or more
Hufner:2000jb ; Floter:2007xv . In addition, in Eq. (4), we did not include a
factor introduced in the VMD model in Ref. Kubarovsky:2015aaa , which takes
into account the difference between polarization degrees of freedom in the
$\gamma p\to J/\psi p$ and $pJ/\psi\to pJ/\psi$ reactions. Such a factor that
equals 2/3 at threshold for the $S$-wave has not been used in the previous
analysis of the scattering lengths; we consider it as a systematic uncertainty
related to the VMD model Strakovsky:2019bev .

In the recent work of Ref. Xu:2021mju  the effect of the VMD assumption was
studied in the formalism of the Dyson-Schwinger equations which one can
consider as an alternative interpretation of the “young age” effect in another
(more formal) language. Their result shows much dramatic effect, up to a
factor of 50 overestimation of the cross sections when using VMD for the two
heaviest vector mesons. Nevertheless, this translates into a factor of 7 for
the scattering lengths and the overall dependence in Fig. 6 remains similar.

Present and future experiments JLab, EIC, and EicC that are aimed to measure
charmonium and bottomonium production on the proton and nuclei will allow
further studies of $J/\psi N$ and $\Upsilon N$ interactions and will allow
access to a variety of other interesting physics aspects that are present in
the near-threshold region. Further studies on both nucleons and nuclei in
heavy vector-meson photo- and electro-production will significantly extend our
knowledge of the gluonic structure of the nuclear matter.

##  V Acknowledgments

We thank Misha Ryskin for useful remarks and continuous interest in the paper,
Yulia Furletova for the detector design for EIC, and Yuxun Guo for predictions
of $\Upsilon$ photoproduction cross sections. This work was supported in part
by the U. S. Department of Energy, Office of Science, Office of Nuclear
Physics under Awards No. DE–SC0016583 and Contract No. DE–AC05–06OR23177.

## References

  * (1) S. W. Herb et al., “Observation of a dimuon resonance at 9.5-GeV in 400-GeV proton-nucleus collisions,” Phys. Rev. Lett. 39, 252 (1977).
  * (2) R. A. Khalek et al. “Science requirements and detector concepts for the Electron-Ion Collider: EIC Yellow Report,” [arXiv:2103.05419 [physics.ins-det]].
  * (3) D. P. Anderle et al. “Electron-Ion Collider in China,” Front. Phys. (Beijing) 16, 64701 (2021).
  * (4) Y. Guo, X. Ji, and Y. Liu, “QCD analysis of near-threshold photon-proton production of heavy quarkonium,” Phys. Rev. D 103, 096010 (2021).
  * (5) M. Gell-Mann and F. Zachariasen, “Form-factors and vector mesons,” Phys. Rev. 124, 953 (1961).
  * (6) J. J. Sakurai, Currents and Mesons, (The University of Chicago Press, Chicago, 1969).
  * (7) N. M. Kroll, T. D. Lee, and B. Zumino, “Neutral vector mesons and the hadronic electromagnetic current,” Phys. Rev. 157, 1376 (1967).
  * (8) M. F. M. Lutz, G. Wolf, and B. Friman, “Scattering of vector mesons off nucleons,” Nucl. Phys. A 706, 431 (2002); Erratum: [Nucl. Phys. A 765, 431 (2006)]; Erratum: [Nucl. Phys. A 765, 495 (2006)].
  * (9) A. I. Titov, T. Nakano, S. Date, and Y. Ohashi, “Comments on differential cross-section of phi-meson photoproduction at threshold,” Phys. Rev. C 76, 048202 (2007).
  * (10) K. G. Boreskov and B. L. Ioffe, “$J/\psi$ meson photoproduction in the peripheral model,” Sov. J. Nucl. Phys. 25, 331 (1977) [Yad. Fiz. 25, 806 (1977)].
  * (11) B. Z. Kopeliovich, I. Schmidt, and M. Siddikov, “Suppression versus enhancement of heavy quarkonia in pA collisions,” Phys. Rev. C 95, 065203 (2017).
  * (12) Y. Z. Xu, S. Chen, Z. Q. Yao, D. Binosi, Z. F. Cui, and C. D. Roberts, “Vector-meson production and vector meson dominance,” [arXiv:2107.03488 [hep-ph]].
  * (13) I. I. Strakovsky et al. [A2 Collaboration at MAMI], “Photoproduction of the $\omega$ meson on the proton near threshold,” Phys. Rev. C 91, 045207 (2015).
  * (14) I. Strakovsky, L. Pentchev, and A. Titov, “Comparative analysis of $\omega p$, $\phi p$, and $J/\psi p$ scattering lengths from A2, CLAS, and GlueX threshold measurements,” Phys. Rev. C 101, 045201 (2020).
  * (15) I. Strakovsky, D. Epifanov, and L. Pentchev, “J/$\psi$-p scattering length from GlueX threshold measurements,” Phys. Rev. C 101, 042201 (2020).
  * (16) E. L. Feinberg, “Hadron clusters and half dressed particles in quantum field theory,” Usp. Fiz. Nauk 132, 225 (1980) [Sov. Phys. Usp. 23, 629 (1980)].
  * (17) P. A. Zyla et al. [Particle Data Group], “Review of Particle Physics,” PTEP 2020, 083C01 (2020).
  * (18) A. Ali et al. [GlueX Collaboration], “First measurement of near-threshold $J/\psi$ exclusive photoproduction off the proton,” Phys. Rev. Lett. 123, 072001 (2019).
  * (19) D. Y. Ivanov, A. Schafer, L. Szymanowski, and G. Krasnikov, “Exclusive photoproduction of a heavy vector meson in QCD,” Eur. Phys. J. C 34, 297 (2004) [Erratum: Eur. Phys. J. C 75, 75 (2015)].
  * (20) O. Gryniuk, S. Joosten, Z. E. Meziani, and M. Vanderhaeghen, “$\Upsilon$ photoproduction on the proton at the Electron-Ion Collider,” Phys. Rev. D 102, 014016 (2020).
  * (21) B. Dey et al. [CLAS Collaboration], “Data analysis techniques, differential cross sections, and spin density matrix elements for the reaction $\gamma p\rightarrow\phi p$,” Phys. Rev. C 89, 055208 (2014).
  * (22) S. Brodsky, E. Chudakov, P. Hoyer, and J. Laget, “Photoproduction of charm near threshold,” Phys. Lett. B 498, 23 (2001).
  * (23) H. Satz, “Quarkonium binding and dissociation: The spectral analysis of the QGP,” Nucl. Phys. A 783, 249 (2007).
  * (24) K. Boreskov, A. Capella, A. Kaidalov, and J. Tran Thanh Van, “Heavy quark and lepton pair production on nuclei,” Phys. Rev. D 47, 919 (1993).
  * (25) J. Hufner, Y. Ivanov, B. Kopeliovich, and A. Tarasov, “Photoproduction of charmonia and total charmonium proton cross-sections,” Phys. Rev. D 62, 094022 (2000).
  * (26) B. Floter, B. Kopeliovich, H. Pirner, and J. Raufeisen, “Charmed meson production in deep inelastic scattering,” Phys. Rev. D 76, 014009 (2007).
  * (27) V. Kubarovsky and M. B. Voloshin, “Formation of hidden-charm pentaquarks in photon-nucleon collisions,” Phys. Rev. D 92, 031502 (2015).
  * (28) O. Gryniuk and M. Vanderhaeghen, Phys. Rev. D 94, 074001 (2016).