All-Heavy Tetraquarks:
The Dynamical Diquark Model
and Other Approaches
Richard Lebed
Lomonosov Conferences
on Elementary Particle Physics
Moscow State University
August, 2021
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Although This Talk Is About Heavy
Tetraquarks in General…
• In this talk, we focus on developments to study interesting
structure observed by LHCb in the di-𝐽/𝜓 channel[Sci. Bull. 65, 1983 (2020)]
ത 𝑏ത state (→ 𝜇+ 𝜇− 𝜇+ 𝜇− ) at 18.4 GeV• After a false alarm of 𝑏𝑏𝑏
using CMS data
[http://meetings.aps.org/Meeting/APR18/Session/U09.6],• LHCb [JHEP 10, 086 (2018)] studied Υ(1𝑆) 𝜇+ 𝜇−
and found no significant structuresത 𝑐,ҧ 𝑏𝑐𝑏
• No structures reported yet in 𝑏𝑏𝑐
ҧ 𝑐,ҧ etc.

2How All-Heavy Tetraquarks Are Made• Mostly gluon-gluon fusion(but both single- and double-parton scattering important forത 𝑄ത production)
nonresonant 𝑄𝑄𝑄[R. Maciuła, W. Schäfer, A. Szczurek, Phys. Lett. B 812, 136010 (2021)]

Production in heavy-ion
collisions also possible:
J. Zhao, S. Shi, P. Zhuang[2009.10319]
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Neutral charmoniumlike sector,August 2021
X(6900)4

Charged charmoniumlike sector,August 2021

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The heavy exotics scorecard: August 2021
• 51 observed exotics– 41 in charmoniumlike sector (incl. pentaquarks)– 1 decaying to di-𝐽/𝜓
– 5 in the (much less explored) bottomonium sector– 1 with a single b quark (and an s, a u, and a d)
– 2 with a single c quark (and an s, a u, and a d)
– 1 with two c quarks• 15 established [PDG] (& none of other 36 disproved)• My naïve count estimates over 100 more exotics
are waiting to be discovered
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The Plot That Launched 100 Theory Papers
LHCb Collaboration, Sci. Bull. 65, 1983 (2020) [2006.16957]

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The Eye Immediately Notices…

X(6900)6.4-6.5 GeV~6.75 GeV

~7.2 GeV

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The Most Important Apparent Features• 𝑋(6900), the only obvious peak, lies about 700 MeV above 2𝑚𝐽/𝜓
but is likely not wider than the 𝜌 (≲ 200 MeV, & perhaps much narrower)• A 𝑐 𝑐𝑐ҧ 𝑐ҧ state, if a traditional di-meson molecule,would be bound through exchanging conventional charmonium—
which for expected ≤ 𝑂 10 MeV molecular binding energies
is very far off mass shell• Typical 𝑐 𝑐ҧ mean charge radii from potential models:
1𝑆: 0.35 fm
1𝑃: 0.63 fm
2𝑆: 0.78 fm⟹ 𝐽/𝜓 exchange in particular would be very short-ranged• So what about nontraditional di-meson molecules?
– Pomeron (multi-gluon) exchanges [C. Gong et al., 2011.11374]– Soft gluons hadronizing into light-meson exchanges (𝜋, 𝐾)[X.-K. Dong et al., 2107.03946]
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The Most Important Apparent Features• 𝐽𝑃𝐶 for 𝐽/𝜓(1−− ) identical boson pair restricted: 𝐶 = + and:0++ , 2++ (𝑆 wave),
0−+ , 1−+ , 2−+ (𝑃 wave)• 𝐽𝑃𝐶 for system made of 𝑐𝑐 and 𝑐ҧ𝑐ҧ identical fermion pairs also restricted:
ഥ, spin-1] or [color-𝟔, spin-0]
In their 𝑆 wave: [color-𝟑10

History of 𝑐𝑐𝑐ҧ 𝑐ҧ Theory Studies• 11 November 1974: Discovery of 𝐽/𝜓
[J. Aubert et al., Phys. Rev. Lett. 33, 1404 (1974);
J. Augustin et al., Phys. Rev. Lett. 33, 1406 (1974)]• First di-𝐽/𝜓 theory paper: Y. Iwasaki, Prog. Theor. Phys. 54, 492 (1975) (!)• 5 theory papers in the 1980s, 1 in the 1990s, 3 in the 2000s(“All the states are unbounded and consequently rather uninteresting”)• 2010: First physics from LHC. Very soon afterwards pointed out that:
Lots of 𝑔𝑔 ⟶ 𝐽/𝜓-𝐽/𝜓 being produced, and can be reconstructed by LHCb:[A. Berezhnoy, A. Likhoded, A. Luchinsky, and A. Novoselov,
Phys. Rev. D 84, 094023 (2011); 86, 034004 (2012)]• And then…nothing else until 2016,
and 12 papers from then until the middle of 202011

History of 𝑐𝑐𝑐ҧ 𝑐ҧ Theory Studies• 16 June 2020: CERN-LHC Seminar, Liupan An (LHCb Collaboration),“Latest Results on Exotic Hadrons at LHCb”• Then 8 more theory papers just in the following two weeks• 30 June 2020: Posting of arXiv:2006.16957,
R. Aaij et al. (LHCb Collaboration), Sci. Bull. 65, 1983 (2020),“Observation of Structure in the 𝐽/𝜓-pair Mass Spectrum”• Since then: 53 theory papers posted on arXiv discussing 𝑐 𝑐𝑐ҧ 𝑐ҧ structure(as of 21 August 2021)

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What Has Been Tried?
(with apologies to the many authors whose names are not listed here!)••••••••••••••

String junction model
Quark model, chromomagnetic interactions
Quark potential model
Chiral quark model
Diquark model
Effective theory with light-meson exchanges
Threshold effects with coupled charmonium channels
Threshold effects plus compact tetraquark
QCD sum rules
Lattice
Regge phenomenology, including Pomeron exchange
Holography
Spin-chain (Bethe Ansatz) algebraic methods
𝑋(6900) might even be a Higgs-like boson!

13The Relevant Charmonium Thresholds
𝜂𝑐 -𝜂𝑐 (2𝑆)𝐽/𝜓-𝐽/𝜓𝜒𝑐0 -𝜒𝑐0𝐽/𝜓-𝜓(2𝑆)Ξത 𝑐𝑐 -Ξ𝑐𝑐

𝜒𝑐1 -𝜒𝑐1𝐽/𝜓-𝜓(3𝑆)

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Any Solid Conclusions/Consensus?
• 𝑿(𝟔𝟗𝟎𝟎) seems to be genuine resonance, even within the presence ofmultiple threshold effects that might explain other 𝑐 𝑐𝑐ҧ 𝑐ҧ structure:[e.g., X.-K. Dong et al., Phys. Rev. Lett. 126, 132001 (2021);
Z.-H. Guo and J.A. Oller, Phys. Rev. D 103, 034024 (2020)]• Others [e.g., J.-Z. Wang, X. Liu, and T. Matsuki, Phys. Rev. D 103, L071503(2021)] suggest 𝑋(6900) itself might be generated by 𝜒𝑐0 -𝜒𝑐1 threshold• Virtually all models predict ground-state 1𝑆 resonances to be much lower
than 𝑋(6900), typically from 6.0-6.4 GeV (ever since Iwasaki [1975])• So then, is 𝑋(6900) …
a 1𝑃 state (e.g., M.-S. Liu et al., 2006.11952)
or a 2𝑆 state [e.g., J.F. Giron and RFL, Phys. Rev. D 102, 074003 (2020);
M. Karliner and J.L. Rosner, Phys. Rev. D 102, 114039 (2020)]?
(Measuring parity will answer this question)

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Any Solid Conclusions/Consensus?
• The broad structure around 𝟔𝟒𝟎𝟎-𝟔𝟓𝟎𝟎 𝐌𝐞𝐕 is about the upper limit ofwhere models predict ground-states (1𝑆) to occur[e.g., B.-C. Yang, L. Tang, and C.-F. Qiao, Eur. Phys. J. C 81, 324 (2021);
Z. Zhao et al., Phys. Rev. D 103, 116207 (2021)]• LHCb’s Model I [Sci. Bull. 65, 1983 (2020)]:
Broad structure is a superposition of (at least) two resonances• And what do we mean by “1𝑆”, which suggests a 2-body description?
Since molecules are problematic for 𝑐 𝑐𝑐ҧ 𝑐ҧ , and no good thresholds are in
the 6400-6500 MeV range, then diquark (𝑐𝑐)𝟑ഥ (𝑐ҧ𝑐)ҧ 𝟑 structure is natural• But not everyone agrees!
C. Deng, H. Chen, J. Ping, Phys. Rev. D 103, 014001 (2021) note that
6-6ത attraction stronger than 3ത -3 (despite quark repulsion in a 6 diquark!),
and find that the ground states mix both configurations,but that 3ത -3 dominates excited states
16Any Solid Conclusions/Consensus?
• The dip around 𝟔𝟕𝟓𝟎 𝐌𝐞𝐕 suggests destructive interference with 𝑋 6900
• LHCb’s Model II [Sci. Bull. 65, 1983 (2020)]:
Interference between broad 6400-6500 MeV structure and a resonance• 𝜒𝑐0 -𝜒𝑐0 threshold effect?
[e.g., M. Karliner and J.L. Rosner, Phys. Rev. D 102, 114039 (2020)]?
• If 𝑋(6900) is 2𝑆 (𝑃 = +),then 6750 MeV is where 1𝑃 states (𝑃 = −) expected[e.g., Giron & RFL]
But of course 𝑃 = + and 𝑃 = − configurations do not have interferencewith each otherAgain, determining the parity of these events is crucial

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Any Solid Conclusions/Consensus?
• LHCb [Sci. Bull. 65, 1983 (2020)] notes structure near 𝟕𝟐𝟎𝟎 𝐌𝐞𝐕• Open-flavor decays of 𝑐 𝑐𝑐ҧ 𝑐ҧ first allowed at Ξത 𝑐𝑐 -Ξ𝑐𝑐 (𝑐𝑐𝑢)(𝑐ҧ𝑐ҧ𝑢ത ) threshold,7242.4(1.0) MeV• Likely no observably narrow 𝑐 𝑐𝑐ҧ 𝑐ҧ structures above this point• Giron & RFL:
“where the color flux tube breaks in a diquark model”• J. Sonnenschein and D. Weissman, Eur. Phys. J. C 81, 1 (2021):“where new string junctions become possible in a holographic model”

18How Many States?
• If both 3ത and 6 diquarks both allowed, one finds a lot of states[M.A. Bedolla, J. Ferretti, C. Roberts, and E. Santopinto,
Eur. Phys. J. C 80, 1004 (2020)]:17 with 𝐶 = + & 𝐽 ≤ 2 predicted below Ξത 𝑐𝑐 -Ξ𝑐𝑐 threshold (see next slide)• Adopt minimal ansatz, of 3ത diquarks only: about half as many states
Take spin couplings to be large only within diquarks[Defining properties of dynamical diquark model: Giron & RFL]:All 𝑆 wave multiplets: 3 degenerate states 0++ , 2++ (and 1+− )In 𝑃 wave multiplets: 7 states (3 with 𝐶 = +),equal-spacing mass rule
if tensor couplings negligible

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Some Parting Thoughts• Desperately need 𝐽𝑃 information to disentangle spectrum• An excellent suggestion: Look at 𝐽/𝜓-𝜓(2𝑆) spectrum[e.g., J.-M. Richard, Sci. Bull. 65, 1954 (2020);
Q.-F. Cao et al., Chin. Phys. C 45, 093113 (2021)], even though its threshold
is 700 MeV higher, and 𝜓(2𝑆) production is lower than that of 𝐽/𝜓
Also note that BESIII sees different 𝑌 states via 𝐽/𝜓 or 𝜓(2𝑆) decays• 𝑔𝑔 producing 𝐽/𝜓 is 𝐶 = +; is there much 𝑔𝑔𝑔 (𝐶 = −) production?
Could find 1+− resonance via 𝐽/𝜓-𝜂𝑐 , although 𝜂𝑐 harder to reconstruct[but note B.R.(𝜂𝑐 → 𝑝𝑝)ҧ = 1.45 × 10−3 ]Alternately, 𝐽/𝜓-𝜒𝑐𝐽 also has 𝐶 = −, but less phase space (> 6512 MeV)ത 𝑏ത production!
• And don’t forget about 𝑐 𝑐𝑏ҧ 𝑏ത and 𝑏𝑏𝑏
𝑐 𝑐𝑏ҧ 𝑏ത (e.g., 𝐽/𝜓-Υ) should have more resonances:evades identical fermions constraint→ Important tests of quark flavor universality
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Backup slides

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