$-0,91$ $(40,39)$ $\frac{1}{2}(1^{+})$ 6. $K^{\ast}(1680)^{0}$ – $1718\pm 18$
$1717,73$ $-0,01$ $(41,40)$ $\frac{1}{2}(1^{-})$ $D^{0}=c\bar{u}$,
$\bar{D}^{0}=\bar{c}u$ 7. $D^{\ast}(2007)^{0}$ – $2006,85\pm 0,05$ $2023,56$
$+0,83$ $(89/2,87/2)$ $\frac{1}{2}(1^{-})$ 8. $D_{1}(2420)^{0}$ – $2422,1\pm
0,6$ $2403,74$ $-0,76$ $(97/2,95/2)$ $\frac{1}{2}(1^{+})$ 9. $D_{1}(2430)^{0}$
– $2412\pm 9$ $2403,74$ $-0,34$ $(97/2,95/2)$ $\frac{1}{2}(1^{+})$ 10.
$D^{\ast}(2600)^{0}$ – $2627\pm 10$ $2606,10$ $-0,79$ $(101/2,99/2)$
$\frac{1}{2}(1^{-})$ 11. $D^{\ast}(2760)^{0}$ – $2781\pm 22$ $2763,23$ $-0,64$
$(52,51)$ $\frac{1}{2}(1^{-})$

| State and mass | $m_{\omega}$ | Error % | $(l,\dot{l})$ |  $q\bar{q}$, $I^{G}(J^{PC})$
---|---|---|---|---|---
|  |  |  |  | $X_{1}=\bar{c}d\bar{s}u$
12. |  $X_{1}(2900)$ – $2904\pm 5$ | $2924,96$ | $+0,72$ | $(107/2,105/2)$ |  $?(1^{-})$,
|  |  |  |  | exotic state
|  |  |  |  |  $B^{0}=d\bar{b}$,
|  |  |  |  | $\bar{B}^{0}=\bar{d}b$
13. |  $B^{\ast}{}^{0}$ – $5324,71\pm 0,21$ | $5297,79$ | $-0,50$ | $(72,71)$ | $\frac{1}{2}(1^{-})$
14. |  $B_{1}(5721)^{0}$ – $5725,9^{+2,5}_{-2,7}$ | $5748,49$ | $+0,39$ | $(75,74)$ | $\frac{1}{2}(1^{+})$
|  |  |  |  |  $B^{0}_{s}=s\bar{b}$,
|  |  |  |  | $\bar{B}^{0}_{s}=\bar{s}b$
15. |  $B^{\ast}_{s}{}^{0}$ – $5415,4^{+1,8}_{-1,5}$ | $5445,98$ | $+0,56$ | $(73,72)$ | $0(1^{-})$
13. |  $B_{s1}(5830)^{0}$ – $5828,70\pm 0,20$ | $5825,40$ | $-0,05$ | $(151/2,149/2)$ | $0(1^{+})$

Table 27. Neutral mesons, spin 2, $\bb K=\bb H$
(Subspace ${\boldsymbol{\sf H}}^{(0,0,2)}_{\rm phys}(\bb H)$, vectors
$\left|\bb K,b,\ell,l-\dot{l}\right\rangle=\left|\bb H,0,0,2\right\rangle$).
State and mass $m_{\omega}$ Error % $(l,\dot{l})$ $q\bar{q}$, $I^{G}(J^{PC})$
$K^{0}=d\bar{s}$, $\bar{K}^{0}=\bar{d}s$ 1. $K^{\ast}_{2}(1430)^{0}$ –
$1427,3\pm 1,5$ $1436,16$ $+0,62$ $(38,36)$ $\frac{1}{2}(2^{+})$ 2.
$K_{2}(1580)^{0}\approx 1580$ $1593,55$ $+0,85$ $(40,38)$ $\frac{1}{2}(2^{-})$
3. $K_{2}(1770)^{0}$ – $1773\pm 8$ $1759,17$ $-0,78$ $(42,40)$
$\frac{1}{2}(2^{-})$ 4. $K_{2}(1820)^{0}$ – $1819\pm 12$ $1801,78$ $-0,94$
$(85/2,81/2)$ $\frac{1}{2}(2^{-})$ 5. $K^{\ast}_{2}(1980)^{0}$
$1994^{+60}_{-50}$ $1977,57$ $-0,82$ $(89/2,85/2)$ $\frac{1}{2}(2^{+})$ 6.
$K_{2}(2250)^{0}$ – $2247\pm 17$ $2256,57$ $+0,42$ $(95/2,91/2)$
$\frac{1}{2}(2^{-})$ $D^{0}=c\bar{u}$, $\bar{D}^{0}=\bar{c}u$ 7.
$D^{\ast}_{2}(2460)^{0}$ – $2461,1^{+0,7}_{-0,8}$ $2452,80$ $-0,34$
$(99/2,95/2)$ $\frac{1}{2}(2^{+})$ 8. $D_{2}(2740)^{0}$ – $2747\pm 6$
$2762,46$ $+0,56$ $(105/2,101/2)$ $\frac{1}{2}(2^{-})$ $B^{0}=d\bar{b}$,
$\bar{B}^{0}=\bar{d}b$ 9. $B^{\ast}_{2}(5747)^{0}$ – $5737,2\pm 0,7$ $5747,73$
$+0,18$ $(151/2,147/2)$ $\frac{1}{2}(2^{+})$ $B^{0}_{s}=s\bar{b}$,
$\bar{B}^{0}_{s}=\bar{s}b$ 10. $B^{\ast}_{s2}(5840)^{0}$ – $5839,86\pm 0,12$
$5824,63$ $-0,26$ $(153/2,149/2)$ $0(2^{+})$

Table 28. Neutral mesons, spin 3, $\bb K=\bb H$
(Subspace ${\boldsymbol{\sf H}}^{(0,0,3)}_{\rm phys}(\bb H)$, vectors
$\left|\bb K,b,\ell,l-\dot{l}\right\rangle=\left|\bb H,0,0,3\right\rangle$).
State and mass $m_{\omega}$ Error % $(l,\dot{l})$ $q\bar{q}$, $I^{G}(J^{PC})$
$K^{0}=d\bar{s}$, $\bar{K}^{0}=\bar{d}s$ 1. $K^{\ast}_{3}(1780)^{0}$ –
$1778\pm 8$ $1757,84$ $-1,13$ $(85/2,79/2)$ $\frac{1}{2}(3^{-})$ 2.
$K_{3}(2320)^{0}$ – $2324\pm 24$ $2303,09$ $-0,88$ $(97/2,91/2)$
$\frac{1}{2}(3^{+})$ $D^{0}=c\bar{u}$, $\bar{D}^{0}=\bar{c}u$ 3.
$D^{\ast}_{3}(2750)^{0}$ – $2763,1\pm 3,2$ $2761,19$ $-0,07$ $(53,50)$
$\frac{1}{2}(3^{-})$

Table 29. Neutral mesons, spin 4, $\bb K=\bb H$
(Subspace ${\boldsymbol{\sf H}}^{(0,0,4)}_{\rm phys}(\bb H)$, vectors
$\left|\bb K,b,\ell,l-\dot{l}\right\rangle=\left|\bb H,0,0,4\right\rangle$).
State and mass $m_{\omega}$ Error % $(l,\dot{l})$ $q\bar{q}$, $I^{G}(J^{PC})$
$K^{0}=d\bar{s},\bar{K}^{0}=\bar{d}s$ 1. $K^{\ast}_{4}(2045)^{0}$ –
$2048^{+8}_{-9}$ $2065,46$ $+0,85$ $(93/2,85/2)$ $\frac{1}{2}(4^{+})$ 2.
$K_{4}(2500)^{0}$ – $2247\pm 17$ $2253,51$ $+0,29$ $(97/2,89/2)$
$\frac{1}{2}(4^{-})$

Table 30. Neutral mesons, spin 5, $\bb K=\bb H$
(Subspace ${\boldsymbol{\sf H}}^{(0,0,5)}_{\rm phys}(\bb H)$, vectors
$\left|\bb K,b,\ell,l-\dot{l}\right\rangle=\left|\bb H,0,0,5\right\rangle$).
State and mass $m_{\omega}$ Error % $(l,\dot{l})$ $q\bar{q}$, $I^{G}(J^{PC})$
$K^{0}=d\bar{s},\bar{K}^{0}=\bar{d}s$ 1. $K^{\ast}_{5}(2380)^{0}$ – $2382\pm
24$ $2397,61$ $+0,65$ $(101/2,91/2)$ $\frac{1}{2}(5^{-})$

####  5.3.3 $\bb R$-subspaces/truly neutral mesons

Unlike the baryon sector, the meson sector contains truly neutral states that
make up the coherent subspaces ${\boldsymbol{\sf H}}^{(s)}_{\rm phys}(\bb R)$.
In this case, the operation of charge conjugation is reduced to an identical
transformation, which leads to the absence of anti-states. In addition,
subspaces ${\boldsymbol{\sf H}}^{(0,0,s)}_{\rm phys}(\bb R)$ contain a large
number of exotic states, the description of which goes beyond the quark model
(see Appendix B).

Table 31. Truly neutral mesons, spin 0, $\bb K=\bb R$
(Subspace ${\boldsymbol{\sf H}}^{(0,0,0)}_{\rm phys}(\bb R)$, vectors
$\left|\bb K,b,\ell,l-\dot{l}\right\rangle=\left|\bb R,0,0,0\right\rangle$).
State and mass $m_{\omega}$ Error % $(l,\dot{l})$ $q\bar{q}$, $I^{G}(J^{PC})$
$\pi^{0},a^{0}=$ $(u\bar{u}-d\bar{d})/\sqrt{2}$ 1. $\pi^{0}$ – $134,9768$
$135,16$ $+0,13$ $(11,11)$ $1^{-}(0^{-})$ 2. $a_{0}(980)^{0}$ – $980\pm 20$
$982,14$ $+0,29$ $(61/2,61/2)$ $1^{-}(0^{++})$ 3. $\pi(1300)^{0}$ – $1300\pm
100$ $1287,97$ $-0,92$ $(35,35)$ $1^{-}(0^{-+})$ 4. $a_{0}(1450)^{0}$ –
$1474\pm 19$ $1475,77$ $+0,13$ $(75/2,75/2)$ $1^{-}(0^{++})$ 5.
$a_{0}(1700)^{0}$ – $1704\pm 5$ $1717,98$ $+0,82$ $(81/2,81/2)$
$1^{-}(0^{++})$ 6. $\pi(1800)^{0}$ – $1800^{+9}_{-11}$ $1802,81$ $-0,39$
$(83/2,83/2)$ $1^{-}(0^{-+})$ 7. $a_{0}(1950)^{0}$ – $1931\pm 26$ $1933,88$
$+0,15$ $(43,43)$ $1^{-}(0^{++})$ 8. $a_{0}(2020)^{0}$ – $2025\pm 30$
$2023,81$ $-0,06$ $(44,44)$ $1^{-}(0^{++})$ 9. $\pi(2070)^{0}$ – $2070\pm 35$
$2069,55$ $-0,02$ $(89/2,89/2)$ $1^{-}(0^{-+})$ 10. $\pi(2360)^{0}$ – $2060\pm
60$ $2354,69$ $-0,22$ $(95/2,95/2)$ $1^{-}(0^{-+})$ $\eta,\eta^{\prime},f=$
$c_{1}(u\bar{u}+d\bar{d})$ $+c_{2}(s\bar{s})$ 11. $f_{0}(500)^{0}$ – $513\pm
32$ $517,39$ $+0,85$ $(22,22)$ $0^{+}(0^{++})$ 12. $\eta^{0}$ – $547,862$
$540,64$ $-1,32$ $(45/2,45/2)$ $0^{+}(0^{-+})$ 13. $\eta^{\prime}(958)^{0}$ –
$957,78\pm 0,06$ $950,71$ $-0,74$ $(30,30)$ $0^{+}(0^{-+})$ 14.
$f_{0}(980)^{0}$ – $990\pm 20$ $982,14$ $-0,79$ $(61/2,61/2)$ $0^{+}(0^{++})$
15. $\eta(1295)^{0}$ – $1294\pm 4$ $1287,97$ $-0,46$ $(35,35)$ $0^{+}(0^{-+})$
16. $f_{0}(1370)^{0}\approx 1370$ $1361,56$ $-0,61$ $(36,36)$ $0^{+}(0^{++})$
17. $\eta(1405)^{0}$ – $1408,8\pm 2$ $1399,12$ $-0,68$ $(73/2,73/2)$
$0^{+}(0^{-+})$ 18. $\eta(1475)^{0}$ – $1475\pm 4$ $1475,77$ $+0,05$
$(75/2,75/2)$ $0^{+}(0^{-+})$

| State and mass | $m_{\omega}$ | Error % | $(l,\dot{l})$ |  $q\bar{q}$, $I^{G}(J^{PC})$
---|---|---|---|---|---
19. |  $f_{0}(1500)^{0}$ – $1506\pm 6$ | $1514,86$ | $+0,59$ | $(38,38)$ | $0^{+}(0^{++})$
20. |  $f_{0}(1710)^{0}$ – $1704\pm 12$ | $1717,98$ | $+0,82$ | $(81/2,81/2)$ | $0^{+}(0^{++})$
21. |  $\eta(1760)^{0}$ – $1751\pm 15$ | $1760,14$ | $+0,52$ | $(41,41)$ | $0^{+}(0^{-+})$
22. |  $\eta(2010)^{0}$ – $2010^{+35}_{-60}$ | $2023,81$ | $+0,68$ | $(44,44)$ | $0^{+}(0^{-+})$
23. |  $f_{0}(2020)^{0}$ – $1992\pm 16$ | $1978,59$ | $-0,67$ | $(87/2,87/2)$ | $0^{+}(0^{++})$
24. | $f_{0}(2060)^{0}\approx 2050$ | $2069,55$ | $+0,95$ | $(89/2,89/2)$ | $0^{+}(0^{++})$
25. |  $\eta(2100)^{0}$ – $2050^{+30}_{-54}$ | $2069,55$ | $+0,95$ | $(89/2,89/2)$ | $0^{+}(0^{-+})$
26. |  $f_{0}(2100)^{0}$ – $2095^{+17}_{-19}$ | $2115,79$ | $+0,99$ | $(45,45)$ | $0^{+}(0^{++})$
27. |  $\eta(2190)^{0}$ – $2190\pm 50$ | $2162,55$ | $-1,25$ | $(91/2,91/2)$ | $0^{+}(0^{-+})$
28. |  $f_{0}(2200)^{0}$ – $2187\pm 14$ | $2162,55$ | $-1,12$ | $(91/2,91/2)$ | $0^{+}(0^{++})$
29. |  $\eta(2225)^{0}$ – $2221^{+13}_{-10}$ | $2209,82$ | $-0,50$ | $(46,46)$ | $0^{+}(0^{-+})$
30. |  $\eta(2320)^{0}$ – $2320\pm 15$ | $2305,89$ | $-0,61$ | $(47,47)$ | $0^{+}(0^{-+})$
31. |  $f_{0}(2330)^{0}$ – $2340\pm 20$ | $2354,69$ | $+0,63$ | $(95/2,95/2)$ | $0^{+}(0^{++})$
|  |  |  |  | $\eta_{c},\chi_{c0}=c\bar{c}$
32. |  $\eta_{c}(1S)$ – $2983,9\pm 0,4$ | $2980,15$ | $-0,12$ | $(107/2,107/2)$ | $0^{+}(0^{-+})$
33. |  $\chi_{c0}(1P)$ – $3414,71\pm 0,3$ | $3438,01$ | $+0,68$ | $(115/2,115/2)$ | $0^{+}(0^{++})$
34. |  $\eta_{c}(2S)$ – $3637,5\pm 1,1$ | $3618,13$ | $-0,53$ | $(59,59)$ | $0^{+}(0^{-+})$
35. |  $\chi_{c0}(3860)$ – $3862^{+50}_{-35}$ | $3865,46$ | $+0,09$ | $(61,61)$ | $0^{+}(0^{++})$
36. |  $\chi_{c0}(3915)$ – $3921,7\pm 1,8$ | $3928,57$ | $+0,17$ | $(123/2,123/2)$ | $0^{+}(0^{++})$
37. |  $R_{c0}(4240)$ – $4239^{+50}_{-21}$ | $4251,77$ | $+0,30$ | $(64,64)$ |  $1^{+}(0^{--})$,
|  |  |  |  | exotic state
38. |  $\chi_{c0}(4500)$ – $4474\pm 4$ | $4451,83$ | $-0,49$ | $(131/2,131/2)$ |  $0^{+}(0^{++})$,
|  |  |  |  | exotic state
39. |  $\chi_{c0}(4700)$ – $4694^{+16}_{-5}$ | $4725,73$ | $+0,67$ | $(135/2,135/2)$ |  $0^{+}(0^{++})$,
|  |  |  |  | exotic state
|  |  |  |  | $\eta_{b},\chi_{b0}=b\bar{b}$
40. |  $\eta_{b}(1S)$ – $9398,7\pm 2,0$ | $9418,75$ | $+0,21$ | $(191/2,191/2)$ | $0^{+}(0^{-+})$
41. |  $\chi_{b0}(1P)$ – $9859,44\pm 0,42$ | $9815,29$ | $-0,44$ | $(195/2,195/2)$ | $0^{+}(0^{++})$
42. |  $\eta_{b}(2S)$ – $9999\pm 4$ | $10016,62$ | $+0,17$ | $(197/2,197/2)$ | $0^{+}(0^{-+})$
43. |  $\chi_{b0}(2P)$ – $10232,5\pm 0,4$ | $10220$ | $-0,12$ | $(199/2,199/2)$ | $0^{+}(0^{++})$
44. |  $H^{0}$ – $125250\pm 17$ | $125195$ | $-0,04$ | $(699/2,699/2)$ |

Table 32. Truly neutral mesons, spin 1, $\bb K=\bb R$
(Subspace ${\boldsymbol{\sf H}}^{(0,0,1)}_{\rm phys}(\bb R)$, vectors
$\left|\bb K,b,\ell,l-\dot{l}\right\rangle=\left|\bb R,0,0,1\right\rangle$).
State and mass $m_{\omega}$ Error % $(l,\dot{l})$ $q\bar{q}$, $I^{G}(J^{PC})$
$\pi^{0},a^{0},\rho^{0},b^{0}=$ $(u\bar{u}-d\bar{d})/\sqrt{2}$ 1.
$\rho(770)^{0}$ – $775,26\pm 0,23$ $772,63$ $-0,34$ $(55/2,53/2)$
$1^{+}(1^{--})$ 2. $b_{1}(1235)^{0}$ – $1229,5\pm 3,2$ $1216,18$ $-1,08$
$(69/2,67/2)$ $1^{+}(1^{+-})$ 3. $a_{1}(1260)^{0}$ – $1230\pm 40$ $1251,69$
$+1,76$ $(35,34)$ $1^{-}(1^{++})$ 4. $\pi_{1}(1400)^{0}$ – $1354\pm 25$
$1361,30$ $+0,54$ $(73/2,71/2)$ $1^{-}(1^{-+})$ 5. $\rho(1450)^{0}$ – $1465\pm
25$ $1475,51$ $+0,72$ $(38,37)$ $1^{+}(1^{--})$ 6. $\rho(1570)^{0}$ –$1570\pm
70$ $1554,21$ $-1,00$ $(39,38)$ $1^{+}(1^{--})$ 7. $\pi_{1}(1600)^{0}$ –
$1661^{+15}_{-11}$ $1676,08$ $+0,91$ $(81/2,79/2)$ $1^{-}(1^{-+})$ 8.
$a_{1}(1640)^{0}$ – $1655\pm 16$ $1634,94$ $-1,21$ $(40,39)$ $1^{-}(1^{++})$

| State and mass | $m_{\omega}$ | Error % | $(l,\dot{l})$ |  $q\bar{q}$, $I^{G}(J^{PC})$
---|---|---|---|---|---
9. |  $\rho(1700)^{0}$ – $1720\pm 20$ | $1717,73$ | $-0,13$ | $(41,40)$ | $1^{+}(1^{--})$
10. |  $\rho(1900)^{0}$ – $1909\pm 17$ | $1889,42$ | $-1,02$ | $(43,42)$ | $1^{+}(1^{--})$
11. |  $a_{1}(1930)^{0}$ – $1930^{+30}_{-70}$ | $1933,62$ | $+0,19$ | $(87/2,85/2)$ | $1^{-}(1^{++})$
12. |  $b_{1}(1960)^{0}$ – $1960\pm 35$ | $1978,33$ | $+0,93$ | $(44,43)$ | $1^{+}(1^{+-})$
13. |  $\rho(2000)^{0}$ – $2000\pm 30$ | $1978,33$ | $-1,08$ | $(44,43)$ | $1^{+}(1^{--})$
14. |  $\pi_{1}(2015)^{0}$ – $2014\pm 20$ | $2023,56$ | $+0,47$ | $(89/2,87/2)$ | $1^{-}(1^{-+})$
15. |  $a_{1}(2095)^{0}$ – $2096\pm 17$ | $2069,29$ | $-1,27$ | $(45,44)$ | $1^{-}(1^{++})$
16. |  $\rho(2150)^{0}$ – $2111\pm 43$ | $2115,54$ | $+0,21$ | $(91/2,89/2)$ | $1^{+}(1^{--})$
17. |  $b_{1}(2240)^{0}$ – $2240\pm 35$ | $2257,34$ | $+0,77$ | $(47,46)$ | $1^{+}(1^{+-})$
18. |  $\rho(2270)^{0}$ – $2265\pm 40$ | $2257,34$ | $-0,34$ | $(47,46)$ | $1^{+}(1^{--})$
19. |  $a_{1}(2270)^{0}$ – $2270^{+50}_{-40}$ | $2257,34$ | $-0,56$ | $(47,46)$ | $1^{-}(1^{++})$
|  |  |  |  | $\omega,\phi,h,f=$
|  |  |  |  | $c_{1}(u\bar{u}+d\bar{d})$
|  |  |  |  | $+c_{2}(s\bar{s})$
20. |  $\omega(782)$ – $782,66\pm 0,13$ | $772,63$ | $-1,28$ | $(55/2,53/2)$ | $0^{-}(1^{--})$
21. |  $\phi(1020)$ – $1019,461\pm 0,016$ | $1013,82$ | $-0,55$ | $(63/2,61/2)$ | $0^{-}(1^{--})$
22. |  $h_{1}(1170)$ – $1166\pm 6$ | $1181,18$ | $+1,30$ | $(34,33)$ | $0^{-}(1^{+-})$
23. |  $f_{1}(1285)$ – $1281,9\pm 0,5$ | $1287,72$ | $+0,45$ | $(71/2,69/2)$ | $0^{+}(1^{++})$
24. |  $h_{1}(1415)$ – $1416\pm 8$ | $1398,86$ | $-1,21$ | $(37,36)$ | $0^{-}(1^{+-})$
25. |  $f_{1}(1420)$ – $1426,3\pm 0,9$ | $1436,93$ | $+0,74$ | $(75/2,73/2)$ | $0^{+}(1^{++})$
26. |  $\omega(1420)$ – $1410\pm 60$ | $1398,86$ | $-0,79$ | $(37,36)$ | $0^{-}(1^{--})$
27. |  $f_{1}(1510)$ – $1518\pm 5$ | $1514,60$ | $-0,22$ | $(77/2,75/2)$ | $0^{+}(1^{++})$
28. |  $h_{1}(1595)$ – $1594^{+12}_{-60}$ | 1594,32 | $+0,02$ | $(79/2,77/2)$ | $0^{-}(1^{+-})$
29. |  $\omega(1650)$ – $1670\pm 30$ | $1676,08$ | $+0,36$ | $(81/2,79/2)$ | $0^{-}(1^{--})$
30. |  $\phi(1680)$ – $1680\pm 20$ | $1676,08$ | $-0,23$ | $(81/2,79/2)$ | $0^{-}(1^{--})$
31. |  $\omega(1960)$ – $1960\pm 25$ | $1978,33$ | $+0,93$ | $(44,43)$ | $0^{-}(1^{--})$
32. |  $h_{1}(1965)$ – $1965\pm 45$ | $1978,33$ | $+0,68$ | $(44,43)$ | $0^{-}(1^{+-})$
33. |  $f_{1}(1970)$ – $1971\pm 15$ | $1978,33$ | $+0,37$ | $(44,43)$ | $0^{+}(1^{++})$
34. |  $\phi(2170)$ – $2162\pm 7$ | $2162,29$ | $+0,01$ | $(46,45)$ | $0^{-}(1^{--})$
35. |  $\omega(2205)$ – $2205\pm 30$ | $2209,56$ | $+0,21$ | $(93/2,91/2)$ | $0^{-}(1^{--})$
|  |  |  |  | $J/\psi,h=c\bar{c}$
36. |  $h_{1}(2215)$ – $2215\pm 40$ | $2209,56$ | $-0,24$ | $(93/2,91/2)$ | $0^{-}(1^{+-})$
37. |  $\omega(2290)$ – $2290\pm 20$ | $2305,63$ | $+0,68$ | $(95/2,93/2)$ | $0^{-}(1^{--})$
38. |  $f_{1}(2310)$ – $2310\pm 60$ | $2303,63$ | $-0,19$ | $(95/2,93/2)$ | $0^{+}(1^{++})$
39. |  $\omega(2330)$ – $2330\pm 30$ | $2354,43$ | $+1,04$ | $(48,47)$ | $0^{-}(1^{--})$
40. |  $J/\psi(1S)$ – $3096,9\pm 0,006$ | $3091,29$ | $-0,18$ | $(55,54)$ | $0^{-}(1^{--})$
41. |  $\chi_{c1}(1P)$ – $3510,67\pm 0,05$ | $3497,28$ | $-0,38$ | $(117/2,115/2)$ | $0^{+}(1^{++})$
42. |  $h_{c}(1P)$ – $3525,38\pm 0,11$ | $3557,32$ | $+0,90$ | $(59,58)$ | $0^{-}(1^{+-})$
43. |  $\psi(2S)$ – $3686,10\pm 0,06$ | $3678,94$ | $-0,19$ | $(60,59)$ | $0^{-}(1^{--})$
44. |  $\psi(3770)$ – $3773,7\pm 0,4$ | $3802,61$ | $+0,76$ | $(61,60)$ | $0^{-}(1^{--})$
45. |  $\chi_{c1}(3872)$ – $3871,65\pm 0,06$ | $3865,20$ | $-0,16$ | $(123/2,121/2)$ |  $0^{+}(1^{++})$,
|  |  |  |  | exotic state
46. |  $Z_{c}(3900)$ – $3887,1\pm 2,6$ | $3865,20$ | $-0,56$ | $(123/2,121/2)$ |  $1^{+}(1^{+-})$,
|  |  |  |  | exotic state
| State and mass | $m_{\omega}$ | Error % | $(l,\dot{l})$ |  $q\bar{q}$, $I^{G}(J^{PC})$
---|---|---|---|---|---
47. |  $\psi(4040)$ – $4039\pm 1$ | $4056,06$ | $+0,42$ | $(63,62)$ | $0^{-}(1^{--})$
48. |  $\chi_{c1}(4140)$ – $4146,5\pm 3,0$ | $4120,70$ | $-0,62$ | $(127/2,125/2)$ |  $0^{+}(1^{++})$,
|  |  |  |  | exotic state
49. |  $\psi(4160)$ – $4191\pm 5$ | $4185,86$ | $-0,12$ | $(64,63)$ | $0^{-}(1^{--})$
50. |  $Z_{c}(4200)$ – $4196^{+35}_{-32}$ | $4185,86$ | $-0,24$ | $(64,63)$ |  $1^{+}(1^{+-})$,
|  |  |  |  | exotic state
51. |  $\psi(4230)$ – $4222,7\pm 2,6$ | $4251,52$ | $+0,68$ | $(129/2,127/2)$ | $0^{-}(1^{--})$
52. |  $\chi_{c1}(4274)$ – $4286^{+8}_{-9}$ | $4317,69$ | $+0,74$ | $(65,64)$ |  $0^{+}(1^{++})$,
|  |  |  |  | exotic state
53. |  $\psi(4360)$ – $4372\pm 9$ | $4384,38$ | $+0,28$ | $(131/2,129/2)$ |  $0^{-}(1^{--})$,
|  |  |  |  | exotic state
54. |  $\psi(4415)$ – $4421\pm 4$ | $4451,57$ | $+0,69$ | $(66,65)$ | $0^{-}(1^{--})$
55. |  $Z_{c}(4430)$ – $4478^{+15}_{-18}$ | $4451,57$ | $-0,59$ | $(66,65)$ |  $1^{+}(1^{+-})$,
|  |  |  |  | exotic state
56. |  $\psi(4660)$ – $4630\pm 6$ | $4656,23$ | $+0,56$ | $(135/2,133/2)$ |  $0^{-}(1^{--})$,
|  |  |  |  | exotic state
57. |  $\chi_{c1}(4685)$ – $4684^{+15}_{-17}$ | $4656,23$ | $-0,59$ | $(135/2,133/2)$ |  $0^{+}(1^{++})$,
|  |  |  |  | exotic state
|  |  |  |  |  $\Upsilon,\chi_{b1}$,
|  |  |  |  | $h_{b}=b\bar{b}$
58. |  $\Upsilon(1S)$ – $9460,30\pm 0,26$ | $9516,86$ | $+0,59$ | $(193/2,191/2)$ | $0^{-}(1^{--})$
59. |  $\chi_{b1}(1P)$ – $9892,78\pm 0,26$ | $9915,44$ | $+0,23$ | $(197/2,195/2)$ | $0^{+}(1^{++})$
60. |  $h_{b}(1P)$ – $9899,3\pm 0,8$ | $9915,44$ | $+0,16$ | $(197/2,195/2)$ | $0^{-}(1^{+-})$
61. |  $\Upsilon(2S)$ – $10023,26\pm 0,31$ | $10016,37$ | $-0,07$ | $(99,98)$ | $0^{-}(1^{--})$
62. |  $\chi_{b1}(2P)$ – $10255,46\pm 0,22$ | $10219,74$ | $-0,35$ | $(100,99)$ | $0^{+}(1^{++})$
63. |  $h_{b}(2P)$ – $10259,8\pm 1,2$ | $10219,74$ | $-0,39$ | $(100,99)$ | $0^{-}(1^{+-})$
64. |  $\Upsilon(3S)$ – $10355,2\pm 0,5$ | $10322,20$ | $-0,32$ | $(201/2,199/2)$ | $0^{-}(1^{--})$
65. |  $\chi_{b1}(3P)$ – $10513,4\pm 0,7$ | $10528,64$ | $+0,14$ | $(203/2,201/2)$ | $0^{+}(1^{++})$
66. |  $\Upsilon(4S)$ – $10579,4\pm 1,2$ | $10528,64$ | $-0,48$ | $(203/2,201/2)$ | $0^{-}(1^{--})$
67. |  $Z_{b}(10610)$ – $10607,2\pm 2,0$ | $10632,63$ | $+0,24$ | $(102,101)$ |  $1^{+}(1^{+-})$,
|  |  |  |  | exotic state
68. |  $Z_{b}(10650)$ – $10652,2\pm 1,5$ | $10632,63$ | $-0,18$ | $(102,101)$ |  $1^{+}(1^{+-})$,
|  |  |  |  | exotic state
69. |  $\Upsilon(10753)$ – $10753\pm 6$ | $10737,13$ | $-0,15$ | $(205/2,203/2)$ | $?^{?}(1^{--})$
70. |  $\Upsilon(10860)$ – $10885,2^{+2,6}_{-1,6}$ | $10842,14$ | $-0,39$ | $(103,102)$ | $0^{-}(1^{--})$
71. |  $\Upsilon(11020)$ – $11000\pm 4$ | $11053,69$ | $+0,49$ | $(104,103)$ | $0^{-}(1^{--})$
72. |  $Z^{0}$ – $91187,6\pm 0,0021$ | $91062,24$ | $-0,14$ | $(597/2,595/2)$ |

Table 33. Truly neutral mesons, spin 2, $\bb K=\bb R$
(Subspace ${\boldsymbol{\sf H}}^{(0,0,2)}_{\rm phys}(\bb R)$, vectors
$\left|\bb K,b,\ell,l-\dot{l}\right\rangle=\left|\bb R,0,0,2\right\rangle$).
State and mass $m_{\omega}$ Error % $(l,\dot{l})$ $q\bar{q}$, $I^{G}(J^{PC})$
$\pi^{0},a^{0},\rho^{0}=$ $(u\bar{u}-d\bar{d})/\sqrt{2}$ 1. $a_{2}(1320)^{0}$
– $1318,2\pm 0,6$ $1323,49$ $+0,40$ $(73/2,69/2)$ $1^{-}(2^{++})$ 2.
$\pi_{2}(1670)^{0}$ – $1670,6^{+2,9}_{-1,2}$ $1675,31$ $+0,28$ $(41,39)$
$1^{-}(2^{-+})$ 3. $a_{2}(1700)^{0}$ – $1698\pm 40$ $1716,96$ $+1,11$
$(83/2,79/2)$ $1^{-}(2^{++})$ 4. $\pi_{2}(1880)^{0}$ – $1874^{+26}_{-5}$
$1888,65$ $+0,78$ $(87/2,53/2)$ $1^{-}(2^{-+})$ 5. $\rho_{2}(1940)^{0}$ –
$1940\pm 40$ $1932,86$ $-0,37$ $(44,42)$ $1^{+}(2^{--})$ 6. $a_{2}(1950)^{0}$
– $1950^{+30}_{-70}$ $1932,86$ $-0,88$ $(44,42)$ $1^{-}(2^{++})$ 7.
$a_{2}(1990)^{0}$ – $2050\pm 10$ $2068,53$ $+0,90$ $(91/2,87/2)$
$1^{-}(2^{++})$ 8. $\pi_{2}(2005)^{0}$ – $1953^{+17}_{-27}$ $1977,57$ $+0,74$
$(89/2,85/2)$ $1^{-}(2^{-+})$ 9. $a_{2}(2030)^{0}$ – $2030\pm 20$ $2022,79$
$-0,35$ $(45,43)$ $1^{-}(2^{++})$ 10. $\pi_{2}(2100)^{0}$ – $2090\pm 29$
$2114,77$ $+1,18$ $(46,44)$ $1^{-}(2^{-+})$ 11. $a_{2}(2175)^{0}$ – $2175\pm
40$ $2161,53$ $-0,62$ $(93/2,89/2)$ $1^{-}(2^{++})$ 12. $\rho_{2}(2225)^{0}$ –
$2225\pm 35$ $2208,80$ $-0,73$ $(47,45)$ $1^{+}(2^{--})$ 13. $a_{2}(2255)^{0}$
– $2255\pm 20$ $2256,57$ $+0,07$ $(95/2,91/2)$ $1^{-}(2^{++})$ 14.
$\pi_{2}(2285)^{0}$ – $2285\pm 20$ $2304,86$ $+0,87$ $(48,46)$ $1^{-}(2^{-+})$
$f,\eta,\omega=$ $c_{1}(u\bar{u}+d\bar{d})$ $+c_{2}(s\bar{s})$ 15.
$f_{2}(1270)$ – $1275,5\pm 0,8$ $1286,95$ $+0,89$ $(36,34)$ $0^{+}(2^{++})$
16. $f_{2}(1430)\approx 1430$ $1436,16$ $+0,43$ $(38,36)$ $0^{+}(2^{++})$ 17.
$f^{\prime}_{2}(1525)$ – $1517,4\pm 2,5$ $1513,84$ $+0,23$ $(39,37)$
$0^{+}(2^{++})$ 18. $f_{2}(1565)$ – $1542\pm 19$ $1553,44$ $+0,74$
$(79/2,75/2)$ $0^{+}(2^{++})$ 19. $f_{2}(1640)$ – $1639\pm 6$ $1634,18$
$-0,29$ $(81/2,77/2)$ $0^{+}(2^{++})$ 20. $\eta_{2}(1645)$ – $1617\pm 5$
$1634,18$ $+1,06$ $(81/2,77/2)$ $0^{+}(2^{-+})$ 21. $f_{2}(1750)$ – $1755\pm
10$ $1759,17$ $+0,24$ $(42,40)$ $0^{+}(2^{++})$ 22. $f_{2}(1810)$ – $1815\pm
12$ $1801,78$ $-0,73$ $(85/2,81/2)$ $0^{+}(2^{++})$ 23. $\eta_{2}(1870)$ –
$1842\pm 8$ $1844,96$ $+0,16$ $(43,41)$ $0^{+}(2^{-+})$ 24. $f_{2}(1910)$ –
$1900\pm 9$ $1888,65$ $-0,59$ $(87/2,83/2)$ $0^{+}(2^{++})$ 25. $f_{2}(1950)$
– $1936\pm 12$ $1932,86$ $-0,16$ $(44,42)$ $0^{+}(2^{++})$ 26.
$\omega_{2}(1975)$ – $1975\pm 20$ $1977,57$ $+0,13$ $(89/2,85/2)$
$0^{-}(2^{--})$ 27. $f_{2}(2000)$ – $2001\pm 10$ $2022,79$ $+1,08$ $(45,43)$
$0^{+}(2^{++})$ 28. $f_{2}(2010)$ – $2011^{+60}_{-80}$ $2022,79$ $+0,58$
$(45,43)$ $0^{+}(2^{++})$ 29. $\eta_{2}(2030)$ – $2030\pm 5$ $2022,79$ $-0,35$
$(45,43)$ $0^{+}(2^{-+})$ 30. $f_{2}(2140)$ – $2140\pm 12$ $2161,53$ $+1,01$
$(93/2,89/2)$ $0^{+}(2^{++})$ 31. $f_{2}(2150)$ – $2157\pm 12$ $2161,53$
$+0,21$ $(93/2,89/2)$ $0^{+}(2^{++})$ 32. $\omega_{2}(2195)$ – $2195\pm 30$
$2208,80$ $+0,63$ $(47,45)$ $0^{-}(2^{--})$ 33. $f_{2}(2240)$ – $2240\pm 15$
$2256,57$ $+0,74$ $(95/2,91/2)$ $0^{+}(2^{++})$ 34. $\eta_{2}(2250)$ –
$2248\pm 20$ $2256,57$ $+0,38$ $(95/2,91/2)$ $0^{+}(2^{-+})$ 35. $f_{2}(2295)$
– $2293\pm 13$ $2304,86$ $+0,52$ $(48,46)$ $0^{+}(2^{++})$ 36. $f_{2}(2300)$ –
$2297\pm 28$ $2304,86$ $+0,34$ $(48,46)$ $0^{+}(2^{++})$ 37. $f_{2}(2340)$ –
$2345^{+50}_{-40}$ $2356,66$ $+0,49$ $(97/2,93/2)$ $0^{+}(2^{++})$

| State and mass | $m_{\omega}$ | Error % | $(l,\dot{l})$ |  $q\bar{q}$, $I^{G}(J^{PC})$
---|---|---|---|---|---
|  |  |  |  | $\chi,\psi=c\bar{c}$
38. |  $\chi_{c2}(1P)$ – $3556,17\pm 0,7$ | $3556,56$ | $+0,01$ | $(119/2,115/2)$ | $0^{+}(2^{++})$
39. |  $\psi_{2}(3823)$ – $3823,7\pm 0,5$ | $3801,84$ | $-0,57$ | $(123/2,119/2)$ | $0^{-}(2^{--})$
40. |  $\chi_{c2}(3930)$ – $3922,2\pm 1,0$ | $3927,54$ | $+0,13$ | $(125/2,121/2)$ | $0^{+}(2^{++})$
|  |  |  |  |  $\Upsilon,\chi_{b2}$,
|  |  |  |  | $h_{b}=b\bar{b}$
41. |  $\chi_{b2}(1P)$ – $9912,21\pm 0,26$ | $9914,68$ | $+0,02$ | $(99,97)$ | $0^{+}(2^{++})$
42. |  $\Upsilon_{2}(1D)$ – $10163,7\pm 1,4$ | $10117,03$ | $-0,46$ | $(100,98)$ | $0^{-}(2^{--})$
43. |  $\chi_{b2}(2P)$ – $10268,65\pm 0,22$ | $10218,98$ | $-0,48$ | $(201/2,197/2)$ | $0^{+}(2^{++})$
44. |  $\chi_{b2}(3P)$ – $10513,4\pm 0,7$ | $10527,88$ | $+0,14$ | $(102,100)$ | $0^{+}(2^{++})$

Table 34. Truly neutral mesons, spin 3, $\bb K=\bb R$
(Subspace ${\boldsymbol{\sf H}}^{(0,0,3)}_{\rm phys}(\bb R)$, vectors
$\left|\bb K,b,\ell,l-\dot{l}\right\rangle=\left|\bb R,0,0,3\right\rangle$).
State and mass $m_{\omega}$ Error % $(l,\dot{l})$ $q\bar{q}$, $I^{G}(J^{PC})$
$a^{0},b^{0},\rho^{0}=$ $(u\bar{u}-d\bar{d})/\sqrt{2}$ 1. $\rho_{3}(1690)^{0}$
– $1688,8\pm 2,1$ $1674,04$ $-0,87$ $(83/2,77/2)$ $1^{+}(3^{--})$ 2.
$a_{3}(1875)^{0}$ – $1874\pm 43$ $1887,38$ $+0,71$ $(44,41)$ $1^{-}(3^{++})$
3. $\rho_{3}(1990)^{0}$ – $1982\pm 14$ $1976,29$ $-0,29$ $(45,42)$
$1^{+}(3^{--})$ 4. $b_{3}(2030)^{0}$ – $2032\pm 12$ $2021,52$ $-0,51$
$(91/2,85/2)$ $1^{+}(3^{+-})$ 5. $a_{3}(2030)^{0}$ – $2031\pm 12$ $2021,52$
$-0,47$ $(91/2,85/2)$ $1^{-}(3^{++})$ 6. $b_{3}(2245)^{0}$ – $2245\pm 50$
$2255,29$ $+0,46$ $(48,45)$ $1^{+}(3^{+-})$ 7. $\rho_{3}(2250)^{0}$ – $2248\pm
17$ $2255,29$ $+0,32$ $(48,45)$ $1^{+}(3^{--})$ 8. $a_{3}(2275)^{0}$ –
$2275\pm 35$ $2303,59$ $+1,25$ $(97/2,91/2)$ $1^{-}(3^{++})$
$f,\phi,\omega,h=$ $c_{1}(u\bar{u}+d\bar{d})$ $+c_{2}(s\bar{s})$ 9.
$\omega_{3}(1670)$ – $1667\pm 4$ $1674,04$ $+0,42$ $(83/2,77/2)$
$0^{-}(3^{--})$ 10. $\phi_{3}(1850)$ – $1854\pm 7$ $1843,69$ $-0,55$
$(87/2,81/2)$ $0^{-}(3^{--})$ 11. $\omega_{3}(1945)$ – $1950^{+30}_{-70}$
$1931,58$ $-0,94$ $(89/2,83/2)$ $0^{-}(3^{--})$ 12. $h_{3}(2025)$ – $2025\pm
20$ $2021,52$ $-0,17$ $(91/2,85/2)$ $0^{-}(3^{+-})$ 13. $f_{3}(2050)$ –
$2048\pm 8$ $2067,25$ $+0,94$ $(46,43)$ $0^{+}(3^{++})$ 14. $\omega_{3}(2255)$
– $2255\pm 15$ $2255,29$ $+0,01$ $(48,45)$ $0^{-}(3^{--})$ 15. $h_{3}(2275)$ –
$2275\pm 25$ $2255,29$ $-0,36$ $(48,45)$ $0^{-}(3^{+-})$ 16.
$\omega_{3}(2285)$ – $2278\pm 28$ $2303,59$ $+1,12$ $(97/2,91/2)$
$0^{-}(3^{--})$ 17. $f_{3}(2300)$ – $2334\pm 35$ $2352,39$ $+0,78$ $(49,46)$
$0^{+}(3^{++})$ $\psi=c\bar{c}$ 18. $\psi_{3}(3842)$ – $3842,71\pm 0,20$
$3863,13$ $+0,53$ $(125/2,119/2)$ $0^{-}(3^{--})$

Table 34. Truly neutral mesons, spin 4, $\bb K=\bb R$
(Subspace ${\boldsymbol{\sf H}}^{(0,0,4)}_{\rm phys}(\bb R)$, vectors
$\left|\bb K,b,\ell,l-\dot{l}\right\rangle=\left|\bb R,0,0,4\right\rangle$).
State and mass $m_{\omega}$ Error % $(l,\dot{l})$ $q\bar{q}$, $I^{G}(J^{PC})$
$a^{0},\rho^{0},\pi^{0}=$ $(u\bar{u}-d\bar{d})/\sqrt{2}$ 1. $a_{4}(1970)^{0}$
– $1967\pm 16$ $1974,50$ $+0,38$ $(91/2,83/2)$ $1^{-}(4^{++})$ 2.
$\rho_{4}(2230)^{0}$ – $2230\pm 25$ $2253,51$ $+1,05$ $(97/2,89/2)$
$1^{+}(4^{--})$ 3. $\pi_{4}(2250)^{0}$ – $2250\pm 15$ $2253,51$ $+0,15$
$(97/2,89/2)$ $1^{-}(4^{-+})$ 4. $a_{4}(2255)^{0}$ – $2237\pm 5$ $2253,51$
$+0,74$ $(97/2,89/2)$ $1^{-}(4^{++})$ $f,\omega,\eta=$
$c_{1}(u\bar{u}+d\bar{d})+c_{2}(s\bar{s})$ 5. $f_{4}(2050)$ – $2018\pm 11$
$2019,73$ $+0,08$ $(46,42)$ $0^{+}(4^{++})$ 6. $\omega_{4}(2250)$ – $2250\pm
30$ $2253,51$ $+0,15$ $(97/2,89/2)$ $0^{-}(4^{--})$ 7. $f_{4}(2300)\approx
2314$ $2301,80$ $-0,53$ $(49,45)$ $0^{+}(4^{++})$ 8. $\eta_{4}(2330)$ –
$2328\pm 38$ $2350,60$ $+0,97$ $(99/2,91/2)$ $0^{+}(4^{-+})$

Table 36. Truly neutral mesons, spin 5, $\bb K=\bb R$
(Subspace ${\boldsymbol{\sf H}}^{(0,0,5)}_{\rm phys}(\bb R)$, vectors
$\left|\bb K,b,\ell,l-\dot{l}\right\rangle=\left|\bb R,0,0,5\right\rangle$).
State and mass $m_{\omega}$ Error % $(l,\dot{l})$ $q\bar{q}$, $I^{G}(J^{PC})$
$\rho^{0}=$ $(u\bar{u}-d\bar{d})/\sqrt{2}$ 1. $\rho_{5}(2350)^{0}$ – $2330\pm
35$ $2348,30$ $+0,78$ $(50,45)$ $1^{+}(5^{--})$ $\omega=$
$c_{1}(u\bar{u}+d\bar{d})+c_{2}(s\bar{s})$ 2. $\omega_{5}(2250)$ – $2250\pm
70$ $2251,21$ $+0,05$ $(49,44)$ $0^{-}(5^{--})$

Table 37. Truly neutral mesons, spin 6, $\bb K=\bb R$
(Subspace ${\boldsymbol{\sf H}}^{(0,0,6)}_{\rm phys}(\bb R)$, vectors
$\left|\bb K,b,\ell,l-\dot{l}\right\rangle=\left|\bb R,0,0,6\right\rangle$).
State and mass $m_{\omega}$ Error % $(l,\dot{l})$ $q\bar{q}$, $I^{G}(J^{PC})$
$a^{0}=$ $(u\bar{u}-d\bar{d})/\sqrt{2}$ 1. $a_{6}(2450)^{0}$ – $2450\pm 130$
$2444,62$ $-0,22$ $(103/2,91/2)$ $1^{-}(6^{++})$ $f=$
$c_{1}(u\bar{u}+d\bar{d})$ $+c_{2}(s\bar{s})$ 2. $f_{6}(2510)$ – $2465\pm 50$
$2444,62$ $-0,83$ $(103/2,91/2)$ $0^{+}(6^{++})$ 3. $f_{6}(3100)$ – $3100\pm
100$ $3082,35$ $-0,57$ $(115/2,103/2)$ $0^{+}(6^{++})$

##  6 Conclusion

It is well known that search for the primary constituents of matter is the
general motivation of theoretical physics. This search, however, still adheres
to the main thesis of atomism and reductionism that matter is an aggregate
consisting of fundamental (i.e., indivisible) subunits. Such position is
opposed to the opinion of Heisenberg who repeatedly stressed that the
“consists of” principle does not work at the microlevel, at least because an
elementary particle may decay in several different ways.

The quark model is actually a modern incarnation of the Democritus hypothesis,
where a two-quark $q\bar{q}$ and a three-quark $qqq$ compositions are
prescribed to mesonic and baryon parts of the matter spectrum. Such
$q\bar{q}$\- and $qqq$-schemes do not account for a number of experimentally
detected states (exotic hadrons, tetraquarks, and pentaquarks); moreover,
there are many other serious problems facing the quark model, such as only a
low accuracy of the baryon spectrum predicted by the
$\operatorname{SU}(6)\otimes\operatorname{O}(3)$-symmetry model, insufficient
understanding of the confinement mechanism, the proton spin crisis (for more
details see [70]). In addition, leptons fall beyond the scope of the quark
model, which fact obviously precludes quarks from being universal subunits of
matter. Being integrated into the Standard Model (SM), the quark model implies
the following list of 17 “fundamental particles”: fermions ($u$, $d$, $s$,
$c$, $b$, $t$, $e$, $\mu$, $\tau$, $\nu_{e}$, $\nu_{\mu}$, $\nu_{\tau}$) and
bosons ($\gamma$, $g$, $Z$, $W$, $H$). This list contains seven fictitious
particles ($u$, $d$, $s$, $c$, $b$, $t$, $g$) that have never been observed in
free state. Five particles ($\mu$, $\tau$, $Z$, $W$, $H$) are certainly not
elementary since they decay by several different channels. Three types of
neutrinos ($\nu_{e}$, $\nu_{\mu}$, $\nu_{\tau}$) transform into each other via
neutrino oscillations, i.e. the lepton number for neutrinos is not conserved.
Within the SM framework, this motley set of 17 entities is interrelated by the
$G_{SM}=\operatorname{U}(1)\times\operatorname{SU}(2)\times\operatorname{SU}(3)$
dynamic symmetry and 18 parameters. It is easily seen that the SM is very far
from the Democritus ideal of a universal primary element of matter. However,
after we exclude the fictitious and decaying particles from the list, each of
the remaining three $\nu$, $e$, and $\gamma$ can claim to be a primary
element.

Concerning the spinor representations of the Lorentz group, note that the
fundamental representation of this group over the field of complex numbers
$\bb F=\bb C$ acts in a two-dimensional spin space whose vector is a two-
component spinor. And further, any finite-dimensional irreducible spinor
representation of the Lorentz group can be factorized as a tensor product of
two-dimensional fundamental representations. In turn, the spin space of the
fundamental representation is a minimal left ideal of the quaternion algebra
(biquaternion algebras $\bb C_{2}\simeq\bb C\otimes\bb H$ in the case of field
$\bb F=\bb C$ and quaternion algebras $C\kern-1.99997pt\ell_{0,2}\simeq\bb H$,
$C\kern-1.99997pt\ell_{1,1}\simeq\bb R(2)$,
$C\kern-1.99997pt\ell_{0,2}\simeq\bb R(2)$ in the case of field $\bb F=\bb R$,
with the division rings $\bb K\simeq\bb H$ for $C\kern-1.99997pt\ell_{0,2}$
and $\bb K\simeq\bb R$ for $C\kern-1.99997pt\ell_{1,1}$ and
$C\kern-1.99997pt\ell_{2,0}$). The elements of minimal left ideals of four-
dimensional quaternion algebras are two-component spinors.

It is remarkable that information bit is structurally identical to a two-
component spinor. On the other hand, two-component spinor describes the lowest
state corresponding to neutrino. Thus, the spinor concept should be a keypoint
in the search for the primary element of matter. A mathematical object of
spinor had appeared due to Cartan [71]. Van der Waerden [10] noted that the
name “spinor” was coined by Ehrenfest after the well-known paper by Goudsmit
and Uhlenbeck [72] on the rotating electron, where they tried to explain the
electron spin mechanical using the model of a spinning top. It is known that
since the work of Uhlenbeck and Goudsmit [72] all attempts to describe the
electron spin classically (i.e., mechanically) have failed. The first theory
that had given a correct mathematical formulation of spin “two-valuedness that
denies classical description” was proposed by Pauli in 1927 [9]. Avoiding
visual kinematic models, Pauli introduced a doubled Hilbert space
${\boldsymbol{\sf H}}_{2}\otimes{\boldsymbol{\sf H}}_{\infty}$ of wave
functions whose vectors are two-component spinors. That was the first
appearance of two-component spinors and the doubling in physics. The next
doubling (bispinors and the space ${\boldsymbol{\sf
H}}_{4}\otimes{\boldsymbol{\sf H}}_{\infty}$) was proposed by Dirac in 1928
[73]. One more doubling leads to hypervistors in the $\bb K$-Hilbert space
${\boldsymbol{\sf H}}_{8}\otimes{\boldsymbol{\sf H}}_{\infty}$, see [63, 64].
Hypertwistors are vectors of the fundamental representation of the Rumer-Fet
group
$\operatorname{SO}(2,4)\otimes\operatorname{SU}(2)\otimes\operatorname{SU}(2)^{\prime}$,
which has a deep relation to the periodic system of elements [60]. Spinors,
bispinors, and twistors are special cases of hypertwistors. In [1] Heisenberg
noted that the process of bifurcation, or dividing in two parts, expands the
space of nature in a very natural way, thus creating the possibility of new
symmetries. This doubling of symmetry is the most essential thing in the
definition of spin. In fact, the spin is the doubling (bifurcation) of
symmetry.

As noted above, the concept of spin stemmed from an attempt to describe the
“classically nondescribable two-valuedness” of an electron viewed as a
particle. However, a particle is a classical concept. Van der Waerden asked
why didn’t Pauli endow electron with a momentum $m_{s}=\pm 1/2$ and a magnetic
momentum $2m_{s}$ [10, p. 243]. But whether a classical particle can have a
nonclassical property? Obviously, no. And this is an answer to the question of
Van der Waerden. An innovative interpretation of spin was given by Yu. Rumer
and A. Fet: “Until now, we believed that one and the same particle, e.g., an
electron can exist in two states: with spin +1/2 and $-1/2$. However, an
electron without a definite spin value have been never observed and is only an
abstract concept. It is quite natural then that there exist two elementary
particles: an electron with spin +1/2 and an electron with spin $-1/2$, while
“just an electron” does not occur in nature” [7, p. 161-162]. The objects like
“just an electron”, “just a proton”, etc., are commonly assumed to live in
spacetime individually, independently of each other, and carrying all their
inherent quantum properties at once. Such an interpretation is incompatible
with the results of the recent experiments to test Bell’s inequalities by
Friedman-Clauser, Aspe, Greenberg-Horn-Zeilinger, and others. The classical
concept of particle is obviously invalid at the microlevel, as well as the
mechanical models associated with this concept. Getting rid of the visual
image (and the mental template) of a “particle”, one can perceive this concept
just as an embodiment of the idea of discreteness, impossibility to divide a
value into infinitely small parts, the denial of continuum. The quantum and
continuum are eternal thesis and antithesis going back to Democritus and
Empedocles. In contemporary physics, however, the quantum concept is based on
the Planck’s fundamental discreteness of energy, in contrast to the purely
speculative process of division to infinity.

In case we identify the minimal structural component
$\left|\mathfrak{q}\right\rangle$ (energy quantum) with neutrino
$\left|\nu\right\rangle$,
$\left|\mathfrak{q}\right\rangle\equiv\left|\nu\right\rangle$, we will come to
a “neutrino theory of everything” with neutrino being the primary element of
matter. That would be another incarnation of reductionism. But the states, be
they neutrinos or any other entities, are secondary with respect to the
substance and do not define it (see the first theorem of Spinoza, Section 4).
This is the main principle of holism. From the holistic viewpoint, the primary
substance, which we call energy or matter, has an infinite spectrum of states,
the lowest of which corresponds to neutrino.

The data presented in this paper show that the mass distribution of states
fits the grid of the weight diagram (known as extended Weyl diagram) with an
average accuracy of 0.41%. According to the mass formula (26), each node
$(l,\dot{l})$ of the weight diagram is associated with the quantum numbers
$l$, $\dot{l}$, which are the eigenvalues of the Casimir operators of the
Lorentz group. Thus, each node of the weight diagram is a state of the
spectrum of matter (a ”particle”) and at the same time a representation of the
Lorentz group in complete compliance with the canonical correspondence of the
GNS construction and Wigner’s interpretation.

In conclusion, this paper was aimed to emphasize the fundamental role of two-
component spinors, which is to describe the minimum structural component of
matter, i.e., a quantum of energy. We assert that the quantum character of the
matter spectrum is deeply connected with the factorization of cyclic vectors
of $\bb K$-Hilbert space (which is separable in this case) in the form of a
tensor product of the fundamental states $\left|\mathfrak{q}\right\rangle$ by
means of the GNS construction and algebraic quantization. Owing to the $\bb
K$-structure, the state $\left|\mathfrak{q}\right\rangle$ splits into two
states $\left|\mathfrak{q}_{a}\right\rangle$ and
$\left|\mathfrak{q}_{s}\right\rangle$. This is true for both the Fermi and
Bose states.

Acknowledgments: I am deeply grateful to Anna Sidorova-Biryukova for her
comprehensive assistance in preparing the acticle.

## Appendix A: Cartan Subalgebra and Weyl Generators

Let $\mathfrak{g}$ be a Lie algebra of dimension $n$ formed by generators
${\boldsymbol{\sf X}}_{i}$ ($i=1\rightarrow n$) that satisfy permutation
conditions

$\left[{\boldsymbol{\sf X}}_{i},{\boldsymbol{\sf
X}}_{j}\right]=\sum^{n}_{k=1}f_{ijk}{\boldsymbol{\sf X}}_{k}\equiv
f_{ijk}{\boldsymbol{\sf X}}_{k},$

where $f_{ijk}$ are structural constants. $n$ generators ${\boldsymbol{\sf
X}}_{i}$ form the basis $\mathcal{B}$ of the Lie algebra $\mathfrak{g}$.

######  Definition 1.

(Maximal Abelian subalgebra). An Abelian subalgebra $\mathfrak{K}$ of a Lie
algebra $\mathfrak{g}$ is called maximal when there are no additional elements
of the algebra $\mathfrak{g}$ that commute with all elements of the subalgebra
$\mathfrak{K}$.

The subalgebra $\mathfrak{K}$ is better known as the Cartan subalgebra of the
algebra $\mathfrak{g}$, and the number $m$ of elements of the Cartan
subalgebra is called the rank of the Lie algebra $\mathfrak{g}$.

######  Definition 2.

(Cartan subalgebra of the Lie algebra). Let $\mathfrak{g}$ be an
$n$-dimensional Lie algebra, then the set of all mutually commuting basis
elements $\left\\{{\boldsymbol{\sf X}}_{i}={\boldsymbol{\sf H}}_{i}\right\\}$
($i=1\rightarrow m$) of the algebra $\mathfrak{g}$ forms the basis of the
maximal Abelian subalgebra $\mathfrak{K}\subset\mathfrak{g}$ of the algebra
$\mathfrak{g}$.

######  Definition 3.

(The rank of the Lie algebra). The dimension $m$ of the Cartan subalgebra
$\mathfrak{K}\subset\mathfrak{g}$ determines the rank of the Lie algebra
$\mathfrak{g}$.

The elements ${\boldsymbol{\sf H}}_{i}$ of the Cartan subalgebra
$\mathfrak{K}$ are called Cartan generators or Cartan elements
(${\boldsymbol{\sf H}}_{i}$ also called Abelian generators or diagonal
operators). Cartan generators satisfy permutation conditions

$\left[{\boldsymbol{\sf H}}_{i},{\boldsymbol{\sf H}}_{j}\right]=0,\quad\forall
i,j=1,\ldots,m.$ (A.1)

This means that all ${\boldsymbol{\sf H}}_{i}$ are simultaneously
diagonalizable. Denoting their eigenvalues by $h_{i}$, we get

${\boldsymbol{\sf
H}}_{i}\left|h_{1},h_{2},\ldots,h_{m}\right\rangle=h_{i}\left|h_{1},h_{2},\ldots,h_{m}\right\rangle,\quad\forall
i=1\rightarrow m.$

The eigenvalues $h_{i}$ are called weights. $h_{i}$ can be considered as
components of an $m$-dimensional vector $\boldsymbol{h}$, which is called a
weight vector. The weights of the Cartan subalgebra are used as quantum
numbers to denote the states of a given multiplet.

Further, from the remaining generators ${\boldsymbol{\sf X}}_{i}$ of the
algebra $\mathfrak{g}$ ($i=1\rightarrow n-m$), which are not elements of the
Cartan subalgebra $\mathfrak{K}$, we form linear combinations, which, in turn,
form a linearly independent set of rising and lowering operators (ladder
operators ${\boldsymbol{\sf E}}_{\alpha}$ – Weyl generators or Weyl elements).
Along with Cartan generators ${\boldsymbol{\sf H}}_{i}$, they form the Cartan-
Weyl basis $\left\\{{\boldsymbol{\sf H}}_{i},{\boldsymbol{\sf
E}}_{\alpha}\right\\}$ of the algebra $\mathfrak{g}$.

Thus, the Lie algebra $\mathfrak{g}$ can be decomposed into a direct sum
consisting of the Cartan subalgebra $\mathfrak{K}$ ($m$ generators
${\boldsymbol{\sf H}}_{i}$) and $n-m$ one-dimensional subalgebras
$\mathfrak{W}_{\alpha}$ formed by the Weyl generators ${\boldsymbol{\sf
E}}_{\alpha}$:

$\mathfrak{g}=\mathfrak{K}\bigoplus^{n-m}_{\alpha=1}\mathfrak{W}_{\alpha}=\mathfrak{K}\oplus\mathfrak{W}_{1}\oplus\mathfrak{W}_{2}\oplus\ldots\oplus\mathfrak{W}_{n-m}.$

There are permutation relations

$\left[{\boldsymbol{\sf H}}_{i},{\boldsymbol{\sf
E}}_{\alpha}\right]=\alpha_{i}{\boldsymbol{\sf E}}_{\alpha},\quad\forall
i=1\rightarrow m,\;\alpha=1\rightarrow n-m.$ (A.2)

The relations (A.2) can be written as follows:

$\begin{bmatrix}\left[{\boldsymbol{\sf H}}_{1},{\boldsymbol{\sf
E}}_{\alpha}\right]\\\ \left[{\boldsymbol{\sf H}}_{2},{\boldsymbol{\sf
E}}_{\alpha}\right]\\\ \vdots\\\ \left[{\boldsymbol{\sf
H}}_{m},{\boldsymbol{\sf
E}}_{\alpha}\right]\end{bmatrix}=\begin{bmatrix}\alpha_{1}\\\ \alpha_{2}\\\
\vdots\\\ \alpha_{m}\end{bmatrix}{\boldsymbol{\sf
E}}_{\alpha}=\boldsymbol{\alpha}{\boldsymbol{\sf E}}_{\alpha}.$

The various $\alpha_{i}$ are called roots of the generator ${\boldsymbol{\sf
E}}_{\alpha}$. The set of roots $\alpha_{i}$ can be considered as components
of the vector $\boldsymbol{\alpha}$ called the root vector, which belongs to
the $m$-dimensional root space. The root vector for each Weyl generator
${\boldsymbol{\sf E}}_{\alpha}$ can be depicted on a diagram called a weight
diagram or Weyl diagram, whose dimension is equal to the rank $m$ of the Lie
algebra $\mathfrak{g}$. Weight diagrams for rank $m\geq 3$ are difficult to
vizualize. An alternative method for depicting weight diagrams for any rank
was proposed by Dynkin in 1946 [74]. It should be noted that by virtue of
(A.1) all Cartan generators ${\boldsymbol{\sf H}}_{i}$ have roots
$\alpha_{i}=0$ and, therefore, are located in the center of the Weyl diagram.

In general, the standard form of commutation relations for generators of the
Lie algebra $\mathfrak{g}$ is written as follows:

$\begin{array}[]{lclcl}\left[{\boldsymbol{\sf H}}_{i},{\boldsymbol{\sf
H}}_{j}\right]&=&0,&&\forall i,j=1,\ldots,m;\\\ \left[{\boldsymbol{\sf
H}}_{i},{\boldsymbol{\sf E}}_{\alpha}\right]&=&\alpha_{i}{\boldsymbol{\sf
E}}_{\alpha},&&\forall i=1\rightarrow m,\;\alpha=1\rightarrow n-m;\\\
\left[{\boldsymbol{\sf E}}_{\alpha},{\boldsymbol{\sf
E}}_{-\alpha}\right]&=&\alpha^{i}{\boldsymbol{\sf H}}_{i};&&\\\
\left[{\boldsymbol{\sf E}}_{\alpha},{\boldsymbol{\sf
E}}_{\beta}\right]&=&N^{\gamma}_{\alpha\beta}{\boldsymbol{\sf
E}}_{\gamma},&&\beta\neq-\alpha.\end{array}$ (A.3)

The quantities $N^{\gamma}_{\alpha\beta}$ can also be expressed in terms of
root vectors, so we know the algebra $\mathfrak{g}$ if its roots are known.
These roots have the property

$\sum_{\alpha}\alpha_{i}\alpha_{j}=\delta_{ij},$

where $\alpha$ can take only $n-m$ values:

$\alpha=\pm 1,\,\pm 2,\,\ldots,\,\pm\frac{1}{2}(n-m).$ (A.4)

Further, the Casimir invariants (operators) ${\boldsymbol{\sf C}}_{\mu}$
commute with all generators ${\boldsymbol{\sf X}}_{i}$ of the algebra
$\mathfrak{g}$:

$\left[{\boldsymbol{\sf C}}_{\mu},{\boldsymbol{\sf
X}}_{i}\right]=0,\quad\forall\mu=1\rightarrow m,\;i=1\rightarrow n.$

The number of Casimir operators for a given Lie algebra $\mathfrak{g}$ is
determined by the rank of this algebra.

######  Theorem 3.

(Racah Theorem) For each $n$-dimensional Lie algebra $\mathfrak{g}$ of rank
$m$, there are a total of $m$ Casimir operators ${\boldsymbol{\sf C}}_{\mu}$
($\mu=1\rightarrow m$) that commute with the generators ${\boldsymbol{\sf
X}}_{i}$ ($i=1\rightarrow n$) of the algebra $\mathfrak{g}$.

As a consequence, all ${\boldsymbol{\sf C}}_{\mu}$ also commute with Cartan
generators ${\boldsymbol{\sf H}}_{i}$:

$\left[{\boldsymbol{\sf C}}_{\mu},{\boldsymbol{\sf
H}}_{i}\right]=0,\quad\forall\mu,i=1\rightarrow m.$

Thus, it is possible to find a complete set of states that are simultaneously
the eigenstates of all ${\boldsymbol{\sf C}}_{\mu}$ and ${\boldsymbol{\sf
H}}_{i}$. Let’s define a ket-vector of the following form:

$\left|c_{1},c_{2},\ldots,c_{m};h_{1},h_{2},\ldots,h_{m}\right\rangle=\left|c_{\mu};h_{i}\right\rangle,$

where $c_{\mu}$, $h_{i}$ are eigenvalues of the operators ${\boldsymbol{\sf
C}}_{\mu}$, ${\boldsymbol{\sf H}}_{i}$. Then

${\boldsymbol{\sf
C}}_{\mu}\left|c_{\mu};h_{i}\right\rangle=c_{\mu}\left|c_{\mu};h_{i}\right\rangle,\quad{\boldsymbol{\sf
H}}_{i}\left|c_{\mu};h_{i}\right\rangle=h_{i}\left|c_{\mu};h_{i}\right\rangle$

for all $\mu,i=1\rightarrow m$. We conclude that the Casimir operators
${\boldsymbol{\sf C}}_{\mu}$ and the generators ${\boldsymbol{\sf H}}_{i}$ of
the Cartan subalgebra $\mathfrak{K}$ allow us to label each state of the
multiplet, while the ladder operators ${\boldsymbol{\sf E}}_{\alpha}$ allow us
to move between states inside the multiplet, as shown on the Weyl diagram.
Thus, when the ladder operator ${\boldsymbol{\sf E}}_{\alpha}$ acts on the
ket-vector $\left|c_{\mu};h_{i}\right\rangle$, it shifts the eigenvalues of
the operators ${\boldsymbol{\sf H}}_{i}$ by the value $\alpha_{i}$ in
accordance with

${\boldsymbol{\sf
E}}_{\alpha}\left|c_{\mu};h_{i}\right\rangle\approx\left|c_{\mu};h_{i}+\alpha_{i}\right\rangle.$

## Appendix B: Quark Model

The quark hypothesis was first introduced by Gell-Mann and Zweig [4] in the
60s in an attempt to build a systematics of the hadron spectra known at that
time. According to the quark model, all existing hadrons (mesons and baryons)
“consist of” some fundamental “subunits” (quarks). To date, the quark model,
endowed with a very specific terminology (flavor, strangeness, charm, beauty,
etc.), considers six types (flavors) of quarks, combined in three generations.
The quantum numbers of quarks are given in Table A.1. (see [75]).

Table B.1. Quantum numbers of quarks  $d$ $u$ $s$ $c$ $b$ $t$ ${\sf Q}$ –
electric charge $-\frac{1}{3}$ $+\frac{2}{3}$ $-\frac{1}{3}$ $+\frac{2}{3}$
$-\frac{1}{3}$ $+\frac{2}{3}$ ${\sf I}$z – isospin $-$$\frac{1}{2}$
$-$$\frac{1}{2}$ $-$$0$ $-$$0$ $-$$0$ $-$$0$ ${\sf I}_{z}$ – $z$-component of
isospin $-\frac{1}{2}$ $+\frac{1}{2}$ $-$$0$ $-$$0$ $-$$0$ $-$$0$ ${\sf S}$ –
strangeness $-$$0$ $-$$0$ $-1$ $-$$0$ $-$$0$ $-$$0$ ${\sf C}$ – charm $-$$0$
$-$$0$ $-$$0$ $+1$ $-$$0$ $-$$0$ ${\sf B}$ – beauty $-$$0$ $-$$0$ $-$$0$
$-$$0$ $-1$ $-$$0$ ${\sf T}$ – truth $-$$0$ $-$$0$ $-$$0$ $-$$0$ $-$$0$ $+1$

The quantum numbers of quarks are related to the electric charge ${\sf Q}$ by
means of the generalized Gell-Mann–Nishijima formula

${\sf Q}={\sf I}_{z}+\frac{B+{\sf S}+{\sf C}+{\sf B}+{\sf T}}{2},$

where $B$ is the baryon number. Quarks are considered to be point-like
particles of spin 1/2 with positive parity. It should be noted that the spin
and parity of quarks are not objects of the $\operatorname{SU}(3)$-theory
(quark model), but are introduced into it from the outside “by hands” (as,
indeed, the quarks themselves). A consequence is the artificial manner to
determine the spin and parity of elementary particles based on the quark
model. Antiquarks have negative parity. The baryon number of quarks is 1/3,
the antiquarks have a negative baryon number ($-1/3$). By convention, the
flavor of a quark has the same sign as its charge ${\sf Q}$. Antiquarks have
opposite signs of flavor [26]. The hypercharge is determined by the formula

${\sf Y}=B+{\sf S}-\frac{{\sf C}-{\sf B}+{\sf T}}{3}.$

Therefore, the hypercharge of $u$\- and $d$-quarks is 1/3, $-2/3$ for the
$s$-quark and 0 for the other quarks.

According to $\operatorname{SU}(3)$-systematics, mesons are $q\bar{q}$-systems
of quark-antiquark pairs ($M=q\bar{q}$), baryons consist of three quarks
($B=qqq$), antibaryons consist of three antiquarks
($\bar{B}=\bar{q}\bar{q}\bar{q}$).

Mesons have a baryon number $B=0$. In the quark model, these states are
represented by a $q\bar{q}$-system (the flavors of the quark $q$ and the
antiquark $\bar{q}$ can be different). The total angular momentum of the
$q\bar{q}$-system (see Figure A.1) is equal to

$\mathbf{J}=\mathbf{L}+\mathbf{S},$ (B.1)

where

$|l-s|\leq J\leq l+s.$

$\bullet$$\scriptstyle
q$$\scriptstyle\bar{q}$$\uparrow$$\downarrow$$\uparrow$$\uparrow$$n$$l$$s=0$$s=1$

Figure B.1. Orbital angular momentum $l$ and radial vibrations of the
$q\bar{q}$-meson.

In this case, $s=s_{q}+s_{\bar{q}}$ is the total spin of quarks, taking the
value $s=0$ (the spins of quarks are antiparallel) or $s=1$ (the spins of
quarks are parallel), $l$ is the orbital angular momentum of the
$q\bar{q}$-system, taking the values $l=0,1,2,\ldots$. Radial excitations
(vibrations) of the $q\bar{q}$-system are denoted by the quantum number $n\geq
1$ (the principal quantum number). Thus, the quantum numbers of the
$q\bar{q}$-meson have the form

$n^{2s+1}l_{J}\quad\text{or}\quad I^{G}(J^{PC}),$

where

$P=(-1)^{l},\quad C=(-1)^{l+s},\quad G=(-1)^{I+l+s}$ (B.2)

respectively, are parity, charge parity and $G$-parity, $I$ – isospin.

The spectrum of excited (radial and orbital) meson states is similar to the
spectrum of a hydrogen atom. So, for the hydrogen atom, the lower vibrational
states are designated $1s$, $2p$, $3d$, $\ldots$, in the quark model, the
designations $1S$, $1P$, $1D$, etc. are used. Mesons are classified by
$J^{PC}$-multiplets. States with $l=0$ are pseudoscalars ($0^{-+}$) and
vectors ($1^{--}$). Orbital excitations with $l=1$ are vectors ($1^{++}$) and
($1^{+-}$), as well as tensors ($2^{++}$), etc.

However, not all experimentally detected mesons fit into the standard
$q\bar{q}$ scheme. The so-called exotic mesons, whose characteristics go
beyond the quark model, are divided into three types [76].
1) Exotics of the first kind. These are states with obviously exotic values of
such basic quantum numbers as electric charge $|{\sf Q}|\geq 2$, or
strangeness $|{\sf S}|\geq 2$, or isotopic spin ${\sf I}>1$. Such states
cannot have usual $q\bar{q}$-type quark structure and must necessarily be
exotic multi-quark states (for example, tetraquarks $qq\bar{q}\bar{q}$).
2) Exotics of the second kind. These are states having exotic combinations of
such quantum numbers as spin $J$, parity $P$ and charge parity $C$, which
mesons with the usual quark $q\bar{q}$-structure cannot have. Thus, for
neutral $q\bar{q}$-mesons with the total spin of quarks $s$ and their orbital
moment $l$, parity and charge parity are determined by the relations (B.2). It
follows that such mesons can only have combinations of quantum numbers
$C=P=(-1)^{J}$ or $(-1)^{J+1}$, as well as $C=(-1)^{J}$, $P=(-1)^{J+1}$. There
cannot be $q\bar{q}$ states with $C=(-1)^{J+1}$ and $P=(-1)^{J}$ or $J=0$ and
$C=-1$ (if $J=0$, then $s=l=0,1$, $C=+1$). Thus, exotic sets of quantum meson
numbers are combinations $J^{PC}=0^{+-}$, $0^{--}$, $1^{-+}$, $2^{+-}$,
$3^{-+}$, etc. For example, vector mesons $\pi_{1}(1400)$ and $\pi_{1}(1600)$
with an exotic $J^{PC}=1^{-+}$ combination were found (for more details, see
[26]). All kinds of exotic mesons can have such $J^{PC}$ values (both multi-
quark states, as well as hybrids and glueballs).
3) Exotics of the third kind. These are states with hidden exotics
(cryptoexotic mesons). Such states have no external exotic features, and their
structure can be determined indirectly by some specific features in their
characteristics (abnormally small widths, abnormal decay channels, special
mechanisms of formation, etc.). Exotic mesons of all kinds can also belong to
the exotics of the third kind.

Further, baryons are fermions with a baryon number $B=1$. In the quark model,
these states are represented by a $qqq$-system (anti-baryons with a
$\bar{q}\bar{q}\bar{q}$-structure have a baryon number $B=-1$). The total
angular momentum of the $q_{1}q_{2}q_{3}$-system (see Figure A.2) is also
determined by the formula (B.1). In this case,
$s=s_{q_{1}}+s_{q_{2}}+s_{q_{3}}$ is the total spin of quarks, taking the
value $s=1/2$ (the spin of one quark is antiparallel to the spins of the other
two) or $s=3/2$ (the spins of all quarks are parallel).

$q_{1}$$q_{2}$$q_{3}$$l_{\rho}$$l_{\lambda}$

Figure B.2. Angular moments $l_{\rho}$ and $l_{\lambda}$ for the $qqq$-baryon.

The total orbital moment for a $q_{1}q_{2}q_{3}$ system is

$\mathbf{L}=\mathbf{l}_{\rho}+\mathbf{l}_{\lambda},$ (B.3)

where

$\boldsymbol{\rho}=\frac{1}{\sqrt{2}}(\mathbf{r}_{1}-\mathbf{r}_{2}),\quad\boldsymbol{\lambda}=\frac{1}{\sqrt{6}}(\mathbf{r}_{1}+\mathbf{r}_{2}-2\mathbf{r}_{3})$

are the Jacobi parameters of a three-body system. Here $\mathbf{r}_{1}$,
$\mathbf{r}_{2}$, $\mathbf{r}_{3}$ are the radius vectors of quarks $q_{1}$,
$q_{2}$, $q_{3}$, the parameter $\boldsymbol{\rho}$ describes the relative
motion of quarks $q_{1}$ and $q_{2}$, and $\boldsymbol{\lambda}$ describes the
motion of the quark $q_{3}$ with respect to the “diquark” system
$q_{1},q_{2}$. The parity of the $qqq$-baryon is determined by the relation

$P=(-1)^{l_{\rho}+l_{\lambda}}.$ (B.4)

Charge parity $C$ and $G$-parity for baryons are not defined. This is due to
the fact that the charge operator $\hat{Q}$ does not commute with the charge
conjugation operator $\hat{C}$. The converse is true only for truly neutral
states or for neutral systems of type $q\bar{q}$. Truly neutral (Majorana)
fermions (baryons) have not yet been found in nature.

Thus, all $qqq$-baryons in the quark model are classified according to
combinations of quantum numbers $J^{P}$. However (as in the case with mesons),
not all experimentally detected baryons fit into $qqq$ scheme. Exotic baryons
[77] whose characteristics go beyond the quark model are states with $|{\sf
Q}|>2$, or ${\sf I}>3/2$, or ${\sf S}>0$. Such states cannot have the usual
$qqq$-type quark structure and must necessarily be exotic multi-quark states
(for example, pentaquarks $qqqq\bar{q}$ or dibaryons $qqqqqq$). In 2015, LHCb
collaboration published the first experimental evidence of the existence of
charmed pentaquark states $c\bar{c}uud$ with a mass of about 4,4 GeV (states
$P_{c}(4312)^{+}$, $P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$, see [78]).

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