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Home , Pionium, Relativistic particle

All modeled using only and with relativistic theory Ray Fleming [email protected]

All mesons were investigated to determine if they can be modeled with the onium model discovered by Milne, Feynman, and Sternglass with only electrons and positrons. They discovered the relativistic solution has the mass of a neutral and the relativistic onium mass increases in steps of me/α and me/2α per which is consistent with known mass quantization. Any pair of or resonances can orbit relativistically and particles and resonances can collocate to form increasingly complex resonances. are positronium, are , D mesons are kaonium, and B mesons are Donium in the onium model. , which are addressed in another paper, have a non-relativistic combined with mesons. The results of this analysis shows that the compo- sition, charge, and mass of all mesons can be accurately modeled. Of the 220 mesons mod- eled, 170 mass estimates are within 5 MeV/c2 and masses of 111 of 121 D, B, charmonium, and bottomonium mesons are estimated to within 0.2% relative error. Since all mesons can be modeled using only electrons and positrons, and theory are unnecessary.

1. Introduction 2. Method This paper is a report on an investigation to find Sternglass and Browne realized that a neutral pion whether mesons can be modeled as combinations of (π0), as a relativistic - pair, can orbit a only electrons and positrons using onium theory. A non- or resonance in what Browne companion paper on baryons is also available. The in- called a trionium compound as shown in figure 1. troduction and method are abbreviated here since there Browne also realized that onium resonances can be is a book and other papers that discuss the method in nested or collocated. A non-relativistic electron or pos- greater detail [1,2,3]. itron can combine with a π0 to form a or pion Milne, Feynman, and Sternglass discovered the rel- [11]. The mass estimates by Sternglass are given in ta- ativistic positronium solution where the electron and ble 1 [7]. 2 positron gain ~70 MeV/c (me/α) each giving it a mass of ~140 MeV/c2 [4,5,6,7]. This is a classical model of the neutral pion that was further refined by Sternglass. A similar derivation by Browne published in Nature is recommended reading [8]. A negative paper was pub- Fig. 1. A Browne type muon as a positron orbited by two electrons. lished where the author left out the centrifugal force term [9], but even the non-relativistic positronium so- From there it is possible to consider resonances with lution would be unstable without the centrifugal force 2, 3, 4, or more pions. For convenience they can be term countering Coulomb attraction. classified as Group 2, 3, 4, and so on, with The relativistic positronium solution is general as it counted as Group 1 resonances along with pions. It is applies to any pair of oppositely charged particles or possible to work toward increasing numbers of pions resonances with any mass. Unstable particle masses and masses in the model and match it with known par- are known to be quantized in factors of me/2α = 35 ticle masses, but it is also important to analyze the de- 2 2 MeV/c and me/α = 70 MeV/c [10]. Therefore, it is cay products so they are accounted for. possible to construct a particle model where all parti- A resonance’s composition is usually found in its cles are modeled using onium theory. most frequent decay modes or the ones with the most pions present. Electron-positron or - quantum fluctuations can be excited to form additional pions or proton-antiproton pairs, but in most cases the There is also the case where only one pion is rela- maximum number of pions in any decay mode corre- tivistic as shown in figure 3. These are referred to as lates with the resonance’s composition. Note that only KD kaons in the author’s papers since they appear to be ± electrons and are referred to as particles from deexcited relative to the regular kaons. A KD has an this point forward. estimated mass of 384.69 MeV/c2, however, they fre- In mesons and baryons each relativistic electron quently have more net mass in other resonances such * ± 2 adds a mass of me/α or me/2α, so the mass can be cal- as the K (892) where they add 397.47 MeV/c . culated by summing the number of electrons in relativ- istic orbit. This mass is then added to the masses of the non-relativistic particles or resonances. Each μ± or π± contains three electrons, so a single relativistic one adds 3 x 35 MeV/c2 ≈ 105 MeV/c2 in relativistic mass Fig. 3. A KD as a stationary pion orbited by a and two in relativistic orbit add ~210 MeV/c2 in rela- relativistic pion. tivistic mass. Note that K*(892)± resonances can decay to four pi- There are three-electron (eee) resonances as in fig- ons and are eKDK resonances in the onium model. The 0 * 0 ure 1, or an eπ resonance where the entire pion is in K (892) is also a KDK resonance. Research needs to orbit as in the Sternglass muon and pion. There are be conducted to explain how some KD resonances get conceivably eμ and eπ resonances that may be nonrela- their extra mass. Note that an ω is a KDKD resonance tivistic or relativistic. With two pions there can be μπ with both masses. The larger mass is used for mass es- and ππ resonances that are non-relativistic or with one timates in this paper unless otherwise stated. or both in relativistic orbit. A basic relativistic μπ res- onance is shown in figure 2 which forms a K-long.

Fig. 4. A possible configuration of an η meson.

Fig. 2. A K-long neutral kaon composed of a muon The η is the only well-known Group 3 meson and it and pion is a πKD resonance as illustrated in figure 4 as those masses sum to 537.04 MeV/c2 which is reasonably The masses of two charged pions plus ~210 2 2 2 close to the measured mass of 547.862 MeV/c . The η' MeV/c in relativistic mass sums to 489.22 MeV/c . with a mass of 957.78 MeV/c2 is the lowest mass group Adding a central electron brings it to 489.73 MeV/c2. 0 2 ± 5 meson as an ηKD resonance which has an estimated These are the K (497.614 MeV/c ) and K (493.677 2 2 mass of 945.33 MeV/c . In both cases the KD picks up MeV/c ) kaons [1,2]. The slight differences in mass are a significant amount of extra energy. The other basic presumed to be due to magnetic, spin, or quantum field Group 5 resonance is a πKDK resonance with an esti- effects. Note that all particle data not otherwise cited mated mass of 1017.94 MeV/c2 which is the φ with a comes from Particle Data Group (PDG) publications known mass of 1019.461 MeV/c2. [12,13]. Kaons may have 6 or 7 kaons so when two kaons combine in a relativistic orbit a total of 6, 7, 13, or 14 electrons may be relativistic. A neutral K-long kaon is shown in figure 2 as they commonly decay to a muon and a pion while K-short Fig. 5. An illustration of a ρ- meson. kaons usually decay to two pions. The different orbits of the μ and π lead to the them having different life- The last of the lower mass resonances upon which most others are built are the ρ mesons with the ρ± hav- times. But their masses are similar since there is no 2 0 ing a mass of 775.11 MeV/c . The combined masses of preferential orbital direction when two π pions are 2 combined. The charge of the muon determines if a K- a ππK, as illustrated in figure 5, is 776.75 MeV/c . long decays preferentially to (e-) or antimatter They have nested orbits, however the μπ orbit is likely (e+) [8,9]. This effect is responsible for all other me- the smaller of the two. The illustrations in the paper sons that have matter and antimatter variants. should be thought of as representations of the compo- sition only. The orbits are not real classical orbits, and

2 may not be in the proper size order, direction, or even a KDKD, KDK, or KK pair. The combined masses of an in the same plane. Those types of details need to be eKDKD resonance as shown in figure 6 with 980.35 worked out for each resonance. MeV/c2 in orbital mass is 1775.80 MeV/c2 which is a In many cases the mass can also change by a factor . This may be surprising to some, but since a tau has of ~35 MeV/c2. This may happen when a pion replaces eight known decay modes to two kaons and 30 decay a muon, or has a similar orbital shift. A KDK, or KK modes to four pions it is necessary to treat for them as resonance may contain 13, 14 or 15 relativistic elec- Group 4 mesons [9]. Both mu and tau particles are me- trons and a 4K or DD resonance usually contains 27, sons in onium theory rather than further sim- 28, 29, or 30 relativistic electrons. Those cases will be plifying particle theory. pointed out, but there was an attempt to keep the rela- tivistic orbits consistent throughout this study to best illustrate the model’s effectiveness. Table 1 summarizes the basic mesons discussed in Fig. 6. A negative tau with a positron orbited by this section. Most mesons are formed from these me- two negative KD kaons. sons plus additional relativistic orbital mass. In onium theory there is also a neutral KDKD reso- Table 1 Composition and mass of common nance with an estimated mass of 1775.29 MeV/c2. This lower mass mesons. is most likely the K2(1770) which has a measured mass 2 Symbol Comp. Mass Est. of 1773 MeV/c . Note, however that the K2(1770) can (MeV/c2) (MeV/c2) decay to more than two kaons, so the data may include ± 2 μ eπ 105.65837 105.62 KKDKD resonances with 490.18 MeV/c in orbital 0 π ee 134.9766 134.4 mass. Onium theory also allows for a KDKD resonance π± eπ 139.57018 139.76 ± with only 13 relativistic electrons giving it a mass of KD eππ 397.47 384.69 2 0 1705.27 MeV/c . This is most likely the K0(1710). KD ππ 396.96 384.18 K± eμπ 493.677 489.73 K-long μπ 497.614 489.22 K-short ππ 497.614 489.22 η πKD 547.682 537.04 ± + ρ ππμπ 775.11 776.75 Fig. 7. A as a positron and a KDK pair ρ0 ππμπ 775.26 772.82 ω KDKD 782.65 782.16 Next are the KDK and the eKDK , illustrated in figure * ± K (892) eKDK 891.66 891.66 7, with estimated masses of 1871.50 and 1872.01 * 0 2 0 ± K (892) KDK 895.81 895.08 MeV/c . These are the D and D with masses of η'(958) ηKD 957.78 945.33 1864.84 MeV/c2 and 1869.61 MeV/c2 respectively. It φ πKDK 1019.461 1017.94 is not clear if there is a lower mass resonance near 1801 To keep this paper brief, the charge details may not MeV/c2 with only 13 relativistic electrons, but the be mentioned in some cases as a central electron can f0(1800) is closest to that mass. always be added or deleted to change the charge so it is 0 or ±1. However, as in baryons the onium model allows for mesons with ±2 charge. For example a π- + + + * ++ orbited by one π and a π π pair would be a K (892) . - Fig. 8. A Ds meson with a positron orbited by a KK Finding ±2 charge mesons is one way to confirm the pair. onium theory. Also note that resonances with the same charge in an orbit are responsible for many rule viola- Then the eKK resonance illustrated in figure 8 has an estimated mass of 1968.22 MeV/c2. That is clearly tions found in resonance decays. ± 2 a DS which has a measured mass of 1968.30 MeV/c . 3. The D mesons Note that it can have oppositely charged kaons as well. 0 There is no clear Ds resonance but it could possibly In onium theory D mesons are two-kaon resonances be the π2(2005) with a PDG estimated mass of 1963 2 with me/α orbits that typically add 980.35 MeV/c in MeV/c2 or one type of resonance listed under the mass to the resonance indicating 14 relativistic elec- a4(1970) as it has several different decay modes indi- 2 trons add ~70 MeV/c each. The resonance may have cating it contains 8 or 9 pions. A KK resonance with 3

13 relativistic electrons has an estimated mass 1897.68 MeV/c2 which could be one resonance listed under the f2(1910) which could also include a Group 8 meson based on its ωω decay mode. * - The next D mesons have increased mass due to a Fig. 11. A D s0(2317) as a pion orbited by muon and a KK pion in the center forming πKDK resonances with esti- pair. 2 0 ± mated masses of 2006.48 and 2011.07 MeV/c for the The D1(2420) and D1(2420) have masses of *0 neutral and charged versions. These are clearly the D 2420.8 and 2423.2 MeV/c2 respectively. The first de- *± and D with measured masses of 2006.96 and 2010.26 cays to a D*(2010)± and a charged pion or a neutral D 2 MeV/c . The next step up in mass from that is the πKK meson and two oppositely charged pions. The charged *± 0 resonance, the Ds , with a measured mass of 2112.1 one decays to a D1(2007) and a pion. They appear to 2 2 MeV/c and an estimated mass of 2007.28 MeV/c . It be Group 7 mesons with a KDK resonance. The sim- is shown in figure 9. A πKDKD resonance would have plest resonance with that mass is an ηKDK as shown in 2 mass of 1914.86 MeV/c which could possibly be the figure 12 with a mass of 2419.36 MeV/c2. * 2 K 0(1950) with a mass of 1917 MeV/c .

0 Fig. 12. A D1(2420) as an eta orbited by a KDK *± Fig. 9. An illustration of a Ds . pair. 0 2 The next D mesons in mass order are not well estab- The D1(2420) similarly has a mass of 2427 MeV/c * 0 lished experimentally as the D 0(2300) was formerly but with much greater measurement error of ±26. It * 0 2 * the D 0(2400) with a PDG average of 2300 MeV/c also decays to a D and a pion but due to the large and a previous measurement of 2407 MeV/c2. There is measurement error it is difficult to know what it is. A * ± * 2 also the D 0(2300) with a measured mass of 2349 KDD resonance has a mass of 2407.73 MeV/c and a MeV/c2. They all decay to a D meson and two pions πKt resonance has a mass of 2414.04 MeV/c2, so those indicating they may be Group 6 mesons. and the ηD shown above are the closest matches. ± 2 The Ds1(2460) has a mass of 2459.5 MeV/c and ± * ± can decay to a Ds and two pions or a D s0(2317) mak- ing it a Group 6 meson with a KK resonance. The mass

* + of a KKK resonance as illustrated in figure 13 with a Fig. 10. An illustration of a D 0(2300) . K orbited by a KK pair with 980.35 MeV/c2 in orbital ± 0 A K and D resonance as shown in figure 10 has a mass totals to 2461.38 MeV/c2. combined mass of 2358.52 MeV/c2 which could ex- * ± 2 plain the D 0(2300) . If it has a 910.33 MeV/c orbit instead, its mass would be 2295.15 MeV/c2 which is * 0 consistent with the D 0(2300) . Then a πKDD reso- + + nance has a combined mass of 2406.65 MeV/c2 which Fig. 13. A Ds1(2460) as K orbited by a KK pair. * 0 * 0 * ± could explain the D 0(2400) observation. There are also the D 2(2460) and D 2(2460) with * ± The D s0(2317) is also in this mass range as it has similar masses of 2460.7 and 2465.4 MeV/c2 respec- 2 * 0 * + a mass of 2317.8 MeV/c . It can decay to a Ds and two tively. The D 2(2460) can decay to a D (2010) plus a * pions or a D s and a pion so it is another Group 6 me- pion indicating they are Group 6 or 7 baryons with a * ± son. The mass of a μπKK resonance like the one illus- KDK resonance. They cannot be KKDK (D 0(2400) ) 2 ± trated in figure 11, where the μ or π has a 105 MeV/c and KKK (Ds1(2460) ) resonances as those are other orbit and the KK pair is in the 980.35 MeV/c2 orbit, has resonances. An ηπt resonance has a mass of 2464.29 a total mass of 2317.97 MeV/c2. The KK pair forms a MeV/c2, and with a π0 as illustrated in figure 14, a mass Ds meson which is consistent with its decay modes. of 2459.70 MeV/c2. Those appear most likely to be the * D 2(2460) resonances.

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one resonance. It has been seen to decay to D and D* mesons plus a pion. A KπD resonance with the π in orbit around the kaon, has a calculated mass of 2607.90 * 0 2 Fig. 14. A D 2(2460) as an eta orbited by an MeV/c matching the lower mass measurements. electron-positron pair and a KDKD pair. The X(2632) has been seen to decay to an ηDs or 0 2 ± 2 KD with masses of approximately 2632 MeV/c . The The Ds1(2536) has a mass of 2535.10 MeV/c and * 0 sum of the masses of a πKDπDs resonance is 2632.13 can decay to D (2007) and a kaon meaning it is likely 2 to be a Group 7 meson. Based on its principal decay MeV/c with the KD in the lower energy state. The πKD accounts for the η in that decay mode. This resonance modes it has a KDK resonance rather than a KK reso- likely accounts for the X(2632). nance, but its third kaon can cause it to occasionally * ± * The next better known D meson is the D (2640) decay to a strange D meson. A πKDD resonance has a 2 2 which has a mass of 2637 MeV/c and decays to a mass of 2534.52 MeV/c with the KD in the lower en- D*(2010)± and two pions, making it a Group 7 or 8 me- ergy state. * 0 son with a D resonance. It turns out that the sum of the The D(2550) has an imprecisely known mass of * 2 * masses of a ππKD resonance as a πKD is 2637.66 2564 MeV/c . It decays to a D and a pion, so it prob- 2 ably contains a D*. The combined masses of an η and MeV/c . Since a ππK is effectively a ρ meson it is il- D* is 2558.12 MeV/c2 so that is probably what it is. lustrated that way in figure 16, however it could be in Note that the list of D mesons skips over the KD* a πKπD configuration instead. resonance which would have a mass near 2500.64 MeV/c2. Out of all the observed resonances that is most likely to be the K4(2500) which has an estimated mass 2 of 2490 ±20 MeV/c . The K4(2500) has been seen to 2 * ± decay to a lambda and antiproton, and a 980.35 MeV/c Fig. 16. A D (2640) as a rho orbited a KDK pair. orbit would help facilitate the excitation of a quantum * ± Next in mass is the D s1(2700) with a mass of proton-antiproton pair. The K3(2320), which has an es- 2 2708.3 MeV/c . It can decay to a D or Ds and a kaon, timated mass of 2324 ±24 MeV/c2 and similarly decays * so it has at least three kaons. A KπDs resonance with to a proton and antilambda, may be a D s0(2317) or a the pion in relativistic orbit has a mass of 2706.59 similar mass D meson. MeV/c2 which is a good match. There should also be a πD* resonance with a mass of 2149.83 MeV/c2. The closest match to that is the X(2150) which decays to a proton-antiproton pair. * Similarly, a KDD resonance has a mass of 2267.08 2 MeV/c which is within the measurement error for the * * + Fig. 17. D 3( )± a K (892) X(2260) which also decays to a proton-antiproton pair. A 2750 as orbited a KDK pair. * 2 The D 3(2750) has a mass of 2763.5 MeV/c and de- cays to a D or D* and a pion which means it probably * + contains a KDK resonance. It turns out that a K (892) orbited by a KDK resonance has a mass of 2763.16 + 2 Fig. 15. A Ds2(2573) as an eta orbited by pion MeV/c , so it appears to be the resonance shown in fig- and a KDKD pair. ure 17. * ± Next is the D sJ(2860) with a mass of 2863.2 Next on the standard list of D mesons is the 2 * 2 MeV/c . It decays to a D or D and a kaon which means Ds2(2573) with a mass of 2569.1 MeV/c . It can decay * to a D0 and a kaon. An ηπt resonance with the pion in it contains at least three kaons. The K (892)KK reso- nance similar to the last resonance has a mass of relativistic orbit as shown in figure 15, has a mass of 2 2 2859.37 MeV/c . 2569.33 MeV/c . 0 * 2 Next is the D(3000) with an approximate mass of The D J(2600) has a mass of 2623 ±12 MeV/c . It 2 was previously listed as having a mass of 2612 MeV/c2 3000 MeV/c based on two earlier measurements with a more recent experiment giving a mass of 3214 ±29 and the current measurements range from 2608.7 to 2 * * * 2681.1 MeV/c2 which obviously represent more than MeV/c . It decays to a D and a pion. A K (892)Ds 5 resonance has a mass of 3003.76 MeV/c2 which is con- Table 2 Continued. sistent with the lower mass measurements. Symbol Comp. Mass Est. Looking at resonances above 3200 MeV/c2, a (MeV/c2) (MeV/c2) 2 * ± KDKD with the KDK in a 490.18 MeV/c relativistic D s0(2317) μπDs 2317.8 2317.97 * ± orbit has a total mass of 3250.93 MeV/c2, and the one D 0(2400) πKDD 2407 2406.65 0 2 D1(2420) ηD 2420.8 2419.36 with a 455.16 MeV/c relativistic orbit has a total mass * 0 2 D 1(2430) ηD 2427 2407.73 of 3215.92 MeV/c matching the higher measurement * 0 D 2(2460) ηπt 2460.7 2459.70 0 * ± listed under the D(3000) . The heavier KDKD reso- D 2(2460) ηπt 2465.4 2464.79 ± nance may be the X(3250) and the X(3350) may be the Ds1(2460) KDs 2459.5 2461.38 * related KKD resonance which has a mass of 3347.14 K4(2500) KD 2490 2500.64 ± * MeV/c2. They both decay to a proton and antiproton Ds1(2536) πKDD 2535.11 2534.52 D(2550)0 ηD* 2564 2558.12 like other D mesons mentioned previously. Ds2(2573) μKDs 2569.1 2570.98 * The most massive D meson from the PDG list that D J(2600) KπD 2608.7 2607.90 ± is analyzed here is the DsJ(3040) with a mass of 3044 X(2632) πKDπDs 2632 2632.13 MeV/c2. It decays to a D* and a kaon. A πK*(892)D* D*(2640)± ππKD 2637 2637.66 * ± 2 D s1(2700) KπDs 2708.3 2706.59 resonance has a mass of 3041.49 MeV/c which is a * * D 3(2750) K (892)D 2763.5 2763.16 good match. * ± * D sJ(2860) K (892)Ds 2863.2 2859.37 0 * * The X0(2900) and X1(2900) made news as potential D(3000) K (892)Ds 3000 3003.76 ± * * but they decay to a kaon and D meson and DsJ(3040) πK (892)D 3044 3041.49 in onium theory they are KD resonances. Their masses X0(2900) KD 2850 2853.46 2 X1(2900) πKDD 2900 2896.83 are roughly 50 MeV/c apart with the X0(2900) having * 2 K(3100) KDs 3100 3095.95 a mass of approximately 2850 MeV/c and the D(3000)0 K KD 3214 3215.92 2 D X1(2900) about 2900 MeV/c . A D meson orbited by a X(3250) KDKD 3250 3250.93 kaon with 490.18 MeV/c2 in orbital mass has a total X(3350) KKD 3350 3347.14 2 mass of 2853.46 MeV/c . A similar πKDD resonance 2 Table 2 summarizes the D mesons and related reso- has a mass of 2896.83 MeV/c . There are also KKD nances discussed in this section showing their compo- and πKDKD resonances with similar masses. Two re- sition, measured mass, and estimated mass. Their com- lated resonances are a Ds orbited by a kaon with a mass 2 * positions make the onium model pattern quite clear. of 2952.15 MeV/c and a Ds orbited by a kaon with a 2 mass of 3095.95 MeV/c . The later could be the 4. The Bottom Mesons K(3100) which is known to decay to a lambda and an- tiproton. The lowest mass bottom meson resonances are the B± and B0 with masses of 5279.34 and 5279.65 MeV/c2 Table 2 Composition and mass comparison of D mesons and related resonances. respectively. Both have numerous decay modes to many types of resonances, with many including two D Symbol Comp. Mass Est. mesons or four kaons. Similar to the D mesons, the (MeV/c2) (MeV/c2) base B mesons can be expected to be DDD resonances f0(1710) KDKD 1704 1705.27 K2(1770) KDKD 1773 1775.29 with the D mesons being KDK resonances as shown in ± ± τ eKDKD 1776.86 1775.80 figure 18. That means the outer D meson is a D with ± 2 D eKDK 1869.61 1872.01 a mass of 1869.61 MeV/c and the inner one is a DD D0 K K 1864.84 1871.5 2 D with a mass of a KDK resonance with 490.18 MeV/c f2(1910) KK 1900 1897.68 in orbital mass totaling 1381.32 MeV/c2. π2(1770) KK 1963 1967.71 ± DS eKK 1968.3 1968.22 * ± D (2010) πKDK 2010.26 2011.07 * 0 D (2007) πKDK 2006.96 2006.48 *± DS πKK 2112.1 2107.28 * X(2150) πD 2150 2149.83 - Fig. 18. A B as a KDKD resonance. X(2260) KDD 2260 2267.08 * 0 0 0 D 0(2300) KD 2300 2295.15 Subtracting those DDD masses from the B leaves * ± 0 D 0(2300) KD 2349 2358.52 2028.38 MeV/c2 which is approximately 29 x 70

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MeV/c2 indicating 29 electrons are in relativistic orbit. mass for the ω, has a total mass of 5698.51 MeV/c2 ± A D has 15 electrons and the DD has 14 relativistic with the extra KD in the lower energy state. electrons, so that fits and is consistent with other bot- tom mesons discussed in this section. A DDD reso- 2 nance composed of a neutral KDK with 490.18 MeV/c in orbital mass, plus a D meson with an additional 2 2030.73 MeV/c in orbital mass totals to 5281.67 0 * 2 0 Fig. 21. A B1(5721) as a KDKDKDD resonance. MeV/c . A B with an extra electron in the KD has an 0 2 estimated mass of 5282.18 MeV/c2. The B1(5721) has a mass of 5725.9 MeV/c and it The related KKt resonance has a mass of 5285.12 decays to a B* and a pion. The next step up in mass has 2 * MeV/c and lower energy KDKDt and KDKt resonances an additional two pions. A KDKDKDD as illustrated in have masses of 5092.71 and 5188.92 MeV/c2 respec- figure 21, has a combined mass of 5723.58 MeV/c2 tively. If those or other bottom mesons predicted by the given the standard orbital masses. onium model are found they would confirm the model.

* 0 Fig. 22. A B 2(5747) as a ηKDKt resonance.

* * * + 2 Fig. 19. A B as a KDKDD resonance. The B 2(5747) has a mass of 5737.2 MeV/c and * 0 2 The B* has a mass of 5324.70 MeV/c2 and decays to the B 2(5747) has a mass of 5739.5 MeV/c . They * a B and a gamma and is 45.05 MeV/c2 more massive both decay to a B or B and a pion. An ηKDKt reso- than the B0. The next step up in mass requires adding a nance as shown in figure 22, has a mass of 5736.78 * 2 pion and changing a K to a KD. A KDKDD resonance MeV/c . Similarly, an ηKDKD resonance has a mass of 2 as shown in figure 19, with the same orbital masses as 5829.53 MeV/c which is a great match for the 0 2 the B0, has a mass of 5326.11 MeV/c2. Bs1(5830) with a mass of 5828.70 MeV/c . It decays * to a B and a kaon, but not a Ds so it is not a true Bs with a KK resonance.

0 Fig. 20. A Bs as a relativistic KDKDs resonance. 0 The bottom strange meson Bs has a mass of 0 * 5366.88 MeV/c2. Strange D mesons are identifiable by Fig. 23. A Bs2(5840) as a πKDωD resonance. 0 0 their KK resonance. A B with an outer KK pair, as The Bs2(5840) is only slightly more massive with a illustrated in figure 20, has a mass of 5366.98 MeV/c2 mass of 5839.86 MeV/c2. It also decays to a B* and a * if the inner KD is in the lower energy state like in an ω. kaon, rather than a Ds, so it likely contains a D . The * * * The bottom strange meson B s has a mass of 5415.4 D resonance which is the best match is the πKDωD MeV/c2 and is 48.51 MeV/c2 more massive than the which has a combined mass of 5838.08 MeV/c2 with 0 * Bs . Like the B the next step up in mass adds a pion the KD in the lower energy state. The πKD is a lower * with a K converted to a KD. A relativistic ωD s reso- energy η-like state as illustrated in figure 23 ± 2 nance with the same orbital masses as used previously The BJ(5840) has a mass of 5851 MeV/c and the 2 0 2 has a mass of 5415.66 MeV/c . BJ(5840) has a mass of 5863 MeV/c . They can both The X(5568) is one of the recent additions to the list decay to a B or B* and a pion. They are about 140 2 2 0 of bottom barons with a mass of 5566.9 MeV/c and it MeV/c more massive than a B1(5721) and a similar * * 2 decays to a Bs and a pion. A πKDKDD s resonance has πKDKDKDD resonance has a mass 5863.15 MeV/c . 2 * a total mass of 5567.52 MeV/c . The related KDπωD resonance has a total mass of ± Another more recent addition is the BJ(5732) which 5850.58 which is a good match for the BJ(5840) . * 2 has a lower measured mass than its name implies at The B sJ(5853) has a mass of 5853 ±15 MeV/c . 5698 MeV/c2. It is known to decay to a B* and a pion. There is little data on this resonance and it is uncon- * 2 A KDωD resonance with 490.18 MeV/c in orbital firmed, but with that mass it could be another 7

* ± πKDKDKDD resonance like the BJ(5840) but with a Table 3 Composition and mass comparison of somewhat different orbital combination. The simplest bottom mesons. * strange resonance with a similar mass is a KDKDD s Symbol Comp. Mass Est. 2 2 2 with the KD pair in a 910.33 MeV/c orbit giving it a (MeV/c ) (MeV/c ) + total mass of 5848.10 MeV/c2. B eKDKD 5279.34 5281.67 0 + 2 B KDKD 5279.65 5282.16 The BJ(5970) has a mass of 5964 MeV/c and the * * 0 2 B KDKDD 5324.70 5326.11 BJ(5970) has a mass of 5971 MeV/c . Both decay to a 0 Bs KDKDs 5366.88 5366.98 * * * B or B and a pion so they are also likely to be based B s ωD s 5415.4 5415.66 * * * on a D rather than a Ds. An ηKDKD resonance as X(5568) πKDKDD s 5566.9 5567.52 * * shown in figure 24 has a total mass of 5970.18 MeV/c2. B J(5732) KDωD 5698 5698.51 + * + *0 B1(5721) KDKDKDD 5725.9 5723.58 The BJ(5970) could have a D giving it an estimated 0 * 2 B1(5721) KDKDKDD 5726.1 5723.58 mass of 5966.70 MeV/c . Also note that a DsDs reso- * + B 2(5747) ηKDKt 5737.2 5736.78 2 * 0 nance with 2030.73 MeV/c in orbital mass has a total B 2(5747) ηKDKt 5739.5 5736.78 2 0 mass of 5967.33 MeV/c , so that resonance should be Bs1(5830) ηKDKD 5828.70 5829.53 0 * found in the same range of masses. Bs2(5840) πKDωD 5839.86 5842.02 * 0 * B sJ(5850) KDπKDKDD 5853 5850.37 + * BJ(5840) πKDωD 5851 5850.86 0 * BJ(5840) πKDKDKDD 5863 5863.15 + * BJ(5970) ηKDKD 5964 5966.70 0 * BJ(5970) ηKDKD 5971 5970.18 + Fig. 24. A B (5970)+ as a relativistic ηK KD* res- Bc KKKDKD 6274.9 6274.56 J D ± onance. Bc(2S) KDKDKtt 6871.6 6873.07 + The bottom charmed meson Bc has a mass of 5. Bottomonium mesons 6274.9 MeV/c2. This meson can decay to a J/Psi and a D meson indicating that it contains 6 kaons. The com- The bottomonium resonances are reported on next bined masses of 2K0 and a B+ is 6274.56 MeV/c2, so because of they are essentially a continuation of the the simplest model is a B+ orbited by two muon-pion bottom mesons. While the bottomonium mesons are pairs as illustrated in figure 25. Some of the decay said to be various onium states of a bottom-antibottom modes contain as many as 16 pions so other combina- quark pair, in onium theory they are based on eight- tions should be explored such as a πηtt with a mass of kaon or four-D-meson resonances. Most heavier reso- 6271.88 MeV/c2. nances from Y(4S) and up decay to two bottom me- sons. In the these are also onium reso- nances, but the compositions and masses are not ac- counted-for as well as they are in the onium model.

+ + Fig. 25. A Bc as a Bc orbited by two K-long ka- ons. ± The last of the bottom baryons studied is the Bc(2S) 2 + Fig. 26. A Y(1S) as a pion orbited by tt and DD with a mass of 6871.6 MeV/c . It decays to the Bc plus pairs. two pions. The combined masses of KDKDKtt with an 2 2 additional 490.18 MeV/c in the KDK orbit has a total The Υ(1S) with a mass of 9460.30 MeV/c is the mass of 6873.07 MeV/c2 which confirms that there is base state of the group, but not the lightest. It decays to at least one resonance with this energy with three two tau particles and numerous different pairs of four- charmed orbits. kaon resonances. Its mass is simple to compute as the Table 3 provides a summary of the bottom reso- combined masses of a πttDD resonance as illustrated nances discussed in this section including their compo- in figure 26, and 2030.73 MeV/c2 in orbital mass totals sition, measured mass, and estimated mass. The trend to 9463.24 MeV/c2. It could also be a πtDtD resonance. in the composition again shows how well the onium This is the same relativistic orbital energy as in the bot- model works and the mass estimates are accurate. tom mesons as lightest resonances in the upsilon group have only one relativistic pair of D mesons.

8

KΨ(2S)Ψ(2S) resonance with 2030.73 MeV/c2 in or- bital mass, the total is 9900.54 MeV/c2. 2 Next is the Xb2(1P) with a mass of 9912.21 MeV/c . It also decays from a Υ(2S) and decays to a Υ(1S). A Fig. 27. A ηb(1S) as a tt pair orbited by a DD pair. 2 KttDDs resonance with 2030.73 MeV/c in orbital Currently, the lowest mass bottomonium meson is mass has a combined mass of 9916.04 MeV/c2. After 2 2 the ηb(1S) with a mass of 9398.7 MeV/c . It is seen in that is the ηb(2S) with a mass of 9999.0 MeV/c . There the decays of a Υ(1S) and is known to decay to two tau is not any useful decay information. The simplest res- 2 * particles. If we subtract the 2030.73 MeV/c in orbital onance with a similar mass is a πKDttDD with 2030.73 mass and divide by two we get 3683.98 MeV/c2 which MeV/c2 in orbital mass which totals to 10001.36 is the mass of a Ψ(2S) (tD) resonance indicating it is a MeV/c2. tDtD resonance. A similar ttDD resonance as illus- trated in figure 27, with 2100.76 MeV/c2 in orbital mass has a total mass of 9393.70 MeV/c2. Most upsilon group resonances have one DD pair with 2100.76 2 Fig. 30. A Y(2S) as a kaon orbited by tt and MeV/c in orbital mass which is indicative of 30 rela- DD*pairs. tivistic electrons. Note that lower mass tttt or πtttt res- onances may exist with masses of 9208.20 and 9277.74 The next resonance in the main upsilon series is the 2 MeV/c2 respectively. Υ(2S) with a mass of 10023.26 MeV/c . It decays most often to a Υ(1S) and two pions indicating an extra kaon is present. A KttDD* resonance as illustrated in figure 30, with 2100.76 MeV/c2 in orbital mass, has a total mass of 10024.72 MeV/c2.

* * 2 Fig. 28. A Xb0(1P) as a D D pair orbited by a DD Next is the Υ2(1D) with a mass of 10163.7 MeV/c . pair. It also decays to a Υ(1S) and two pions indicating an extra kaon is present. It is approximately 140 MeV/c2 Next on the list is the Xb0(1P) with a mass of 2 more massive that the Υ(2S) likely meaning that a D is 9859.44 MeV/c . It decays from a Υ(2S) and decays to * * 2 a Υ(1S). Subtracting 2030.73 MeV/c2 in orbital mass now a D . A πKttDD resonance with 2100.76 MeV/c 2 in orbital mass has a combined mass of 10164.29 and dividing by two equals 3914.36 MeV/c , which is 2 * * * MeV/c . the mass of a X(3915) a DD resonance. A D D DD 2 The Xb0(2P) has a mass of 10232.5 MeV/c . It de- resonance as illustrated in figure 28, with 2100.76 * * 2 cays from a Υ(3S) and decays to a Υ(2S). A πKttD D MeV/c in orbital mass has a total mass of 9860.50 2 MeV/c2. resonance with 2030.73 MeV/c in orbital mass has a total mass of 10231.62 MeV/c2 which may account for this resonance. The Xb1(2P) has a similar mass of 10255.46 MeV/c2 and a KtDD*D* resonance with 2100.76 MeV/c2 in orbital mass has a mass of 2 Fig. 29. A Xb1(1P) as a K orbited by tt and DD 10254.82 MeV/c . Then the hb(2P) has a mass of 2 pairs. 10259.8 MeV/c and decays to a ηb(2S). A step up in * * 2 mass from the πKDttDD ηb(2S) is a πKtDDD reso- Next is the Xb1(1P) with a mass of 9892.78 MeV/c . 2 It also decays to a Υ(1S). The combined masses of a nance which has a mass of 10260.34 MeV/c . The Xb2(2P) is only slightly heavier with a mass of KttDD resonance as illustrated in figure 29, with 2 2 10268.65 MeV/c . It can decay to an omega and a 2100.76 MeV/c in orbital mass has a total mass of * * 2 Υ(1S) or a Xb2(1P) and two pions. Another KtDD D 9891.31 MeV/c . 2 2 resonance with 2100.76 MeV/c in orbital mass has a The hb(1P) has a mass of 9899.3MeV/c and it de- combined mass of 10265.36 MeV/c2. It is illustrated in cays to an ηb(1S), a ttDD resonance. It also appears to be a KtDtD resonance but slightly different from the figure 31. The last three resonances show that slightly different KtDD*D* orbital combinations can account Xb1(1P). In the next section it is shown that a Ψ(2S) is for the different masses of similar resonances. a tD resonance so if the hb(1P) mass is calculated as a

9

2 MeV/c and a BBs resonance has a mass of 10646.22 MeV/c2. So there are two BB resonances with approx- imately the same mass.

* * Fig. 31. A Xb2(2P) as a kaon orbited by D D and tD pairs. The Υ(3S) is next with a mass of 10355.2 MeV/c2. It decays to a Υ(2S) and two pions. A KDDD*D* reso- Fig. 34. A Υ(10753) as a BsBs pair. nance as illustrated in figure 32, with 2100.76 MeV/c2 The Y(10753) has a mass of 10752.7 MeV/c2. It can in orbital mass has a total mass of 10354.17 MeV/c2. decay to a Y(3S) and two pions, but given its mass it is probably a BB resonance. The combined masses of a BsBs pair as illustrated in figure 34, is 10733.76 MeV/c2. Which is about 20 MeV/c2 too low much like * * the Y(4S). A B B s. combination has a mass of Fig. 32. A Y(3S) as a kaon orbited by D*D* and DD 10740.05 MeV/c2 which is closer and it should also ex- pairs. ist. There are certainly ways to get a closer approxima- * * Next in mass order are the Xb1(3P) and Xb2(3P) with tion using the onium method, but the BsBs or B B s so- masses of 10513.42 and 10524.02 MeV/c2 respec- lutions seem most plausible. * tively. They can both decay to a Υ(3S). A KDKtDDD Table 4 Composition and mass comparison of resonance with 2100.76 MeV/c2 in added orbital mass bottomonium mesons. 2 has a combined mass of 10514.94 MeV/c . A similar Symbol Comp. Mass Est. * KKttDD resonance has a combined mass of 10525.64 (MeV/c2) (MeV/c2) 2 MeV/c . ηb(1S) ttDD 9398.7 9393.70 Υ(1S) πttDD 9460.3 9463.24 * * Xb0(1P) DD DD 9859.44 9860.50 Xb1(1P) KttDD 9892.78 9891.31 hb(1P) KtDtD 9899.3 9900.54 Fig. 33. A Y(4S) as a BB pair. Xb2(1P) KttDDs 9912.21 9916.04 * ηb(2S) πKDttDD 9999 10001.36 The first BB resonance is the Υ(4S) as illustrated in Υ(2S) KttDD* 10023.26 10024.72 2 * figure 33 with a mass of 10579.40 MeV/c . It decays Υ2(1D) πKttDD 10163.7 10164.29 * * to a pair of B mesons greater than 96% of the time. Xb0(2P) πKtD tD 10232.5 10231.62 * * There is no additional relativistic mass associated with Xb1(2P) KtDD D 10255.46 10254.82 * hb(2P) πKtDDD 10259.8 10260.34 the BB orbit, so it should be considered to be a non- * * Xb2(2P) KtDD D 10268.65 10265.36 relativistic BB resonance. Twice their mass of 5279.34 Υ(3S) KDDD*D* 10355.2 10354.17 2 2 * MeV/c is 10558.68 MeV/c . That BB resonance is al- Xb1(3P) KDKtDDD 10513.42 10514.94 * most certainly the Υ(4S) with an increase in mass from Xb2(3P) KKtDtD 10524.02 10525.64 an unidentified source. However, a πωttDs resonance Υ(4S) BB 10579.4 10558.68 * with 2030.73 MeV/c2 and 2100.76 MeV/c2 in addi- Zb(10610) BB 10607.2 10604.04 Z (10650) B*B* 10652.2 10649.40 tional orbital mass has a total mass of 10575.73 b Υ(10753) BsBs 10752.7 10733.76 2 MeV/c so it is possible to model the mass of the Y(4S) Υ(10860) B*X(5568) 10889.9 10891.60 more precisely, it is just not as neat nor as plausible. Υ(11020) BB(5721) 11000 11005.24 The next more massive bottom meson is the 2 Another known BB resonance is the Υ(10860) Zb(10610) with a mass of 10607.2 MeV/c . It decays * which is also known as the Y(5S) as it decays primarily from a Y(5S) and primarily decays to a BB , which 2 to a BB or BsBs pair. It has a mass of 10889.9 MeV/c . also supports it being a BB resonance. The combined * * 2 The estimated mass of a B X(5568) combination is masses of a BB resonance is 10604.04 MeV/c which 10891.60 MeV/c2 which is the closest simple BB reso- is almost certainly what it is. * * 2 nance. A B sB s resonance has a mass around 10830.8 The Zb(10650) has a mass of 10652.2 MeV/c . It de- MeV/c2 and it should also exist and may have been cays from the Y(5S) producing an extra pion during classified as a Y(10860). decay. A B*B* resonance has a mass of 10649.40 10

The heaviest of the well-established mesons is the Next are the Χco(1P) and Xc1(1P) with masses of Υ(11020) with a mass of 11000 MeV/c2. It has been 3414.75 MeV/c2 and 3510.66 MeV/c2 respectively. An seen to decay to an Xb1(1P) plus extra pions rather than ηKDKDs resonance has the combined mass of 3411.24 2 two B mesons. But at this mass there is a good chance MeV/c while the related ηKKDs resonance has the it is composed of two B mesons and a BB(5721) com- combined mass of 3511.38 MeV/c2. 2 bination has a mass of 11005.24 MeV/c . Two possibly related resonances are the hc(1P) and 2 Table 4 provides a summary of the bottomonium ηc(2S) with masses of 3525.38 MeV/c and 3637.5 mesons with their likeliest composition, measured MeV/c2 respectively. They are approximately 105 and 2 mass, and estimated mass. These mesons clearly fol- 210 MeV/c more massive than a η'KDs resonance. low the onium model pattern and the model provides That could mean that one or two pions within the KD excellent mass estimates. kaons in the η' gain orbital mass yielding estimated masses of 3528.73 MeV/c2 and 3633.77 MeV/c2. That 6. Charmonium Mesons essentially makes them Xco(1P) excitation states. Another large group of mesons is said to be charmo- nium resonances composed of various onium states of charm-anticharm quark pairs. But in this onium model Fig. 37. A Xc2(1P) as a tt resonance. they are four-kaon or two-D-meson resonances. These 2 are essentially an extension of the D mesons the way The Xc2(1P) with a mass of 3556.20 MeV/c . Ap- bottomonium resonance are an extension of B mesons. pears to be a tt resonance as illustrated in figure 37. A A few of the resonances discussed in the D meson sec- non-relativistic tt pair has a combined mass of 3553.72 tion may be four-kaon resonances. MeV/c2, so that is almost certainly what it must be.

Fig. 38. A Ψ(2S) as a tD resonance. Fig. 35. A J/Ψ(1S) as a pion orbited by two kaons The next Ψ resonance is the Ψ(2S) with a mass of and a KK pair. 3686.097 MeV/c2. A tD resonance as shown in figure The base four-kaon meson is the J/Psi (J/Ψ(1S)) 38, with 35 MeV/c2 extra mass, has the combined mass with a mass of 3096.916 MeV/c2. It has many decay of 3681.48 MeV/c2. The extra mass indicates there are modes with some to four kaons meaning there are at 29 relativistic electrons like in most bottom mesons. least 8 pions in them. The combined mass of a πKKDs However, it is notable that a πtt resonance has a calcu- resonance is 3095.22 MeV/c2. So, the J/Psi is most lated mass of 3693.29 MeV/c2 or 3728.29 MeV/c2. simply a four-kaon resonance with an extra pion as il- After that is the Ψ(3770) with a mass of 3773.13 lustrated in figure 35. MeV/c2. A DD resonance with 35 MeV/c2 in extra mass indicating 29 relativistic electrons, has the com- bined mass of 3774.23 MeV/c2. This follows the pat- tern of the Ψ(2S). The Ψ2(3823) is next in mass with a mass of 3822.2 MeV/c2. The simplest way for a reso- Fig. 36. A ηc(1S) as a φKK resonance. nance with this mass to appear is a pion with a Ψ(2S) The lowest mass meson normally put in this group as their masses sum to 3825.67 MeV/c2, so it is likely 2 is, however, the ηc(1S) with a mass of 2983.4 MeV/c . to be a πtD resonance. The eta group mesons have a central pion which is con- Next is the Ψ3(3842) with a mass of 3842.71 sistent with the J/Psi above. A πKDKDs resonance with MeV/c2. It decays to a pair of D mesons; however, its the KD in the lower mass state has a total mass of mass is equivalent to a DDs resonance as they sum to 2 2986.24 MeV/c . This is essentially a φKK resonance 3840.31 MeV/c2 using the estimated D meson mass. as illustrated in Figure 36. Note that there could also 2 Then the Xc1(3872) with a mass of 3871.69 MeV/c be the related η'Ds resonance with a mass of 2926.08 appears to be the DDs resonance with 29 relativistic 2 MeV/c . electrons as that resonance has a mass of 3872.92 MeV/c2. 11

mass of 4043.94 MeV/c2. So that is a good possibility that is a step up in energy from the X(3940). ± Next in mass is the X(4050) with a mass of 4051 2 Fig. 39. A Xc0(3860) as a πDD resonance. MeV/c . It commonly decays to a Xc1(1P) and the re- A less well characterized resonance follows as the lated Xc2(1P) is a tt resonance. The mass of a Ktt reso- 2 nance is equal to 4051.33 MeV/c2 so that is an excel- Xc0(3860) has a mass of 3862 MeV/c but with a large uncertainty of +26/-32. The simplest DD resonance lent match. ± that fits within the error bars is a πDD with a combined That is followed by the X(4100) with a mass of 2 mass of 3878.79 MeV/c2, so it may be the πDD reso- 4096 ±20 MeV/c . It has been seen to decay to an nance illustrated in figure 39. ηc(1S) indicating that it may contain an η. The mass of 2 Then the DD* resonances with 28 and 29 relativistic an ηtt resonance is equal to 4101.58 MeV/c so that is electrons have the combined masses of 3882.27 an excellent match given the limited information. Then 2 MeV/c2 and 3917.28 MeV/c2 using the estimated D the Xc1(4140) has a mass of 4146.8 MeV/c . It decays to a J/Psi and phi, but notably the combined masses of meson mass. The Zc(3900) has a mass of 3887.2 2 MeV/c2 while the X(3915) has a mass of 3918.4 a KtD resonance equals 4144.08 MeV/c . MeV/c2 which are good matches for the DD* reso- nances. The Xc2(3930) follows with a mass of 3922.2 MeV/c2. It is known to decay to two D mesons or an Fig. 42. A X(4160) and Ψ(4160) as πD*D* reso- nances. ηc(1P) which has a KD in the lower energy state. A 2 2 KDhc(1P) has a combined mass of 3922.85 MeV/c . The X(4160) with a mass of 4156 MeV/c and de- Also, a πDD resonance with 29 relativistic electrons cays to a D*D* pair. Since it is almost exactly 140 has a mass of 3913.80 MeV/c2 giving another onium MeV/c2 heavier than a D*D* pair so it is likely to be a resonance near this mass. πD*D* resonance which has an estimated mass of The X(3940) has a mass of 3942 MeV/c2 and a DD* 4157.93 MeV/c2. The Ψ(4160) has a mass of 4191 2 decay mode. A possible match is a KDtt with the lower MeV/c but there are three measurements that average 2 mass KD which has a combined mass of 3938.41 4155 MeV/c that led to its name. Some of those could 2 * * MeV/c . However, there may also be a μDDs with a be πD D resonances like the X(4160) illustrated in 2 * * mass of 3943.57 MeV/c or a DsDs resonance with a figure 42. Then the πD D resonance with 29 relativ- mass of 3936.60 MeV/c2, although they should have istic electrons has a mass of 4195.10 MeV/c2. Decays * decay modes that include a Ds meson. to a D sDs have been seen with the Ψ(4160) so there may be another resonance that contains one or two Ds resonances. 2 Next in mass is the 4196 MeV/c Zc(4200), but its * * Fig. 40. A X(4020) as a D D s resonance. mass is not known very precisely as it is +31/-29. It Similarly, the X(4020)± has a mass of 4024.1 decays to a J/Ψ and π which is also not very helpful at MeV/c2 and the X(4055)± has a mass of 4054 MeV/c2. determining an onium model for it. But using those de- ± * * * * cay products, the simplest resonance with that mass is The X(4020) decays to a D D pair and a D D reso- 2 nance as illustrated in figure 40 has a combined mass a 4194.25 MeV/c πη'J/Ψ. A more stable resonance 2 * * like a KΨ(2S) (KtD) which has a mass of 4179.77 of 4020.52 MeV/c . The D D resonance with 29 rela- 2 tivistic electrons has a mass of 4055.53 MeV/c2. MeV/c is probably more likely. Next is the Ψ(4230) with a mass of 4220 ±15 2 MeV/c . It has an ωXco(1P) decay mode so it may con- tain a Ds. A KDDDs with the lower energy KD has a 2 mass of 4222.60 MeV/c . Slightly more massive is the Fig. 41. A Ψ(4040) as a KDtD resonance. Rc0(4240) (Formerly X(4240)) with a mass of 4239 2 2 MeV/c . It decays to a Ψ(2S). A KDD resonance has The Ψ(4040) has a mass of 4039 MeV/c and it com- 2 monly decays to two D mesons of various types. A the combined mass of 4236.83 MeV/c and it should be fairly stable and thus observable. KDtD resonance as shown in figure 41 has a combined 12

The X(4250)± has a mass of 4248 MeV/c2 but it is not well known having error bars of +44/-29. It decays to a Xc1(1P) so it may contain an η and Ds as well. That * 2 said, a KDDD with a mass of 4264.56 MeV/c may be Fig. 44. A Ψ(4660) as a η'tD resonance. the most stable resonance near this mass, while an ηπtt The next to last most massive resonance in the cur- 2 is closer in mass at 4241.15 MeV/c . rent PDG lists is the Ψ(4660) with a mass of 4643 MeV/c2. It has been seen to decay to a Ψ(2S) and the mass of a Ψ(2S) and an η' equals 4643.88 MeV/c2. It is illustrated as an η'tD resonance in figure 44. Then 2 Fig. 43. A Ψ(4260) as a ηtD resonance. lastly is the Xc0(4700) with a mass of 4704 MeV/c . An η'DD resonance has the combined mass of 4701.80 The Ψ(4260) has a somewhat lower mass than its 2 name implies of 4230 MeV/c2. The key to understand- MeV/c using the estimated D meson masses. ing this resonance is the decay to an η and Ψ(2S) as Table 5 lists the best known charmonium reso- their combined masses are 4229.34 MeV/c2. An ηtD nances from the current PDG listings with the symbol, resonance is illustrated in figure 43. composition, measured mass, and estimated mass based on the onium model. The compositions are con- After that is the Xc1(4274) with a mass of 4274 MeV/c2. It has a J/Psi and phi decay mode. Its mass is sistent with the model and most of the masses are ac- equivalent to a Ψ(3770) and a K0 which equals 4270.74 curately estimated. MeV/c2 making it a KDD resonance. Then the X(4350) Table 5 Composition and mass comparison of has a mass of 4350.6 MeV/c2. The simplest resonance some eight pion and greater mesons. with that mass is a KDKDtt resonance with a total mass Symbol Comp. Mass Est. of 4348.66 MeV/c2. (MeV/c2) (MeV/c2) 2 The Ψ(4360) is next with a mass of 4368 MeV/c . It ηc(1S) πKDKDs 2983.4 2986.24 can decay to a Ψ(2S) (tD) and two pions. Its mass is J/Ψ(1S) πKKDs 3096.9 3095.22 Xco(1P) ηKDKDs 3414.75 3411.24 equivalent to a Xc1(3872) (DDs) and a kaon which X (1P) ηKKD 3510.66 3511.38 2 c1 s sums to 4369.30 MeV/c making it a KDDs. hc(1P) η'KDs 3525.38 3528.73 2 Next is the Ψ(4390) with a mass of 4391.5 MeV/c . Xc2(1P) tt 3556.2 3553.72 It has been seen to decay to an hc(1P) and possibly a ηc(2S) η'KDs 3637.5 3633.77 Ψ(3770) (DD) and two pions. That is indicative of a Ψ(2S) tD 3686.097 3681.48 Ψ(3770) DD 3773.13 3774.23 central pion such as the one in an η plus a kaon with a Ψ (3823) πΨ(2S) 3822.2 3825.67 * 2 DD pair. A KtD s resonance has the combined mass of Ψ3(3842) DDs 3842.71 3840.31 2 * * 4386.57 MeV/c with the pion in a D s. A KDD has Xc0(3860) πDD 3862 3878.79 2 slightly less mass at 4373.55 MeV/c . Xc1(3872) DDs 3871.69 3872.92 * 2 Zc(3900) DD 3887.2 3882.27 The Ψ(4415) has a mass of 4421 MeV/c and a D * * ± X(3915) DD 3918.4 3917.28 and D 2(2460) decay mode that gives us a good clue X (3930) K h (1P) 3922.2 3922.85 * ± c2 D c about its composition since the D 2(2460) is a πKDπt. X(3940) KDtt 3942 3938.41 The πKDπtDs resonance has a combined mass of X(4020) D*D* 4024.1 4020.52 2 4421.77 MeV/c . Following that is the Zc(4430) with a Ψ(4040) KDtD 4039 4043.94 ± mass of 4478 MeV/c2. It can decay to a Ψ(2S) and the X(4050) Ktt 4051 4051.33 X(4055)± D*D* 4054 4055.53 combined masses of a KDKDΨ(2S) resonance is X(4100)± ηtt 4096 4101.58 2 4481.04 MeV/c or slightly less if calculated as a X(4140) KtD 4146.8 4144.08 KDKDtD resonance. X(4160) πD*D* 4155 4157.93 * * After that is the Xc0(4500) with a mass of 4506 Ψ(4160) πD D 4191 4195.10 MeV/c2. It is known to have a J/Psi and phi decay mode Zc(4200) πη'J/Ψ(1S) 4196 4194.25 Ψ(4230) KDDDs 4218 4222.60 with an intermediate Xco and K decay. It turns out that R (4240) KDD 4239 4236.83 0 c0 the related Xc1(1P) mass plus 2K equals 4505.88 X(4250)± ηπtt 4248 4241.15 MeV/c2 which is consistent with its mass and decay Ψ(4260) ηtD 4230 4229.34 products. Xc1(4274) KΨ(3770) 4274 4271.84

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Table 5 Continued. Table 6 Continued. Symbol Comp. Mass Est. Comp. Est Symbol Mass. (MeV/c2) (MeV/c2) (MeV/c2) (MeV/c2) * X(4350) KDKDtt 4350.6 4348.66 πωK (892) 1813.88 K2(1820) 1819 Ψ(4360) KXc1(3872) 4368 4369.3 ηωKD 1824.19 X(1835) 1826.5 * * Ψ(4390) KtDs 4391.5 4386.57 K (892)η' 1849.44 φ3(1850) 1854 * Ψ(4415) DsD 2(2460) 4421 4429.0 η'KDK 1848.93 η2(1870) 1842 Zc(4430) KDKDΨ(2S) 4478 4481.04 ηηω 1878.37 π2(1880) 1874 Xc0(4500) KKXc1(1P) 4506 4505.88 KKK*(892) 1879.01 K(1830) 1874 * Ψ(4660) η'Ψ(2S) 4643 4643.88 φK (892) 1911.12 f2(1910) 1900 * Xc0(4700) η'DD 4704 4701.80 ηKK (892) 1937.13 f2(1950) 1936 πKDKK 2013.48 f2(2010) 2011 * * πφK (892) 2050.69 K 4(2045) 2045 7. Other Eight or Greater Pion mesons † ωω 2055.48 f4(2050) 2050 Since the charmonium mesons have a 980.35 ωK*(892)† 2164.49 φ(2170) 2170 MeV/c2 relativistic orbit, there are many related reso- KπKKK 2219.32 η(2225) 2221 † nances with eight pions that do not have pairs of kaons ωη' 2230.61 f0(2200) 2231.1 ρKK 2252.69 K2(2250) 2247 in relativistic orbit or have the 455.16 or 490.18 K*(892)η'† 2339.62 f (2330) 2330 2 0 MeV/c mass orbits. There are more than 20 reso- KDKKKK 2359.40 f2(2340) 2345 † * nances that can be identified as eight-pion resonances KKK*(892) 2377.06 K 5(2380) 2382 because they have decay modes such as a ρηπ, ρρ, ωω, † Includes a 490.18 MeV/c2 orbit. * * ηη', or K (892)K (892). Note that there are many mesons in the D meson Other resonances do not have an eight-pion or mass range from 1900 to 2500 MeV/c2 that have so greater decay mode but their mass and known decay many potential onium models that it is improper to products match an onium resonance well enough that guess what they may be without better data. it can be identified as such. In that case it is best to calculate masses using the model first and match up the 8. Two to Seven Pion mesons mesons that seem to fit the best. In table 6 the masses of the eight-pion or greater resonances are calculated There are additionally many mesons with two to and mesons matched to them on a best fit basis. seven pions. These are examined and shown in table 7 using the same approach as the last section. Interest- Table 6 Composition and mass comparison of mesons containing eight or more pions. ingly there are a number of two-pion onium resonances identified as a 2π, σ, f0(500), or f0(600) in the literature, Comp. Est Symbol Mass. that have been investigated but have not been accepted (MeV/c2) (MeV/c2) as particles in the standard model. Some possible ηηππ 1374.86 f0(1370) 1370 ηηππ 1409.88 η(1405) 1408.8 matches are shown here. ρηπ 1462.54 ρ(1450) 1465 Table 7 Composition and mass comparison of ηηKD 1469.08 a0(1450) 1474 mesons containing two to seven pions. ηη' 1505.64 f0(1500) 1500 ρρ 1550.22 f2(1565) 1542 Comp. Est Symbol Mass. 2 2 ρω 1557.76 f2(1565) 1542 (MeV/c ) (MeV/c ) ωω 1565.3 f2(1565) 1542 πee 279.14 ππ * ηπK (892) 1579.09 K2(1580) 1580 ππ 384.18 X(360) 360 πηη' 1645.21 f2(1640) 1639 μπ 455.30 f0(500) 500 ρKDK 1653.48 a1(1640) 1655 μππ 489.84 f0(500) 500 * ρK (892) 1666.82 π2(1670) 1670.6 πππ 523.75 f0(500) 500 ρK*(892) 1666.82 K*(1680) 1680 μK 603.27 f0(600) 600 ωK*(892) 1674.31 φ(1680) 1680 πK 637.18 f0(600) 600 * * 2KDK (892) 1686.6 ρ3(1690) 1688.8 ππKD 676.61 K (700) 700 * πωω 1704.87 f0(1710) 1704 KDKD 794.94 K (800) 800 πωω 1704.87 a2(1700) 1705 KK 987.35 f0(980) 990 ρη' 1732.94 ρ(1700) 1720 μKDK 984.03 a0(980) 980 ωη' 1740.43 η(1760) 1751 πKDη 1072.12 X(1070) 1072 ρKK 1762.51 f2(1750) 1755 μKK 1100.88 X(1110) 1107 14

Table 7 Composition and mass comparison of 111 mass estimates were with 0.2% relative and 94 are mesons containing two to seven pions. within 0.1%. This report shows that it is possible to Comp. Est Symbol Mass. model all mesons using onium theory and account for (MeV/c2) (MeV/c2) their decay products, charge, and mass. All meson ρKD 1172.63 h1(1170) 1170 mass is relativistic beyond the electron and positron KπK 1231.96 a1(1260) 1230 rest masses. μπω 1237.95 b (1235) 1229.5 1 The onium model is also flexible enough to account ρK 1272.77 K1(1270) 1272 πKπK 1274.36 f2(1270) 1275.5 for the few remaining mesons that were not modeled ηKπ 1286.15 f1(1285) 1286.9 in this paper due to there being insufficient data to KDKDK 1292.55 η(1295) 1294 make a conclusive judgement about their composition ρπKD 1313.79 a2(1320) 1316.9 § or mass. Quarks are completely unnecessary to the un- ηρ 1323.02 f0(1323) 1323 † derstanding of mesons. KKD 1381.32 h1(1380) 1380 ρπK 1403.81 K1(1400) 1403 This paper makes it clear that electrons and posi- ρπK 1412.34 a1(1420) 1411 trons are the only particles needed to model all mesons. † * πω 1412.40 K 1(1410) 1414 And since it was shown in a separate paper that all bar- πωK 1415.90 h1(1415) 1416 † * yons can be modeled with only electrons, protons, and πKDKD 1424.69 K 2(1430) 1425.6 * , it is clear that all unstable resonances can be ηKDK 1426.23 K 0(1430) 1425 * πKDK (892) 1428.70 f1(1420) 1426.3 accounted for with onium theory. KK† 1477.53 K(1460) 1482.40 ηηKD 1480.41 a0(1450) 1474 References KKK 1481.03 η(1475) 1475 πKDKK 1515.55 f1(1510) 1518 [ 1 ] Fleming, R. (2019), “An onium model of particles with only † πKDK 1520.89 f2(1525) 1525 electrons and protons,” GSJounal.net, 19 Nov. ηKK 1543.08 X(1545) 1545 [ 2 ] Fleming, R. (2020), Goodbye Quarks: The Onium Theory, πKK† 1617.10 K(1630) 1629 Amazon.com, 27 May. πKKK 1620.60 η2(1645) 1617 [ 3 ] Fleming, R. (2020), “Tau, D, and other kaonium mesons in ρKDK 1653.53 a1(1640) 1655 the onium particle theory,” GSJounal.net, 28 Jun. ρKDK 1653.53 X(1650) 1652 [ 4 ] Milne, E.A. (1948) Kinematic Relativity, Oxford: Claren- † ωKD 1670.30 ω1(1670) 1667 don, pp 216-9. † πKπK 1752.08 X(1750) 1753.5 [ 5 ] Sternglass, E. J., (1993) Before the Big Bang: Origins of the † * KDKDK 1778.79 K 3(1780) 1776 Universe, Four Walls Eight Windows Publishing, pp 134- † * KKK 1940.13 K 0(1950) 1945 135. † * KKK 1971.21 K 2(1980) 1974 [ 6 ] Sternglass, E. J. (1961) “Relativistic electron-pair systems † 2 Includes a 490.18 or 455.16 MeV/c orbit. and the structure of the neutral meson,” Phys. Rev., 123, § A f0(1200-1600) resonance. 391. Particles that were investigated as part of this study [ 7 ] Sternglass, E.J. (1965) “Electron-positron model for the but not modeled due to insufficient data include; charged mesons and pion resonances,” Il Nuovo Cimento, 1 Gennaio, Volume 35, Issue 1, pp 227-260. π(1300), ω(1420), f2(1430), ρ(1570), h1(1595), [ 8 ] Browne, P.F. (1966) “Rotating fields and particle-like π1(1600), ω(1650), π(1800), f2(1810), ρ(1900), states of electron-positron systems,” Nature 211, pp 810– a0(1950), a4(1970), ρ3(1990), π2(2005), f0(2020), 813. π2(2100), f0(2100), ρ(2150), f2(2150), ρ3(2250), [ 9 ] Schild, A. (1963) “Electromagnetic two-body problem,” f2(2300), f4(2300), ρ5(2350), f6(2510), K1(1650), and Phys. Rev., 131, 2762. other resonances listed on the PDG list of selected fur- [ 10 ] MacGregor, M.H., (2007) The Power of α: Electron Ele- mentary Particle Generation with α-Quantized Lifetimes ther states that are not mentioned. and Masses.” World Scientific Pub Co Inc, 14 Mar. [ 11 ] Browne, P.F. (1981) “Relativistic states of positronium and 9. Conclusion structure of fundamental particles,” J. Phys. A: Math. Gen. A total of 220 mesons were modeled in this study 14 597-619. [ 12 ] Tanabashi. M., et al. (Particle Data Group) (2018) Phys. using onium theory with only electrons and positrons Rev. D 98, 030001. as elementary components. Of those, 170 were within [ 13 ] Zyla, P.A., et al. (Particle Data Group), (2020) Prog. 5 MeV/c2 of the measured value and of the 121 D, B, Theor. Exp. Phys. 2020, 083C01. charmonium, and bottomonium resonances modeled,

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