Variation of hadronic and nuclei mass level oscillation periods for different spins

TL;DR

This paper investigates regular oscillations in hadronic and nuclear mass levels, revealing a cosine pattern in mass differences that suggests underlying symmetries across different spins.

Contribution

It introduces a systematic analysis showing oscillatory behavior in hadronic and nuclear masses, highlighting a common cosine pattern across different spins.

Findings

Oscillations in hadronic masses fit a cosine function.

Mass differences versus mean mass reveal symmetry.

Similar oscillatory patterns observed in nuclear level masses.

Abstract

A systematic study of hadronic masses shows regular oscillations that can be fitted by a simple cosine function. This property can be observed when the difference between adjacent masses of each family is plotted versus the mean mass. This symmetry of oscillation is also observed for the nuclear level masses of given spin.

Tables

Table 1. Table 1: Quantitative information concerning the oscillation behavior of some mesons and baryons analysed previously. P is the period (in MeV).

name	q.c.	fig.	J	mass	P
$K_0$	$q\overline{s}$	6(a)	0	493.7	691
$\eta$	$q\overline{q}$	2(b)	0-+	547.9	622
$f_0$	$q\overline{q}$	5(a)	$0^{++}$	475	647
$K_1$	$q\overline{s}$	6(b)	1	892	408
$f_2$	$q\overline{q}$	5(b)	$2^{++}$	1275	415
$K_2$	$q\overline{s}$	6(c)	2	1425	201
charm.	$c\overline{c}$	5(c)	$1^{--}$	2981.5	358
botto.	$b\overline{b}$	5(d)	$1^{--}$	9391	452
$N^{\ast }$	qqq	1(b)	1/2	939	390
$N^{\ast }$	qqq	1(c)	3/2	1520	201
$N^{\ast }$	qqq	2(a)	5/2	1675	390
$\Lambda$	qqs	4(a)	1/2	1115.7	220
$\Lambda$	qqs	4(b)	3/2	1519.5	396
$\Delta$	qqq	3(a)	1/2	1620	201
$\Delta$	qqq	3(b)	3/2	1232	201
$\Sigma$	qqs	4(c)	1/2	1189.4	396
$\Sigma$	qqs	4(d)	3/2	1385	176

Table 2. Table 2: Quantitative information concerning the oscillation behavior of some J=2 nuclei levels analysed previously. P is the period (in MeV).

nucleus	fig.	P(MeV)
4He	13(d)	5.03
10B	10(c)	2.2
14N	11(c)	2.0
16N	13(b)	2.07
16O	14(c)	3.08
20Ne	15(c)	1.885
26Mg	18(a)	1.885
56Fe	16(b)	1.76
62Zn	19(a)	0.817
80Se	19(b)	0.452
92Nb	19(d)	0.377
100Ru	19(c)	0.352
132Ce	19(f)	2.14
146La	19(e)	0.817
154Gd	18(d)	0.490
194Pt	18(b)	0.490
214Po	18(c)	0.446
230Th	18(e)	0.427

Table 3. Table 3: Quantitative information concerning the oscillation behavior of some nuclei levels with spin different from 2, analysed previously. P(m) is the period (in MeV).

Spin	nucleus	fig.	P(MeV)
0	16O	14(a)	5.03
	20Ne	15(b)	1.885
	56Fe	16(a)	2.07
1/2	15N	12(b)	1.885
1	10B	10(b)	2.2
	14N	11(b)	2.01
	16N	13(a)	1.82
	16O	14(b)	3.33
3/2	15N	12(c)	1.885
5/2	15N	12(d)	1.633
	25Al	17(a)	2.45
	27Al	17(a)	2.45
	155Tb	17(b)	0.575
	159Tb	17(b)	0.575
	165Dy	17(c)	0.547
	165Er	17(c)	0.314
3	10B	10(d)	2.2
	14N	11(d)	2.04
	16N	13(c)	2.39
	16O	14(d)	2.51
	20Ne	15(d)	1.885
	56Fe	16(c)	1.63
	208Pb	20(a)	0.94
4	16O	14(e)	3.96
	56Fe	16(d)	1.005
	208Pb	20(b)	0.754
5	208Pb	20(c)	0.88
6	208Pb	20(d)	0.50

Equations

$$m_{(n+1)}-m_n=f[(m_{(n+1)}+m_n)/2]$$

$$\Delta M={\alpha }_0+{\alpha }_{1}cos((M-M_0)/M_1)$$

Full text

Hadronic masses, mesons baryons,symmetry

I Introduction

A new property of particle masses was recently shown when studying the mass variation versus the mass increase for adjacent meson and baryon masses of given families boris . The investigated function is:

[TABLE]

where $m_{(n+1)}$ corresponds to the (n+1) hadron mass value. The difference of two successive masses was plotted versus the mean value of the two nearby masses. Such studies were restricted to hadron families holding at least five masses. Regular oscillations were observed giving rise to a new symmetry, the symmetry of oscillation. It was noticed in boris that ”the existence of composite hadrons, results from the addition of several forces, related to strong interaction, that combine in, at least, one attractive and one repulsive force. The equilibrium among these forces allows the hadron to exist, otherwise the composite mass will either disintegrate, or mix into a totally new object with loss of the individual components”. As in classical physics, these opposite forces may generate oscillating behaviour.

The obtained data are fitted using a cosine function:

[TABLE]

where M0 /M1 is defined within 2$\pi$. All coefficients, and masses used to draw the figures are in MeV units. The quantitative information is given in Tables I and II presented below. The oscillation periods are P = 2 $\pi M_{1}$. Both $\alpha_{0}$ and $\alpha_{1}$ are adjusted on the extreme values on all figures.

Whereas smaller periods than those given in the Tables may also reproduce the data, we show in the following figures, the largest possible values.

The discussion concerned the oscillatory periods, and not the oscillation amplitudes which need theoretical study outside the scope of previous and present papers.

For the same reason, the existence of substructures in hadrons, we expect to observe oscillations in nuclei made with nucleons. Such study will be considered after the hadronic masses.

II Application to hadronic masses

The masses and widths are read from the Review of Particle Physics olive , taking into account all the data reported, even if, in some cases, they are omitted from the summary table.

In our previous paper boris data were shown for the following meson families with fixed quantum numbers: $f_{0}$ in fig. 1(a), $f_{2}$ in fig. 1(b), charmonium (${\it c}{\bar{c}}$) $0^{-}(1^{--})$ in fig. 4(a) and bottomonium (${\it b}{\bar{b}})$ $0^{-}(1^{--})$ in fig. 4(b). Several other data were also shown in the same paper without restriction to given quantum numbers for charmed in fig. 2(a), charmed strange in fig. 2(b), (${\it c}{\bar{c}}$) in fig. 3(a), (${\it b}{\bar{b}}$) in fig. 3(b) mesons. The corresponding figures displayed several data outside the fitting curves, mainly for (${\it c}{\bar{c}}$) but also for charmed strange mesons.

Resulting data were also shown for several baryon families without selection of given quantum numbers, contrary to the indication reported in column $J^{PC}$ of boris . They are: $N^{*}$ in fig. 5(a), $\Delta$ in fig. 6(a), $\Lambda$ in fig. 6(b), $\Lambda_{C}$ in fig. 5(b), $\Xi$ in fig. 7(a) and $\Xi_{C}$ in fig. 7(b).

All previous data were fitted with oscillations with use of a few first masses. The fit gets often spoiled over a few MeV. The comparison between the selection of charmonium and bottomonium data with fixed quantum numbers, and without spin selection, suggests the relevance to restrict the study to particle families with given spins. Of course the necessity to have still at least five known masses remains. This condition will reduce the possibilities of application.

This study is done below, where the figures shown in boris with such criteria of given spin, are repeated here in purpose of consistency. The masses reported in olive are used independently of the number of attributed stars. When the name is different from the mass, the mass is used.

Fig. 1 shows in inserts (a), (b), (c), and (d) the data for $N^{*}$ (1/2*+), $N^{*}$ (1/2+* + 1/2*-), $N^{*}$ (3/2+* + 3/2*-), and $N^{*}$ (1/2+* +1/2*-* + 3/2*+* + 3/2*-*). Although two data in insert (b) lie outside the curve, both fits in inserts (a) and (b) are obtained with the same period P = 390 MeV. We will use that to add subsequently the masses having the same spin but different parities, and allow therefore to get more data with five, or more masses analysed simultaneously. Fig. 1(c) shows nice fit for $N^{*}(3/2^{+}$) + $N^{*}(3/2^{-})$. The period here is P =201 MeV. Fig. 1(d) shows the result for $N^{*}$ baryon masses, having both parities and both spins (1/2) and (3/2). These data are fitted with P = 201 MeV.

Therefore we will later on add the data having different parities and study them separately for different spins.

Fig. 2 shows in insert (a) the mass difference between successive masses, plotted versus both corresponding mean masses for $N^{*}$ (5/2*+) + $N^{*}$ (5/2-*). The corresponding period P = 390 MeV is the same as the one obtained to describe the variation of the data for $N^{*}$ (J=1/2) fig. 1(b) and $N^{*}$ (J=3/2) fig. 1(c) baryons.

The data and fit for the $\eta$ meson are shown in fig. 2(b). The very precised masses are well fitted with the period P = 622 MeV. A mass at M = 762 MeV is introduced, following the recent suggestion BTETG .

Fig. 3 shows the data for $\Delta$ baryons J = (1/2) in insert (a) and J = (3/2) in insert (b), without distinguishing the parities. Both data are fitted with the same period P = 201 MeV.

Fig. 4 shows the the data for strange baryons $\Lambda~{}(1/2^{+})$ and ($1/2^{-}$) in insert (a), $\Lambda~{}(3/2^{+})$ and ($3/2^{-}$) in insert (b), $\Sigma(1/2^{+})$ and $(1/2^{-})$ in insert (c), $\Sigma(3/2^{+})$ and $(3/2^{-})$ in insert (d). The data in fig. 4(a) are fitted by an oscillating curve, although here more simple functions are possible.

Fig. 5 shows the results for $f_{0}$ ($0^{++}$) and $f_{2}$ ($2^{++}$) light unflavoured mesons in inserts (a) and (b). The $\sigma$ or $f_{0}$(500) meson is broad and its mass (which unprecision is taken to $\Delta$M = 125 MeV), is badly determined. The reasonable fit allows to extrapolate the masses of the next $f_{0}$ not extracted experimentally up to now. They are M $\approx$ 2670 and 2760  MeV. In the same way, the masses of the next $f_{2}$ mesons can be tentatively predicted to be: M $\approx$ 2380, 2450, and 2625 MeV. Inserts (c) and (d) show the results for (${\it c}{\bar{c}}$) $0^{-}(1^{--})$, and (${\it b}{\bar{b}})$ $0^{-}(1^{--})$ mesons. The mass of the last quoted meson used in fig. 5(c), X(4660) $?^{?}(1^{--})$ fits perfectly in this distribution, and is therefore kept, assigning tentatively the same quantum numbers. The extrapolation allows to predict tentatively the next corresponding $\Psi$ masses: M $\approx$ 4805 and 5080 MeV. In the same way, the tentatively extrapolated $\Upsilon$ masses are: M $\approx$ 11330 and 11560 MeV.

Fig. 6 shows the data for strange kaons, K ($0^{+-}$) in insert (a), K ($1^{+-}$) in insert (b), and K ($2^{+-}$) in insert (c). Oscillatory shapes must be used for fits. The corresponding periods decrease regularly from P = 691 MeV, to P=408 MeV, and finally to P=201 MeV for increasing spins in inserts: (a), (b), and (c).

All obtained periods fitting the previous data are reported in table I and fig. 7. Meson (baryon) periods are plotted with full red circles (full blue squares) versus the lower mass of each family. They all are located in three well defined ranges, the same for mesons and baryons. Three general properties are observed:

The periods of the three (meson) families with the lowest spin (J = 0): $f_{0}$, $K^{0}$, and $\eta$ are located in the highest range close to P$\approx$655 MeV.

The periods corresponding to the other families distributed in the two other ranges, favor the intermediate range for lower spin. This is true for periods of K  (J=1) compared to K (J=2), N $(J=1/2)$ compared to N $(J=3/2)$, $\Sigma$ $(J=1/2)$ compared to $\Sigma~{}$(J=3/2). However the opposite is observed for periods corresponding to $\Lambda~{}(J=1/2)$ and $\Lambda~{}(J=3/2)$. And also the period of oscillation corresponding to the masses of N (J=5/2) is larger than that of N (J=3/2) and is equal to that of N (J=1/2), suggesting here again an oscillatory behaviour. Such behaviour is indeed observed in fig. 7 with P = 357 MeV, better adjusted to meson than to baryon results. So the periods of $\Delta~{}(J=1/2)$ and $\Delta~{}(J=3/2)$) lie outside the distribution.

We notice that the distribution reported in fig. 7, fits the period correponding to (${\it c}{\bar{c}}$) $0^{-}(1^{--})$ mesons and also the period corresponding to (${\it b}{\bar{b}})$ $0^{-}(1^{--})$ mesons not plotted on fig. 7 since the very large gap between masses. The distribution reported in fig. 7 differs from that reported in Tatischeff:2016psc. New data are analysed in the present paper, when several data, without spin selection, were used in Tatischeff:2016psc.

The mean values of the three ranges shown in fig. 7, are pointed out by dashed lines (in green on line) at M(0) $\approx$ 655 MeV, M(1) $\approx$ 385 MeV, and M(2) $\approx$ 195 MeV. The meson oscillating periods, shown in red full circles, are gathered together following the relation: J - $\left|{S}\right|$ + 2*I which value equals 0, 1, and 2 for the three ranges from up to down. This relation does not apply for ${\it f_{2}}$ mesons.

III Application to nuclei masses

It is reasonable to expect oscillations in the nuclei mass levels for the same reason as before for hadrons. The nucleons in nuclei are bound by opposite forces. This property is studied below using data from Lederer Ajzenberg-Selove:1979uqf; Ajzenberg-Selove:1968ztr; Ajzenberg-Selove:1976uid when not specified. We start the analysis without spin selection, considering all level masses.

Fig. 8 shows the mass difference between successive masses plotted versus the corresponding massses for 7Li (red) P=2.45 MeV, 8Li (blue) P=3.02 MeV, 9Be (green) P=2.76 MeV, and 10B (purple) P=1.88 MeV respectively in inserts (a), (b), (c), and (d). We observe an increase of the level number with increasing mass. We observe also that the fit between data and calculated curves spoils after the five-six first MeV in the case of 10B nucleus. For heavier nuclei, these properties are amplified as seen in fig. 9.

Fig. 9 shows the mass difference between successive masses plotted versus the corresponding masses for 17N P=1.70 MeV, 17O P=1.88 MeV, 17F P=2.39 MeV, and 12B P=1.76 MeV nuclei. In spite of the large mass differences for all data, emphasized by the log scale, the first data are rather well fitted, then followed by a large number of spread data.

So the situation is comparable to the one observed for hadrons, and therefore brings us to separate the nuclei level masses by their spins.

The next figures will study the oscillation properties of nuclei level masses having the same spin. Although a large number of level masses are known for the majority of nuclei, rather few have a number of known quantum numbers allowing the same studies as previously done (five or more level masses with the same spin).

Fig.10 shows the mass difference between successive masses plotted versus the corresponding masses for 10B nucleus. The four inserts (a), (b), (c), and (d) show data with: all spins P = 1.885 MeV, J=1 P=2.2 MeV, J=2 P=2.2 MeV, and J=3 P=2.2 MeV. The data for separated spins (inserts (b), (c), and (d)) are well fitted with the same period of oscillation. However the small number of levels with spin J=3 (insert (d)) involves an arbitrary fit.

Fig. 11 shows the mass difference between successive masses plotted versus the corresponding masses for 14N nucleus. The four inserts (a), (b), (c), and (d) show data with: all spins P = 1.82 MeV, J=1 P=2.01 MeV, J=2 P=2.01 MeV, and J=3 P=2.04 MeV. The data for separated spins (inserts (b), (c), and (d)) are well fitted with almost the same period of oscillation.

Fig. 12 shows the mass difference between successive masses plotted versus the corresponding masses for 15N nucleus. The four inserts (a), (b), (c), and (d) show data with: all spins P = 1.885 MeV, J=1/2 P=1.885 MeV, J=3/2 P=1.885 MeV, and J=5/2 P=1.63 MeV. The data for separated spins are better fitted in inserts (b) and (d) than in (c).

Fig. 13 shows the mass difference between successive masses plotted versus the corresponding masses for 16N nucleus Tilley:1993zz. The three inserts (a), (b), and (c) show data with: J=1 P = 1.82 MeV, J=2 P=2.07 MeV, and J=3 P=2.39 MeV. Insert (d) shows the result for the levels J=2 P=5.03 MeV of the 4He nucleus Tilley:1992zz. The data are well fitted.

Fig. 14 shows the mass difference between successive masses plotted versus the corresponding masses for 16O nucleus Tilley:1993zz. The five inserts (a), (b), (c), (d), and (e) show data with: J=0 P = 5.03 MeV, J=1 P=3.33 MeV, J=2 P=3.08 MeV, J=3 P=2.51 MeV, and J=4 P=3.96 MeV. One data with J=2, close to 16 MeV, is outside the fit. For all nuclei, at large excitation energy, the spins of some levels are unknown, therefore these levels are ignored.

Fig. 15 shows the mass difference between successive masses plotted in log scale versus the corresponding masses for 20Ne nucleus. The four inserts (a), (b), (c), and (d) show data with: all spins P = 1.885 MeV, J=0 P=1.885 MeV, J=2 P=1.885 MeV, and J=3 P=1.885 MeV. The data for separated spins (inserts (b), (c), and (d)) are rather well fitted with the same period of oscillation, the same as obtained for 15N (J=3/2).

Fig. 16 shows the mass difference between successive masses plotted versus the corresponding masses for 56Fe nucleus 56Fe . The four inserts (a), (b), (c), and (d) show data with: J=0 P =2.07 MeV, J=2 P=1.76 MeV, J=3 P=1.63 MeV, and J=4 P=1.00 MeV. The data are well fitted; here the oscillatory periods decrease with increasing spins. The fit in insert (a) is undetermined due to small number of data. There is two data outside the fit in insert (b) and one in insert (c).

Fig.17 shows the mass difference between successive masses plotted versus the corresponding masses for several nuclei with spin J=5/2. The data in insert (a) correspond to 25Al (full blue squares) and 27Al (full red circles) fitted with the period P=2.45 MeV. The data in insert (b) correspond to 155Tb (full blue squares) and 159Tb (full red circles) fitted with the period P=0.575 MeV. The fit is obtained using 155Tb data. Three red data over four, corresponding to 159Tb, lie close to the same curve. The data in insert (c) correspond to 165Er (full blue squares) fit with the period P=0.314 MeV, and 165Dy (full red circles) 165Dy fit with the period P=0.547 MeV. Both nuclei 165Er and 165Dy differ by only by one proton (and one neutron), therefore the large difference between their oscillating periods is unclear.

Fig. 18 shows the mass difference between successive masses plotted versus the corresponding masses for several nuclei with spin J=2. The data in insert (a), (b), (c), (d), and (e) correspond to 26Mg P=1.885 MeV, 194Pt P=0.49 MeV, 214Po P=0.446 MeV, 154Gd P=0.49 MeV, and 230Th P=0.427 MeV nuclei respectively. Here again the periods decrease with increasing nuclei masses.

Fig. 19 shows the mass difference between successive masses plotted versus the corresponding masses for several other nuclei with spin J=2. The data in insert (a), (b), (c), (d), (e), and (f) correspond to 62Zn 62Zn P=0.817, 80Se 80Se P=0.452 MeV, 100Ru 100Ru P=0.352 MeV, 92Nb P=0.377 MeV, 146La P=0.817 MeV, and 132Ce P=2.14 MeV nuclei respectively.

Fig. 20 shows the mass difference between successive masses plotted versus the corresponding masses for 208Pb 208Pb in log scale. Inserts (a), (b), (c), and (d) correspond respectively to data having the following spins: J=3 P=0.942 MeV, J=4 P=0.754 MeV, J=5 P=0.88 MeV, and J=6 P=0.503 MeV. The extracted periods do not fullfil the trend observed previously, namely to decrease with increasing spins. However only J=4 or preferably J=5 data are concerned with this comment. We observe that both corresponding inserts (b) and (c) exhibit one data outside the fit which remains eventually doubtful, asking eventually for more data.

IV Discussion and Conclusion

Table II shows the periods of oscillation of the nuclei levels J=2 studied previously. Their variation versus the mass number A is displayed in fig. 21. Fast increases are observed for 4He, 16O, and 132Ce, with an abrupt fall between 56Fe and 62Zn. The two first high data are related to doubly magic number nuclei. We observe larger periods for closed shells or subshells, followed by smaller periods. The fall between 56Fe and 62Zn should then be attributed to the passing through the magic number Z=28. The 132Ce has 58 protons which close the ${\it 1f_{7/2}}$ subshell.

The missing of enough known spins for the nuclei levels with neutron (or proton) numbers close to other magic numbers, prevents to study these mass regions.

Table III and fig. 22 show the periods of oscillation of the other nuclei level periods studied previously. J=0 data are shown with black empty squares, J=1 with red full circles, J=3 with blue full squares, J=4 with green full stars, and J=5/2 with purple full triangles. The periods concentrate between both dashed lines and decrease with increasing masses, with a fast jump for all 16O periods J=0, 1, 3, and 4. Table III shows that the period of variation of different spin levels remain constant for light nuclei like 10B, 14N, and 20Ne. This is also the case for 16O except for the J = 4 levels. For heavier nuclei, the periods decrease for increasing spins. For 56Fe nucleus for increasing spins J=0, 2, 3, and 4, the periods are respectively: P=2.07, 1.8, 1.63, and 1.005 MeV. For 208Pb, the period for J=5 is somewhat larger than expected for a regular decrease.

When these studies considered all spins Tatischeff:2016psc, the figures of period variation versus the masses exhibited nice shapes in agreement with a clear oscillation for the first several masses only. In the present study done for given spins, the agreement of data versus calculations is good in all ranges where almost all levels have a known spin.

This study should be extended for higher mass hadrons, not known presently. It was already mentioned that a minimum of five masses of given quantum numbers must exist.

The same remark holds for nuclear levels. Whereas a lot of nuclear levels is known, the spin of many of them is ignored. Moreover, there are few levels with unknown spin, which masses are located between the masses of known spin levels. This may then alter the data of higher mass levels.

In conclusion the paper shows that the oscillating periods of mesons and baryons follow the same variation. This symmetry of oscillation is observed for the masses of hadrons and masses of nuclei levels which display oscillatory behaviours well observed using the relation (1) and well fitted with the cosine function (2). Such behaviour requires the need for a theoretical study to describe the oscillating distributions and particularly the oscillation amplitudes.