The Naked Truth about Hadronic Regge Trajectories



A.E. Inopin

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1. INTRODUCTION

Regge trajectories (RT) in hadron physics have been known for some 40 years. Initially
they were introduced by Tullio Regge [1,2], who simply generalized the solution for scat-
tering amplitude by treating the angular momentum L as a complex variable. He proved
that for a wide class of potentials the only singularities of the scattering amplitude in the
complex L plane were poles, now called Regge poles. If these poles occur for positive inte-
ger values of L they correspond to the bound states or the resonances, and they are also im-
portant for determining certain technical aspects of the dispersion properties of the ampli-
tudes. Regge interpreted the simple poles of al(k2) on the complex L-plane to be either
resonances or bound states. Chew and Frautschi [3] applied the Regge poles theory to in-
vestigate the analyticity of al(k2) in the case of strong interactions. They simply postulated
that all strongly interacting particles are self-generating (the bootstrap hypothesis) and that
they must lie on Regge trajectories (Chew-Frautschi conjecture) [4]. At first, linearity was
just a convenient guide in constructing the Chew-Frautschi plots, because data were scarce
and there were few a priori rules to direct the mesons and baryons into the same trajectories
arXiv: 19 Dec 2000 [5]. Once linearity was found to be a good working hypothesis, justification was given
through certain assumptions in the Regge poles theory as follows: For ReL -1/2, the par-
tial-wave components of the scattering amplitude f have only simple poles and are functions
of k2


al(k2) (k2) / (L - (k2)), (1)

where is the residue and the position (Regge trajectory) of the simple poles.
By the end of the 1960s quarks were discovered experimentally and quark-parton
model emerged almost immediately [6]. In the 1970s quantum chromodynamics (QCD) got
a firm ground as a theory of strong interactions. This theory, QCD, has nothing to do with
the original framework of Regge and Chew-Frautschi, since it's dealing with different dy-
namical equations.
The aim of the present paper is to dissect the naked truth about hadronic Regge trajec-
tories. The whole issue is an eclectic mix of confusions, partly because of a huge number
of quark models, which are in many ways contradictory to each other. This leads to am-


1


biguous conclusions about the true nature of Regge trajectories. In this situation the
hadronic data itself presents the purest imprint of the hadronic world. Therefore we will
scrutinize the last issue of Review of Particle Physics 2000 [7], and reconstruct all possible
RT for mesons and baryons. Then we will extract all the slopes, characterizing the given
RT and examine how they deviate from the standard recipe = 0.9 GeV-2. The best pa-
rameters which describe wild deviations of RT from linear and parallel lines are the disper-
sion and the average value of slope for the given RT, <>.

2. MESONS


We will work with the full listings of PDG [7]. First on the list is the light unflavored
mesonic sector, where massive experimental discoveries were made during the last decade.
The most extensively investigated are scalar-isoscalar f mesons, total 28 states. From these
data [7,8] we could construct four radial and two orbital f trajectories. Radial RT for f2 (Jpc
= 2++) is a 13-plet and it is the most nonlinear hadronic RT, with =7.91 GeV-2 and <> =
6.37 GeV-2 (see Fig.1). This trajectory has two peak slopes of 17.54 GeV-2 and 27.47GeV-2.
The f0 radial RT is nonet and it is also quite nonlinear , with =1.69 GeV-2, <> = 2.30
GeV-2 with peak slope value of 6.10 GeV-2. The f4 radial RT has only three states, but it's
essentially nonlinear, with =2.85 GeV-2, <> = 4.40 GeV-2 and peak slope value of 6.41
GeV-2. All the essentially nonlinear mesonic RT will be assembled in Table1. With new
PWA just coming from Crystal Barrel data [8], it is possible to construct radial RT for the
f1 mesons (Jpc = 1++). This f1 is a quartet including the newly discovered f1(1971). It is es-
sentially nonlinear with =1.55 GeV-2, <> = 2.29 GeV-2 with peak slope value of 3.69
GeV-2. Orbital RT for f mesons include parent f0 and daughter f0. Parent f0 is a quartet with





2
Fig.1. Slopes versus radial quantum number Nr for f0 and f2 radial RT.


= 0.65 GeV-2, <> = 1.29 GeV-2 and with peak slope value of 2.03 GeV-2 and it is es-
sentially nonlinear. Daughter f0 RT is a triplet with = 0.82 GeV-2, <> = 1.26 GeV-2 and
peak slope value of 1.84 GeV-2. These two trajectories are essentially nonlinear and
nonparallel.
We now turn to the a-mesons. From the data [7,8] we could construct 4 radial and 3
orbital a-trajectories. The a0 radial RT is a triplet with = 0.20 GeV-2, <> = 0.69 GeV-2.
This trajectory is only slightly nonlinear. The a1 radial RT is a quartet with = 0.19 GeV-2,
<> = 0.79 GeV-2 and it's only slightly nonlinear. The a2 radial RT is a sextet with =
0.90 GeV-2, <> = 1.73 GeV-2 with peak slope value of 3.18 GeV-2. This trajectory is es-
sentially nonlinear. The a3 radial RT is a doublet with = 1.73 GeV-2.
Orbital a-trajectories are unique, because a0 has parent, daughter and granddaughter RT.
The a0 parent RT is a quartet with = 0.96 GeV-2, <> = 1.50 GeV-2, with peak slope
value of 2.61 GeV-2. The a0 daughter RT is a triplet with = 1.05 GeV-2, <> = 2.32 GeV-
2, with peak slope value of 3.06 GeV-2. The a0 granddaughter RT is a triplet with = 1.43
GeV-2, <> = 3.88 GeV-2, with peak slope value of 4.88 GeV-2. It's interesting that slopes
are increasing successively from parent to daughter to granddaughter a0 RT, with all three
RT being nonparallel and essentially nonlinear.
Next on our list will be h1 mesons. Combining data [7,8] with the just discovered at
BNL h1(1594) [9], we have a quintet of states lying on one radial RT. This trajectory has
= 0.55 GeV-2, <> = 1.25 GeV-2 with peak slope value of 1.81 GeV-2, which makes it es-
sentially nonlinear.

Table1: Slopes for essentially nonlinear meson RT (, average <>, mean square deviation , in
GeV-2)
RT for mesons Slopes for neighbor pairs <>
f0(0++) parent 3.00 0.78 0.94 1.58 1.01
f0(0++) daughter 1.84 0.68 1.26 0.82
f0(0++) radial 3.13 1.16 2.34 1.45 0.88 6.10 1.72 1.66 2.30 1.69
f1(1++) radial 2.56 3.69 0.63 2.29 1.55
f2(2++) radial 2.39 3.56 17.54 3.33 1.64 1.82 4.95 27.47 1.61 3.60 3.36 5.13 6.37 7.91
f4(4++) radial 2.38 6.41 4.40 2.85
a2(2++) radial 0.98 3.18 1.18 2.04 1.27 1.73 0.90
a0(0++) parent 2.61 0.86 1.03 1.50 0.96
a0(0++) daughter 3.06 1.58 2.32 1.05
a0(0++) gr.daugh 4.88 2.87 3.88 1.43
radial 0.72 2.65 0.96 1.06 1.10 3.70 1.70 1.20
h1 radial 1.81 1.61 0.69 0.87 1.25 0.55
K(0-) parent 0.73 1.14 0.34 1.25 0.87 0.42
K(0-) daughter 1.69 2.38 2.04 0.49
K(0-) radial 0.53 1.92 1.44 1.30 0.71
K(1+ ) radial 2.90 1.32 2.11 1.12
K (2- ) radial 1.54 6.49 0.57 2.87 3.18
J/ radial 0.25 1.60 0.47 1.03 0.46 0.76 0.55
c(1P) parent 1.51 3.11 2.31 1.13
b(1P) parent 1.54 2.54 2.04 0.71
b(2P) parent 2.11 3.66 2.89 1.10
radial 0.09 0.15 0.21 0.16 0.30 0.18 0.08



3


There is no big news in pion sector. We have three radial RT for mesons. The ra-
dial RT (Jpc = 0-+) is a triplet with = 0.03 GeV-2, <> = 0.62 GeV-2 and it's practically
linear. The 1 radial RT is a doublet with = 1.55 GeV-2, which is quite large. The 2 ra-
dial RT is a doublet with = 0.64 GeV-2. A plot of versus M2 shows that all three RT
are not parallel.
Let us consider mesons. With newly discovered states [8] we can construct two radial
RT and one orbital RT. The radial RT is a septet with = 1.20 GeV-2, <> = 1.70 GeV-2,
with peak slope value of 3.70 GeV-2. It is clearly essentially nonlinear RT. The 2 radial
RT is a quartet with = 0.31 GeV-2, <> = 1.20 GeV-2, with peak slope value of 1.46
GeV-2. This is fairly nonlinear RT, nonparallel to . The parent orbital RT is a triplet
with = 0.08 GeV-2, <> = 0.79 GeV-2, and so it belongs to the class of linear trajectories.
We now turn to mesons. It is possible to construct one radial and one orbital RT.
Radial RT is a quartet with = 0.30 GeV-2, <> = 0.81 GeV-2, with peak slope value of
1.16 GeV-2. This is fairly nonlinear RT. Parent orbital RT is a triplet with = 0.07 GeV-2,
<> = 0.83 GeV-2. This is almost linear RT.
Next on our list are mesons. Using data [7,8] we can construct only one radial RT for
(Jpc = 1--). This is a quintet with = 0.29 GeV-2, <> = 1.07 GeV-2, with peak slope
value of 1.42 GeV-2. So it is slightly nonlinear radial RT.
The -meson sector has quite sparse data with only one radial RT. It is a doublet with
= 0.56 GeV-2.
Now we have finished an analysis of light unflavored mesons and turn to strange mes-
ons (S=  1, C=B=0). The K-mesons sector is very rich, encompassing 24 states so far.
Even without appealing to exchange-degeneracy (EXD) we could construct a large number
of trajectories. We have four radial and two orbital RT for K-mesons. K(0-) radial RT is a
quartet with = 0.71 GeV-2, <> = 1.30 GeV-2, with peak slope value of 1.92 GeV-2. This
is essentially nonlinear RT. K(1-) radial RT is a triplet with = 0.16 GeV-2, <> = 0.94
GeV-2, with peak slope value of 1.06 GeV-2. This is slightly nonlinear RT. K(2-) radial RT
is a quartet with = 3.18 GeV-2, <> = 2.87 GeV-2, with peak slope value of 6.49 GeV-2.
This is one of the most nonlinear RT in nature. K(1+) radial RT is a triplet with = 1.12
GeV-2, <> = 2.11 GeV-2, with peak slope value of 2.90 GeV-2. This is essentially nonlin-
ear RT. All radial RT for K mesons, except K(1-) are essentially nonlinear.
Orbital K-meson's RT started from K(0-) parent spin singlet. It is a quintet with =
0.42 GeV-2, <> = 0.87 GeV-2, with peak slope value of 1.25 GeV-2. This is clearly essen-
tially nonlinear RT. K(0-) daughter RT is a triplet with = 0.49 GeV-2, <> = 2.04 GeV-2,
which is essentially nonlinear. As we see, parent and daughter K(0-) trajectories are nonlin-
ear and nonparallel. Vector K*(1-) parent RT is a quintet with = 0.13 GeV-2, <> = 0.84
GeV-2, and it is just slightly nonlinear RT.

Now we turn to charmed mesons (C = 1) sector. There is insufficient data to construct
RT there. The same situation persist in charmed-strange sector (C=S=1). Things are much
more interesting in the charmonium sector. We have here one radial RT and one orbital
RT. The radial trajectory, which started from famous J/, is a sextet with = 0.55 GeV-2,
<> = 0.76 GeV-2, with peak slope value of 1.60 GeV-2. This is essentially nonlinear RT.
The orbital trajectory is a parent c(1P) and it is a triplet. It has = 1.13 GeV-2, <> =
2.31 GeV-2, with peak slope value of 3.11 GeV-2. This is essentially nonlinear RT. As we
see, all the charmonium trajectories are essentially nonlinear.


4


The data on bottom mesons (B=1), bottom, strange mesons (B=1, S=1) and bottom,
charmed mesons (B=C=1) are still insufficient to construct RT.
Next we consider bottomonium sector. From the data [7] we can construct one radial
and two orbital RT. The radial RT is a sextet with =0.08 GeV-2, <> = 0.18 GeV-2.
Although dispersion is small, this is strict consequence of small slopes, which are very dif-
ferent. This is essentially nonlinear RT. b(1P) orbital RT is a triplet with = 0.71 GeV-2,
<> = 2.04 GeV-2. This is essentially nonlinear RT. The b(2P) orbital RT is a triplet with
= 1.10 GeV-2, <> = 2.89 GeV-2, with peak slope value of 3.66 GeV-2. This is essen-
tially nonlinear RT, which is nonparallel to b(1P).
We conclude that out of a total of 32 mesonic RT, 22 belong to the category of essen-
tially nonlinear. Seven RT are fairly nonlinear, and only three RT are linear, which
amounts to 9% share. (We did not account for doublets RT, which don't have a curvature).

3. BARYONS

3.1 N -


In the baryonic sector we have many more trajectories than for mesons. Our strategy
will be to discuss the most interesting cases, leaving the rest for the tables and figures.
The nonstrange sector is very rich, comprising 23 N and 22 states [7]. We will in-
clude in the analysis three new resonances, just discovered at ELSA, SAPHIR [7]:
D*13(1895), S*11(1897) and P*11(1986)1. We will also include in analysis the so-called
N(~3000 Region) and (~3000 Region), which are mostly the results of PWA by Hendry
[11]2. So, altogether we have 31 N and 28 resonances. N, and spectra exhibit a very
interesting clustering structure. In nucleon sector we see the following four clusters: sextet
S11(1650)-D15(1675)-F15(1680)-D13(1700)-P11(1710)-P13(1720) is squeezed within 70 MeV
interval; triplet D*13(1895)-S*11(1897)-P13(1900) is squeezed within 5 MeV interval; triplet
D13(2080)-S11(2090)-P11(2100) is squeezed within 20 MeV interval, and quartet G17(2190-
D15(2200)-H19(2220)-G19(2250) is squeezed within 60 MeV interval. First cluster is split
into three parity doublets: S11(1650)-P11(1710), D13(1700)-P13(1720), D15(1675)-F15(1680).
Second cluster has one parity doublet: D*13(1895)-P13(1900). Third cluster has one parity
doublet: S11(2090-P11(2100). Fourth cluster has one parity doublet: H19(2220)-G19(2250).
In sector we see the following two clusters: septet S31(1900)-F35(1905)-P31(1910)-
P33(1920)-D35(1930)-D33(1940)-F37(1950) is squeezed within 50 MeV interval and triplet
F37(2390)-G39(2400)-H311(2420) is squeezed within 30 MeV interval. First cluster is split
into three parity doublets plus one extra state: S31(1900)-P31(1910), P33(1920)-D33(1940),
F35(1905)-D35(1930). Second cluster has no parity doublets. This clustering pattern and the
precise mechanism of parity doubling in N, and spectra remained the challenges for the
current quark models. One promising approach introduced the so-called Rarita-Schwinger
(RS) clusters [12]. As a result, the author found three RS clusters both in N and spectra,
as opposed to the experimentally seen four clusters in N, and two in sectors.
Now we turn to the analysis of N, and Regge trajectories. Major parent nucleon tra-
jectory P11(938)-F15(1680)-H19(2220)-K113(2700)-L117(3500) is moderately nonlinear quin-
tet, with = 0.28 GeV-2, <> = 0.81 GeV-2. Major parent trajectory P33(1232)-



1 We will mark by asterisk, "*", all the tentatively defined resonances throughout this paper.
2 We include these data in our original papers [10], where we used NRQM to fit all the spectra.

5


F37(1950)-H311(2300)-L315(3700) is also just moderately nonlinear quartet, with = 0.18
GeV-2, <> = 0.78 GeV-2.
Nevertheless there are plenty of nonlinear RT in the nonstrange sector (see Fig. 2). P31
parent RT: P31(1750)-F35(1905)-H39(2300)-K313(3200) is essentially nonlinear quartet with
= 1.63 GeV-2, <> = 1.71 GeV-2, with peak slope value of 3.53 GeV-2. S31 parent RT:
S31(1620)-D35(1930)-G39(2400)-I313(2750)-L317(3300)-N321(4100) stretches to the highest
possible mass 4.1 GeV. It is an essentially nonlinear sextet, with = 0.56GeV-2, <> =
0.97 GeV-2, with peak slope value of 1.82 GeV-2. S11 parent RT: S11(1535)-D15(1675)-
G19(2250) is extremely nonlinear triplet with = 2.51 GeV-2, <> = 2.67GeV-2, and peak
slope value of 4.44 GeV-2. D13 parent RT: D13(1520)-G17(2190)-I111(2600)-L115(3100)-
N119(3750) is essentially nonlinear quintet, with = 0.24 GeV-2, <> = 0.74 GeV-2.





1/2+ parent





1/2-parent






J/2

Fig. 2. Slopes for essentially nonlinear baryonic orbital RT.


Among the radial RT in N, sector there are few essentially nonlinear. S11 radial RT is
a quartet: S11(1535)-S11(1650)-S11*(1897)-S11(2090). It is essentially nonlinear RT with
= 0.87 GeV-2, <> = 1.72 GeV-2 and peak slope value of 2.72GeV-2. D13 radial RT is a
quintet: D13(1520)-D13(1700)-D13*(1895)-D13(2080)-D13(2600). It is essentially nonlinear
RT with = 0.57 GeV-2, <> = 1.23 GeV-2 and peak slope value of 1.72GeV-2. P11 radial
RT is a quintet: P11(939)-P11(1440)-P11(1710)-P11*(1986)-P11(2100). It is essentially non-
linear RT with = 0.59 GeV-2, <> = 1.29 GeV-2 and peak slope value of 2.15GeV-2.
Some of the RT are too short (doublets) to judge on nonlinearity, but they have rather
large slopes. F35 radial RT is a doublet with = 2.7 GeV-2. P31 daughter orbital RT is a
doublet with = 5.68 GeV-2 and it is nonparallel to P31 parent RT. P13 parent orbital RT is


6


a doublet with = 2.0 GeV-2. We conclude that in N, and sector we have five nucleon
and two delta essentially nonlinear RT.

Table 2: Slopes for essentially nonlinear baryon RT (, average <>, mean square deviation , in
GeV-2)
RT for baryons Slopes for neighbor pairs <>
N1/2- parent 4.44 0.89 2.67 2.51
N3/2- parent 0.80 1.02 0.70 0.45 0.74 0.24
N1/2+ radial 0.84 1.18 0.98 2.15 1.29 0.59
N1/2- radial 2.72 1.14 1.30 1.72 0.87
N3/2- radial 1.72 1.43 1.36 0.41 1.23 0.57
1/2+ parent 3.53 1.20 0.40 1.71 1.63
1/2- parent 1.82 0.98 1.11 0.60 0.34 0.97 0.56
1/2+ radial 0.76 1.40 1.08 0.45
1/2- radial 1.23 2.22 1.32 1.59 0.55
3/2- radial 1.83 0.39 1.11 1.02
1/2+ radial 0.83 7.58 2.65 2.49 3.39 2.91
1/2- radial 6.06 3.65 1.07 3.60 2.50
3/2- radial 3.41 1.03 2.22 1.68


3.2
-


We turn now to the - sector. There are many interesting features in this qqs sector.
One of them is exchange degeneracy (EXD) hypothesis, which happened to hold quite well
in the - sector. As we will see later, EXD lead to trajectories with negative slopes,
which never arise in N- sector. Another feature is the clustering in the - sector, which
is qualitatively different from clustering in the N- sector. Third feature is the existence of
parity doublets in the - sector. Because - has only one strange quark, their shape is
still not so deformed. If we can imagine that we have three balls in a bag, and two of them
are of almost the same weight, while the third a bit heavier than the two, the bag will get the
form of a pear. It will be reflectionally asymmetric. Near the rest the deformation is per-
haps still not so dramatic and the lowest excitations are similar to those of the nonstrange
baryons. For that reason we observe also in -spectrum the same sequence 1/2+, 1/2-, 3/2-
as in nonstrange baryons. If the heavier ball starts to gain rotational energy, the deforma-
tion will increase. The pear shape gets more pronounced and when the pear oscillates it
gives rise to parity doublets, which we already see.
Full listings [7] give to us 18 and 26 resonances. Some of the states are lacking
the JP assignments. Let's take a closer look at this. State (2000) does not have JP, but
data from Cameron78 (see full listings [7]) allowed tentatively, the JP = 1/2- assignment.
Further evidence came from the recent paper by Iachello [13] and older one by Capstick-
Isgur [14]. Therefore we assign JP = 1/2- to the (2000). The -states with highest
masses, (2350) and (2585) were not described theoretically and there are no clear claims
from the experiments. For this reason we will not include (2350), (2585) in our Regge
analysis, and we have total of 16 resonances to work with.
The situation with hyperons is even more interesting. Two low-lying states, (1480)
and (1560) do not have any JP assignments from the experiment and theory can't predict
them either. We will exclude (1480), (1560) from our analysis. The production experi-
ments [7] give strong evidence for (1620), tentatively claiming JP = 1/2+. This claim is in

7


accord with fresh calculations by Iachello [13]. So with newly defined *1/2+(1620), we
form an exact parity doublet 1/2-(1620) - *1/2+(1620). The production experiments [7]
give strong evidence for (1670) bumps without JP assignments. Using predictions by
Iachello [13] and Isgur [14], we clearly get JP = 1/2- for (1670). It's interesting that this
way we have two resonances with the same mass and different JP (see [7]). Such a degener-
acy waits proper theoretical explanation. The state (1690) has most likely claim from the
data [7] as JP = 5/2+. We will assign JP = 5/2+ to (1690) in our analysis. Next hyperon
without JP assignment will be (2250). Using the results from Iachello [13] and Isgur [14],
we assigned JP = 5/2- to (2250). Last few bumps, (2455), (2620), (3000) and (3170)
has no experimental claims for JP, and there are no theoretical predictions so far for such a
high masses. For this reason we will not include (2455), (2620), (3000), (3170) in
our analysis. Finally, we have total of 22 hyperons for our analysis.
Clustering pattern in spectrum is very nontrivial. We clearly see three clusters there.
Quartet P11(1660)-D13(1670)-S*11(1670)-F*15(1690) is squeezed within 30 MeV interval.
There is one parity doublet within this cluster: P11(1660)-S*11(1670). Triplet S11(1750)-
P11(1770)-D15(1775) is squeezed within 25 MeV interval. There is one parity doublet
within this cluster: S11(1750)-P11(1770). Triplet F15(2070)-P13(2080)-G17(2100) is
squeezed within 30 MeV interval. There are no parity doublets in this cluster. It's amazing
that we have three 1/2+ - 1/2- parity doublets in the sector: S11(1620)-P*11(1620),
P11(1660)-S*11(1670) and S11(1750)-P11(1770).
In sector we witness only one cluster. This quartet S01(1800)-P01(1810)-F05(1820)-
D05(1830) is squeezed within 30 MeV interval. The whole cluster is split into two parity
doublets: S01(1800)-P01(1810) and F05(1820)-D05(1830). Note that an author [12] suggested
four clusters in sector. It is a big puzzle for current quark models to explain this differ-
ence in clustering and parity doubling between N- and - sectors.
Now we turn to Regge analysis of - sector. We will mostly concentrate on essen-
tially nonlinear trajectories. The 1/2- radial RT is a quartet S01(1405)-S01(1670)-
S01(1800)-S*01(2000). It is essentially nonlinear RT with = 0.55 GeV-2, <> = 1.59
GeV-2 and peak slope value of 2.22 GeV-2. The 3/2- radial RT is a triplet D03(1520)-
D03(1690)-D03(2325). It is essentially nonlinear RT with = 1.02 GeV-2, <> = 1.11 GeV-
2 and peak slope value of 1.83GeV-2. The 1/2+ radial RT is a triplet P01(1116)-P01(1600)-
P01(1810). It is essentially nonlinear RT with = 0.45 GeV-2, <> = 1.08 GeV-2. Many
other -trajectories possess some degree of nonlinearity. The 1/2+ radial RT is a quintet
P11(1193)-P*11(1620)-P11(1660)-P11(1770)-P11(1880). It is essentially nonlinear RT with
= 2.91 GeV-2, <> = 3.39 GeV-2 and peak slope value of 7.58 GeV-2. The 1/2- radial RT
is a quartet S11(1620)-S*11(1670)-S11(1750)-S11(2000). It is essentially nonlinear RT with
= 2.91 GeV-2, <> = 3.39 GeV-2 and peak slope value of 6.06 GeV-2. The 3/2- radial
RT is a triplet D13(1580)-D13(1670)-D13(1940). It is essentially nonlinear RT with = 1.68
GeV-2, <> = 2.22 GeV-2 and peak slope value of 3.41 GeV-2.

Amazingly, all essentially nonlinear radial RT in - sectors are mirroring each other:
1/2+ - 1/2+; 1/2- - 1/2-; 3/2- - 3/2-. There are no essentially nonlinear RT among
the orbital - trajectories.

3.3 EXD in
-
-





8


It has been known for years that exchange degeneracy seems to exist experimentally at
least for strange hyperons [5]. We will construct appropriate RT with tentatively assigned
resonances. Major -trajectory in this scheme will be a quartet: P11(1193)-P13(1840)-
F*
15(1690)-F17(2030). This essentially nonlinear RT has one negative slope 2 = -1.89
GeV-2, = 1.47 GeV-2, <> = -0.20 GeV-2. Corresponding daughter RT is a triplet:
P*11(1620)-P13(2080)-F15(1915). This essentially nonlinear RT also has one negative slope
2 =-1.52 GeV-2, = 1.49 GeV-2, <> = -0.47 GeV-2, and it's nonparallel to the parent
RT. Major negative parity -trajectory is a quartet: S11(1620)-D13(1580)-D15(1775)-
G
17(2100). This essentially nonlinear RT has one negative slope 1 = -7.81 GeV-2, =
5.32 GeV-2, <> = -1.67 GeV-2. Corresponding 1/2- daughter RT is a triplet: S*11(1670)-
D13(1940)-D*15(2250). This is moderately nonlinear RT with = 0.18 GeV-2, <> = 0.90
GeV-2 and it's nonparallel to the parent RT.
We start sector with major 1/2+ trajectory, which is a quintet: P01(1116)-P03(1890)-
F
05(1820)-F07(2020)-H09(2350). This essentially nonlinear RT has one negative slope 2
=-3.85 GeV-2, = 2.76 GeV-2, <> = -0.71GeV-2. Major negative parity 1/2- trajectory is
a quartet: S01(1405)-D03(1520)-D05(1830)-G07(2100). This is essentially nonlinear RT
with = 1.20 GeV-2, <> = 1.64 GeV-2 and peak slope value of 3.03 GeV-2.
As we see, EXD in - sector leads to a new class of trajectories, which are character-
ized by negative average slopes. Five out of six - trajectories are essentially nonlinear.

Table3: Slopes for baryonic essentially nonlinear EXD RT
RT for baryons Slopes for neighbor pairs <>
1/2+ parent 0.51 -1.89 0.79 -0.20 1.47
1/2+ daughter 0.59 -1.52 -0.47 1.49
1/2- parent -7.81 1.53 1.26 -1.67 5.32
1/2+ parent 0.43 -3.85 1.30 0.69 -0.71 2.76
1/2- parent 3.03 0.96 0.94 1.64 1.20
1/2+ parent 0.48 3.14 1.81 1.88


3.4
,,
, Charmed, Beauty Baryons
We still have to analyze double-strange hyperons, (qss). Full listings [7] give to us
11 's. Some of the states are lacking the JP assignments. State (1620) is a bump, which
does not have JP assignment from the experiment, and it also could not be predicted by the-
ory so far. For this reason, we will exclude (1620) from our analysis. The state (1690)
does not have JP assignment from the experiment, but Iachello [13] predict this state with JP
=1/2+. The state (1950) does not have JP assignment from the experiment, but Iachello
[13] predict this state with JP =3/2+. The state (2030) has tentative assignment JP=5/2?
from the experiment [7], and we will consider it as a *5/2+(2030). Resonance (2120)
does not have JP assignment from the experiment, but Iachello [13] predicts this state with
JP =3/2-. Next bump, (2250) does not have JP assignment from the experiment, but
Iachello [13] predicts JP =1/2+. The state (2370) does not have JP assignment from the ex-
periment, but Isgur [14] predict this state with JP =7/2-. The last resonance, (2500), does
not have JP assignment from the experiment, and it also could not be predicted by theory so
far. For this reason, we will exclude (2500) from our analysis. Finally, we have nine
resonances to work with: P11(1315)-P13(1530)-P*11(1690)-D13(1820)-P*13(1950)-
F*15(2030)-D*13(2120)-P*11(2250)-G*17(2370). There is no clustering in sector. Major


9


parent RT is a doublet P11(1315)-F*15(2030), with = 0.84 GeV-2, = 0. Another or-
bital RT is a parent 3/2-. It is a doublet D13(1820)-G*17(2370), with = 0.87 GeV-2, =
0. 1/2+ radial RT is a triplet P11(1315)-P*11(1690)-P*11(2250). It has = 0.31 GeV-2,
<> = 0.67 GeV-2 and it is fairly nonlinear RT. The 3/2+ radial RT is a doublet
P13(1530)-P*13(1950), with = 0.68 GeV-2, = 0. The 3/2- radial RT is a doublet
D13(1820)-D*13(2120) with = 0.85 GeV-2, = 0. As we see, basically all RT's in sec-
tor are too short to make a conclusions about their linear/nonlinear nature.
If we assume EXD for hyperons, we can construct parent 1/2+ triplet RT:
P11(1315)-P*13(1950)-F*15(2030). This RT happens to be essentially nonlinear with =
1.88 GeV-2, <> = 1.81 GeV-2 and peak slope value of 3.14 GeV-2. We will compile all
EXD-essentially nonlinear RT in , , sectors in Table 3.
If we consider , charmed baryons and beauty baryons, there are not enough data to
construct and analyze Regge trajectories.
We conclude that out of total 21 baryonic RT 13 are essentially nonlinear (62%). Five
RT are fairly nonlinear (24%), and only 3 trajectories (14%) are in fact linear. (We don't
account for doublet RT here, because they have no curvature).


4. CONCLUSIONS


We have constructed and scrutinized here all possible RT for the full listings PDG2000
[7] and even accounted for the newest data on mesons [8,9]. We want to stress that this ap-
proach leads to minimal bias in the interpretation of the results, unlike the results from any
available quark models.
We have shown that in the mesonic sector out of total 32 RT, 22 trajectories are essen-
tially nonlinear (or 69%), and seven trajectories are fairly nonlinear (or 22%). Only three
trajectories could be classified as linear, which amounts to 9% share. Among essentially
nonlinear meson RT 10 are orbital and 12 are radial.
In baryonic sector out of total 21 RT, 13 trajectories are essentially nonlinear (or 62%),
and five trajectories are fairly nonlinear (or 24%). Only three trajectories could be classi-
fied as linear, which amounts to 14% share. Among essentially nonlinear baryon RT four
are orbital and 12 are radial.
Appropriate dispersion, , for nonlinear mesonic and baryonic RT span the range 1~7.9
GeV-2, and slopes span range 1~27.5 GeV-2.
We have four doublet RT in mesonic sector and 31 doublet RT in baryon sector, which
don't have a curvature, and so this massive sector is used only for the evaluation of paral-
lelism between different RT.
So, our results strongly disagree with general opinion, that hadron RT are straight and
parallel lines. As the data shows, in the currently available resonance energy region,
mesonic and baryonic RT are grossly nonlinear, and only small part (~12%) of all RT could
be classified as linear, with ~0, 0.9 GeV-2.
The existence of clusters in baryon spectra and their absence in mesonic sector is a big
puzzle. In N-sector we have four clusters: sextet, quartet and two triplets. In -sector we
have only two clusters: septet and triplet. In -sector we have three clusters: quartet and
two triplets. In -sector we have one cluster, a quartet. The N, clusters have average
spacing between the levels of 8.9 MeV, and , clusters have very similar average spacing
of 8 MeV.



10


First nucleon cluster is split into three parity doublets: 1/2- -1/2+, 3/2- -3/2+, 5/2- - 5/2+.

First cluster is also split into three parity doublets: 1/2- -1/2+, 3/2- - 3/2+, 5/2--5/2+. So
they are exactly mirroring each other. Within nucleon clusters, parity doublets (PD) 1/2- -
1/2+ and 3/2- - 3/2+ occurred twice.
Second nucleon cluster has one PD: 3/2- - 3/2+. Second cluster has no PD at all.
Third nucleon cluster has one PD, 1/2- - 1/2+ and fourth nucleon cluster has one PD,
9/2- - 9/2+.
Within clusters we find two 1/2- - 1/2+ PD, which are exactly mirroring two nucleon
1/2- - 1/2+ PD.
In total, in N, , , and sectors we have six 1/2- - 1/2+ PD, three 3/2- - 3/2+ PD, three
5/2- - 5/2+ and one 9/2- - 9/2+ PD. The exact dynamical reasons for such clustering and
parity doubling patterns in baryons remains a big puzzle for theory.


Acknowledgements

Author is very grateful to G.S. Sharov and S. King for the help with manuscript.


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