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Are the - and A2-Trajectories Linear.

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Annalen der Phyeik. 7. Folge, Band 32, Heft 4,1976, S. 297 -303
J. A. Berth, Leipig
Are the e- and A?-Trajectories Linear?
Institut fur Hochenergiephysik der Akademie der Wiseenschaften der DDR, Borlin-Zeuthenl)
Institute for Theoretical Physics, Academy of Sciences of tho Ukranian SSR, Kiev (UdSSR)
With 8 Figures
We analyse in this paper to what extent the rise of the real part of Regge trajectories is due
to contributions from baryon-antibaryon thresholds and to what extent it is connected with the
presence of a linear term.It is shown that sewing the trajectory from the scattering to the resonance
region results in “switching off” the linear term; the remaining contributions of various thresholds
guarantee the necessary linearity of the trajectories.
The Regge trajectory concept has for long been used as an ercsential element in
hadronic physics. Nevertheless recent progress in the strong interaction physics renews
the activity towards this object. This is connected, in particular, with the developments of narrow resonance and analytic dual models (the properties of the two can be
found, for example, in review articles [l]and [2], respectively), in which Regge trajectories appear as certain dynamical variables. On the other hand, much more information on the behaviour of Regge trajectories is now available due to recent charge exchange experimental data [3--51.
Therefore the desire to have a good model or at least a persuasive parametrization
of Regge trajectories is justified. It is widely believed that a trajectory is n linear
function of the energy squared. However, attractive as this hypotheuis may seem at
first sight, there is no particular reason why this should be so. lteally,
i) none of the meson trajectories contains more than two resonances (three for
baryon trajectories) firmly established [6] ;
ii) the resonance spectrum allows for alternative parametrizations of Re a ( S )
[7--91 different from the linear one;
iii) dificulties appear when a linear extrapolation from the resonance region to thc
scattering one is carried out [9, 101;
iv) it is shown that the Regge trajectories in the analytic approach increaae asymptotically as 1/;F [2,111.
If we rely on bounds obtained in theanalytic approach, i. e. a(#)
at large S,
and assume that the main contribution to the trajectory comes from the threshold
at S = 4mz then the simplest form for the trajectory obviously is
-1 5
I) On leave of absence from and address after November 15th 1974: Institute for Theoretical
Physics, Kiev, Ukranian SSR..
2 98
There can be stable states (or “narrow” resonances) with spill J < A and inass
M < 2m on trajectory (1) but i t does not contain resonances with M > 2m [9, 121.
If m + 00 and siniultaneousely A --t 00, y + 00 such that
A - 2m y + a,,
y/4m +-b,
A - y 1/4ma- S + a + bS.
Thus if the main contribution to the trajectories conies froin very heavy thresholds,
then for a limited range in S the trajectories will be practically linear.
Look now at data froni the scattering [3-5] and the resonance [6] region on the
e- and A2-trajectoriea: in the region 1 5 S 5 3 GeV2 these trajectories are remarkably close to a straight line. But, on t.he other hand, baryon-antibaryon thresholds are
those among the heavy thresholds whose contributions should be included in trajectories and which in the hierarchy of heavy threaolds should be, apparently, the first
ones. However, they seem to be too light to provide for nearly linear behaviour of the
trajectories in so wide a domain (though no resonances are confirmed firnily above
the baryon-antibaryon thresholds). Therefore we keep the linear term in the trajectory
but include also the cqntributions froni various two-particle thresholds,
~ ( 8=)A
+ bS - 2 y , I/S, - S .
NOW we analyse the existing data on trajectories in order to clarify
to what extent the growth of the real part of the trajectoiy is due to the contribution
from heavy thresholds and to what extent it conies froin the linear term.
Consider first the e-trajectory. We include in (2) all two-particle thresholds, corN fi, AE,
and e” 2-threshold s
responding to stable particles, i. e. the nn,K
Two resonances are known to lie on the e-trajectory, the e and g with paranietera [6]
7n, = 770 MeV,
= I50 MeV,
= 180 MeV.
m, = 1686 MeV,
Fig.1 Contribution to the e-trajmtory (2) of the linear term as 8 function of the
intercept or of the p i t i o n of WSNZ point a(&) = 0 end the same for the sum of
beryon-antibaryon thresholds
Are the e- and A,-TraJectoriesLinear ?
The niesonic thresholds - mz and K K determine mainly resonance widths, while
the baryon-antibaryon thresholds, together with the term (bS) in (2), are responsible
for resonance masses.
Assume that all heavy thresholds enter in (2) with equal constants y. Then (3) imposes 4 conditions on 5 parameters: A, b, y , ynn,and y g ~while,
aa already mentioned,
the behaviour of Re a($)is determined mainly by parameters b and y. Therefore we
s'tudy the behaviour of the trajectory for different sets of these parameters.
Important characteristics of a Regge trajectory in the scattering region are the
intercept a(0)and the value S = So a t which a(rS,)= 0. In Fig. 1 those values of parameters b and y are shown which should be chosen in order that the e-trajectiory in the
form (2) would generate resonances with parameters (3) and would have definite
values of a(0)and So.It can be seen from Fig. 1 that for a(0) 0.62 the linear term in
(2) should disappear. Similar is the situation for So -0.64.
Look now at experimental data on u(0)and S., The following values of a(0)can be
obtained from the behaviour of the differential cross section of the process n-p + non
a(0) = 0.56 f 0.02 [3],
= 0.53
= 0.51
The valuea of a(0)obtained from the difference of the total cross sections of n+p and
n-p scattering are somewhat higher than those presented above,
a(0)= 0.67 f 0.06 [14],
= 0.57 f 0.01 [15].
As to So, we have from the behaviour of the e-trajectory, determined from the energetic dependence of the nN charge exchange differential cross section :
EJ0 = -0.68 f 0.03 [3],
= -0.64
= -0.64GeV2
If determined from the position of the dip in the differential croaa section, So,aa seen
in Fig.2, drops a little bit with energy (this effect is well described in the model 1131)
and assumes the value 2: -0.65 GeVa.
These values of a(0)and Sosuggest that once we have included in the trajectory the
contributions from baryon-antibaryon thresholds the linear term is switched off
when we sew t.he trajectory from two different regions (scattering and resonance).
' PL (6eV/c)
Fig.2 Poeition of the dip in the differential c r w section in n- --t nen scattering
for various values of energy. Data for 3 < p < 16 @V/c are lorn (161 and for
p > 20 GeV/c from ref. [3]. The curve is drawn by eye
A. I. B U ~ R and
The behavior of the e-trajectory without the linear term is shown in Figs. 3-5.
It can be seen from Fig. 3 that besides the e and g resonances there is a wide resonance
with J = A, m -N 2 130 MeV and F = 470 MeV on the e-trajectory. A candidate for
this resonance is the e (2100) or T (2200) [6]. The real part of the e-trajectory passes
a t AS'* 2560 MeV the value Re a = 7. However, one will not find a resonance with
-6 -5
3 5
- -3
Fig. 3
Fig. 4
Fig.3 Real part of the e-trajectory in the resonance region. The trajectory is defined
by expreasion (2) with b = 0 and with values of other parameters listed in Table 1
Fig.4 Real part of the @trajectory for -7.5
S 2 GeV/c)*. The data for
8 2 - 1.8 (GeV/c)*are from ref. [17]and those for S 2 - 1.2 (GeV/~)~arefromref.
[3, 51. The dashed line is a straight one passing through the Q- and the A,-resonances
< <
such spin in the model since the contribution from thresholds placed below this point
are too large and the amplitude with such a trajectory will not have poles in this region
(a more detailed discussion of this point can be found in [9]).
Figs. 4 and 5 show the behaviour of ~ ~ in
( the
8 )scattering region. The model fits the
existing data on the e-trajectory better than a linear trajectory drawn through the e,
A, and g-mesons. A still better fit is possible after a more scrupulous parametrization
Fig.6 The @-trajectoryin the scattering region. The dashed line is a straight one
assing through the e- and A,-resonancea. By , we denote data from ref. [3] and
those from ref. [6].
Are the e- and A,-Trajectoriee Linear ?
30 1
of various thresholds. (It should be remembered that the data used here come from an
effective trajectory for the process n-p + non; for the reconstruction of the autentic
trajectory the contributions of other singularites and of the backgra und to the cross
section should also be included).
In the case of the A,-trajectory we include the same baryon-anti1 aryon thresholds
and also nq- and KE-thresholds (the latter accounts effectively fc r the nearby in:
elastic en-threshold, contributing essentially to the A,-meson width The value of So
for the A,-trajectory according to [5] is close to that for the @-trajectory.By putting
So = -0.645GeVa, as in the case of the e-trajectoryof the type (2) without the term
[as],and using the parameters of the A,-meson [6]
r n A a = 1310 MeV,
r A a = 100 MeV,
we find that the A,-trajectory behaves as shown in Figs. 6 and 7.
The model for the A,-trajectory under discussion describes scattering data better
than a linear one drown through thee- and the A,-mesons (see Fig. 7). There are three
resonances on the A,-trajectory (see Fig. 6) with upins 2, 4, 6 and masses 1310, 1850
and 2 250 MeV, respectively; however, there are no resonances with J 2 8. Typically,
three resonances appearing on the trajectory are practically on a straight line. We have
met a similar situation in the case of the @-trajectory.
Fig. G
Fig. 7
Fig.6 Real part of tho &-trajectory in the reeonance region. The positions of the
thresholds accounted for are shown. Appropriate parameters are listed in Table 1
Fig.7 The &trajectory in the scatbring re ion.
= t i e data from NAL [6]
0 = the data from SERPUKHOV
[3, 63,
Thus, the account in the trajectory for contributions from the baryon-antibaryon
thresholds allows for a satisfactory description of all available data on meson trajectories. Also, it turns out that the account for these contributions allows one to switch
off in the trajectory the linear term, now a rudiment. Two particle stable thresholds
already guarantee sufficient linearity. The account for other thresholds can further
linearize the trajectory (these contributions can also raise the value of max [Re a ( S ) ] ;
however this would not lead to new resonances with spin J 2 7 because of the considerable contribution from baryon-antibaryon thresholds).
It has been known for a long time that very heavy thresholds, for example quarkantiquark ones, give almost linear trajectories in limited S-range. However, an estiinate [18] predicts the maases of these quarks to be z 10 GeV, i. e. the leading threshold
A. I. B U ~ R Iand
in the trajectory would be at 8-v 400 GeV2.Our analysis shows that theleading thresholds may be much lower, namely at S 3 + 7 Gev2.
As seen from Table 1, containing the paranieters of the e- and A,-trajectories, the
meson thresholds contribution to the trajectories is much smaller than that of baryons.
The appearence of reaonance8 is connected with the considerable contribution from
heavy thresholds. Therefore, it is natural to assume that the Pomeron trajectory, which
Table 1 Parametera of the e and A2-trajectmies
max [Re a(s)]
Memn thresholds
(e - A2)-trajectory a&) = 0.47
+ 0.89 . 8
Y: = Y
is nonresonant, (a discussion of its properties can be found, for example, in 1191) is
determined mainly by contributions from meson thresholds. Retaining the mt,K B
and rpj threshold6 and putting
ynn= ygg = yS9 = 0.35
we find that the P-trajectory in such a model does not contain resonances
max [Rea(S)] = 1.82,
and behaves as shown in Fig. 8. Here we also present data on the effective trajectory
of pp-scattering taken from [20].
- 0
c -2
-76 -74
S (6eV/cl
Fig.8 The Pomeron trajectory when meson thresholds are included only with
y, = 0.35. Data on the effective trajectory of ppscattering are from ref. [20]
Flnally, we stress that various models of Regge trajectories are connected with
specific models for the scattering amplitudes, in which they play the role of a sort
of dynamical variables. Linear trajectories are connected with narrow resonance dual
models [l]. Another class of dual models [21] is based on logarithmic trajectories. A
main feature of these trajectories is the existence of a “ionization point”, above which
there are no resonances. As to the model of Regge trajectories being developed in the
present paper, it is intimately connected with a class of dual model8 [2] analytic in the
sense of CHEW-MANDELSTAM[22]. These model amplitudes are claimed to account
correctly for all main singularities and therefore important for their development is the
construction of consistent models for Regge trajectories.
Are the e- and &Trajectories Linear ?
For discussions we are indebted to L. L. JENKOVSZKY
and to the participants of t h e
Theoretical Seminar of the Jnstitut fur Hochenergiephysik. One of us (A. B.) is pleased
to acknowledge the hospitality of the Institut. fiir Hochenergiephysik, Akadeniie der
Wissenschaften der DDR.
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29 (1973). .
Fortschr. P h p . 91, 427
(1973); L. GONZALEZMESTRES, R.HONQTUAN.Preprint LPTHE 72/20, Onay (1972).
[3] v. h'. BOLOTOV,
Yedernaya Fiz. 18,1046 (1973).
Yadernaya Fiz. 18, 1262 (1973).
[4] V. N. BOLOTOV,
Plenary Seasion Talk at the XVIl International Conference on High Energy
Physica in London, July 1974.
[GI Review of Particle Pro rtiea, Phys. Lett. WB, No. 1 (1974).
[I] P. J. KELEPEN,Phys. E v . Lett. 28. 998 (1969).
[8] C. T. GRANT,E. L. LOMON,
Phys. Rev. D8,260 (1971).
Ukr. Fiz. J. a@,164 (1976); A. I. BUORIJ,N. A. KOBYLINSKY,
ITP-74-63& Kiev (1974).
[lo] D. 1'. SHIBKOV,
Uzpekhi Fiz. Nauk 109, 87 (1970).
[ l l ] IC. N.KHURI,Phys. Rev. Lett. 18, 1094 (1967); R. CHILDERS,Phys. Rev. Lett. 21, 868.
lCG9 (1968); J. C. BOTKE,Nucl. Phys. B40,141 (1972).
Lett. Nuovo Cimento 6, 389 (1972).
[13] E. LEADER,
Phys. Rev. D7. 836 (1973).
1141 U. P. GORIN,Yadernaya Fiz. 17. 309 (1953).
[la] R . E. HEUDRICK,
Preprint COO-2232B-58, 1974.
Preprint LBL-63, April 1973.
1171 N. W. DEAN,Phys. Rev. D6, 133 (19i2).
[18] P. D. B. COLLINS, Phys. Rep. 1, 103 (1971).
V. P. SHELEST,Preprint ITP-71-103E, Kiev (1971);
Lett. Nuovo Cimento 8, 76 (1972);
V. P. SHELEST,Preprint RIFP-139,
Kyoto (1972).
Talk presented at IXth Balaton Symposium on Particle Physics,
Hungary, June 1974.
[21] D. D. COON,Phys. Lett. 29B,669 (1969); M. BAKER,
D. D. COON,Phys. Rev. DZ, 2349 (1970).
[22] G . F. CHEW, The Analytic S-Matrix, New York 1966.
Bei der Kedaktion eingegnngen am 18. November 1974.
Ansclir. d. Verf.: Dr. A. I. BUORIJand Dr. N.,A. KOBYLINSW
Inst. for Theoretical Physics Academy of Sciences of the Ukranian SSR
Kiev (UdSSR)
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