a r
Xi
v
:h e
p
-
p
h
/
9
9
1
1
5
3v
1
2
6
N
o
v
1
9
9
9
Analytic model of a Regge trajectory in the space-like and time-like regions 1R.Fiore a ,L.L.Jenkovszky b ,V.Magas b,c ,F.Paccanoni d ,A.Papa a a Dipartimento di Fisica,Universit`a della Calabria,Istituto Nazionale di Fisica Nucleare,Gruppo collegato di Cosenza I-87036Arcavacata di Rende,Cosenza,Italy b Bogolyubov Institute for Theoretical Physics,Academy of Sciences of Ukraine 252143Kiev,Ukraine c Department of Physics,Bergen University,Allegaten 55,N-5007,Norway d Dipartimento di Fisica,Universit`a di Padova,Istituto Nazionale di Fisica Nucleare,Sezione di Padova via F.Marzolo 8,I-35131Padova,Italy Abstract A model for a Regge trajectory compatible with the threshold behavior required by unitarity and asymptotics in agreement with Mandelstam analyt-icity is analyzed and confronted with the experimental data on the spectrum of the ρtrajectory as well as those on the π−p →π0n charge-exchange re-action.The fitted trajectory deviates considerably from a linear one both in the space-like and time-like regions,matching nicely between the two.PACS num
bers:12.40.Nn,13.85.Ni.Regge trajectories may be considered as building blocks in the framework of the analytic S -matrix theory.We dedicate this contribution to the late N.N.Bo-golyubov,whose contribution in this field is enormous,on the occasion of his 90-th anniversary.The model to be presented is an example of the realization of the ideas of the analytic S -matrix theory.There is a renewed interest in the studies of the dynamics of the Regge trajec-tories [1,2,3].There are various reasons for this phenomenon.
The hadronic string model ([4])was successful as a mechanical analogy,generating a spectrum similar to that of a linear trajectory,but it fails to incorporate the interaction between the strings.Although intuitively it seems clear that hadron production corresponds to breakdown of the strings,the theory of interacting strings faces many problems.Paradoxically,the final goal of the hadronic string theory and,in a sense,of the modern strong interaction theory,is the reconstruction of the dual (e.g.Veneziano)amplitude from the interacting strings,originated by the former.Non-linear trajectories were derived also from potential models.The saturation of the spectrum of resonances was shown [5]to be connected to a screening quark-antiquark potential.
A relatively new development is that connected with various quantum deforma-tions,although the relation between q deformations and non-linear(logarithmic) trajectories wasfirst derived by Baker an
d Coon[6].q-deformations of the dual am-plitudes(or harmonic oscillators)resulted[7,8]in deviations from linear trajectories, although the results are rather ambiguous.By a different,so-called k-deformation, the authors[3]arrived at rather exotic hyperbolic trajectories.
All these developments were preceded by earlier studies of general properties of the trajectories[9],that culminated in classical papers of the early70-ies by E.Predazzi and co-workers[10],followed by the paper of late A.A.Trushevsky[11], who were able to show,on quite general grounds,that the asymptotic rise of the Regge trajectories cannot exceed|t|1/2.This result,later confirmed in the frame-work of dual amplitudes with Mandelstam analyticity[12],is of fundamental im-portance.Moreover,wide-angle scaling behavior of the dual amplitudes imposes an even stronger,logarithmic asymptotic upper bound on the trajectories.The com-bination of a rapid,nearly linear rise at small|t|with the logarithmic asymptotics may be comprised in the following form of the trajectory:
α(t)=α(0)−γln(1−βt),(1) whereγandβare constants.
The threshold behavior of the trajectories is constrained by unitarity:
I mαn(t)∼(t−t n)R eα(t n)+1/2,(2) where t n is the mass of the n−th threshold.The combination of this t
hreshold behavior with the square-root and/or logarithmic behavior is far from trivial,unless one assumes a simplified square root threshold behavior that,combined with the logarithmic asymptotics,results in the following form[13]
α(t)=α0−γln(1+β√
Fig.1shows the Chew-Frautschi plot with the trajectory(3),(4)and four thresh-olds[14]included.This trajectory matches well with the scattering data[16,17,18], as shown in Fig.2,wherefits to the scattering data based on the model[19]are presented.
The construction of a trajectory with a correct threshold behaviour and Man-delstam analyticity,or its reconstruction from a dispersion relation is a formidable challenge for the theory.This problem can be approached by starting from the fol-lowing simple analytical model where the imaginary part of the trajectory is chosen as a sum of terms like
I mαn(t)=γn
t−t n
√√t
n
θ(t n−t)+
+2
π nγn
Γ(λn+3/2)
t n2F1 −λn,1;3/2;t n
01234567
t [GeV 2]01
2
3
4
5
6
7
R e α(t )ρ (770)
a 2 (1320)
ρ3 (1690)a 4 (2040)ρ5 (2350)
a 6 (2450)
Figure 1:Chew-Frautschi plot for the six low-lying I =1parity even mesons (ρ-trajectory).The masses of the resonances were taken from [20].References [1]L.Burakovsky,String model for analytic nonlinear Regge trajectories ,
hep-ph/9904322;M.M.Brisudova,L.Burakovsky and T.Goldman,Effective functional form of the Regge trajectories ,hep-ph/9906293.
[2]F.Filipponi,G.Pancheri,Y.Srivastava,Phys.Rev D 59,076003(1999).
[3]J.Dey,P.L.Ferreira,L.Tomio and R.R.Choudhury,Phys.Lett.B 331,355
(1994);A.Delfino,J.Dey and M.Malheiro,Phys.Lett.B 348,417(1995);J.Dey,M.Dey,P.L.Ferreira and L.Tomio,Phys.Lett.B 365,157(1996).
[4]B.M.Barbashov and V.V.Nesterenko,Relativistic String Model in Hadron
Physics (in Russian),(Energoatomizdat,Moscow,1987).
[5]F.Paccanoni,S.S.Stepanov and R.S.Tutik,Mod.Phys.Lett.A 8,549(1993);
A.V.Kholodkov et al.,J.Phys.G 18,985(1992).
[6]D.D.Coon,S.Yu and M.Baker,Phys.Rev.D 5,1429(1972).
[7]M.Chaichian,J.F.Gomes and P.Kulish,Phys.Lett.B 311,93(1993).
[8]L.L.Jenkovszky,M.Kibler and A.V.Mishchenko,Mod.Phys.Lett.A 10,51
(1995).
[9]V.N.Gribov and I.Ya.Pomeranchuk,Nucl.Phys.38,516(1962).
[10]A.Degasperis and E.Predazzi,Nuovo Cimento A 65,764(1970);H.Fleming
and E.Predazzi,Lett.Nuovo Cimento 4,556(1970).
[11]A.A.Trushevsky,Ukr.Fiz.Zh.22353(1977).
[12]A.I.Bugrij et al.,Fortschritte der Phys.21,427(1973).
[13]R.Fiore,L.L.Jenkovszky,V.Magas,F.Paccanoni,Phys.Rev.D 60,116003
(1999).
4
[14]See:N.A.Kobylinsky,V.Timikhin,Acta Physica Polonica B9,149(1977)and
earlier references therein.
[15]Tables of Integral Transforms Vol.II,Ed.A.Erdelyi et al.(McGraw-Hill,New
York,1953)
[16]Serpukhov-CERN Collaboration(W.D.Apel et al.),Phys.Lett.B72,132
(1977),JETP Lett.26,502(1977),Pisma Zh.Eksp.Teor.Fiz.26,659(1977).
[17]A.V.Barnes,D.J.Mellema,A.V.Tollestrup,R.L.Walker,O.I.Dahl,R.A.John-
son,R.W.Kenney,M.Pripstein,Phys.Rev.Lett37,76(1976)
[18]Serpukhov-Brussels-Annecy(LAPP)Collaboration(F.Binon et al.),Z.Phys.
C9,109(1981)
[19]F.Arbab and C.B.Chiu,Phys.Rev.147,1045(1966).
[20]C.Caso et al.,Eur.Phys.J.C3,1(1998).
5
0.00.10.20.30.40.5−t [GeV 2]050
100150200
d σ/d t [µb /G
e V 2]
Serpukov−CERN (1977)0.00.10.20.30.40.5
−t [GeV 2]050
100150
d σ/d t [µb /G
e V 2]Barnes et al. (1976)
0.00.10.20.30.40.5
−t [GeV 2]050
d σ/d t [µb /G
e V 2
]Serpukhov−Brussels−Annecy (1981)
Figure 2:Differential cross section dσ/dt [µb/GeV 2]versus −t [GeV 2]for the process π−p →π0n .The solid curves represent the result of the fit with the model by Arbab and Chiu [19]using the trajectory defined in Eqs.(3)and (4).Data are taken from Ref.[16](top-left),Ref.[17](top-right)and Ref.[18](bottom).
6
豫公网安备41160202000603
站长QQ:729038198
关于我们
投诉建议