The explicit procedures for reconstruction of full set of helicity amplitudes in elastic proton-proton and proton-antiproton collisions

0.04 MB·English

by  V. A. Okorokov

The Explicit Procedures for Reconstruction of
Full Set of Helicity Amplitudes in Elastic
Proton-Proton and Proton-Antiproton Collisions
V.A. Okorokov and S.B. Nurushev†
∗
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0 MoscowEngineeringPhysicsInstitute(StateUniversity),115409,Moscow,Russia
∗
0 †InstituteforHighEnergyPhysics,142281Protvino,MoscowRegion,Russia
2

n
Abstract. The explicit proceduresare described for reconstruction of the full set of helicity am-
a
plitudesinproton-protonandproton-antiprotonelasticscattering.Theproceduresarebasedonthe
J
derivativerelationsforthe helicity amplitudesin s-channel,on the explicitparametrizationof the
7
leadingspinnonflipamplitudesandcrossing-symmetryrelations.Asymptotictheoremsareused
1
fordefinitionoffreeparametersinderivativerelations.WealsostudytheOdderoninfluenceonthe
2 helicity amplitude reconstruction. Reconstruction procedures are valid at extremely wide energy
v domainandbroadrangeofmomentumtransfer.Theseproceduresmightbeusefulinstudyingthe
5 spinphenomenainproton-protonandproton-antiprotonelasticscattering.
4
Keywords: proton,antiproton,elasticscattering,helicityamplitude
0
PACS: 11.80-m 11.80.Cr 13.75.Cs 13.85.Dz
2
1
6
0 INTRODUCTION
/
h
p
Elastic scattering of hadrons has always been a crucial tool for study the dynamics of
-
p strong interaction. In the absence of the strong interaction theory the predictions or
e
interpretation of the experimental observables are furnished by the phenomenological
h
: approaches.Oneofthereliableapproachesistheasymptoticalmodel.Othermodels(like
v
i Reggemodel)sufferfromnecessitytointroducemanyfree parameterswhichshouldbe
X
defined by fitting to the experimental data. In such a situation the direct reconstruction
r
a of the scattering matrix from the complete set of the experimental data will be the
appropriate method. Unfortunately such set of experiments were never been fulfilled in
thehighenergyregion.OnemayhopethatsuchprogramwillberealizedatRHIC,FAIR
(PAXproject)andotherfacilitiesinnearfuture.Inordertomakethepredictionsforthe
measurable observables in those facilities one needs to have a method which should
be well justified, contains a small number of free parameters and applicable at wide
range of the kinematics variables. Below we propose such a technique. Its applicability
may be tested by the joint consideration of proton-proton and proton-antiproton elastic
scatteringdata.
RECONSTRUCTION PROCEDURE
Two methods have been suggested for building the full set of helicity amplitudes for
elastic pp¯-collisionsin [1, 2]. We describe in details themethod fordeducing the helic-
ityamplitudesbasedoncrossing-symmetryrelationandthederivativerelationshere.The
amplitudes for the binary reaction A+B C+D in s , t , and u channels all de-
→ − − −
pend upon the Mandelstam variables and are described by just one set of analyticfunc-
tions evaluated in different regions of variables s,t,u. Thus the following preliminary
expression for full set of helicity amplitudes of pp¯ elastic scattering via set of helicity
amplitudesfor pp elasticreactionhavebeen derivedin[2]:
F pp¯ =1 2 sin2y F pp+F pp+F pp + 1+cos2y F pp
1 1 2 3 4
F pp¯ =1(cid:14)2(cid:2)sin2y (cid:0)F pp+F pp F pp(cid:1) (cid:0)1+cos2y (cid:1)F pp(cid:3) 
2 1 3 − 4 − 2 

F 3pp¯ =1(cid:14)2(cid:2)sin2y (cid:0)F 1pp+F 2pp−F 4pp(cid:1)−(cid:0)1+cos2y (cid:1)F 3pp(cid:3)  (1)

F pp¯ =1(cid:14)2(cid:2) 1+co(cid:0)s2y F pp sin2y (cid:1) F (cid:0)pp+F pp (cid:1)F pp (cid:3)+2F ppsiny 
4 1 − 3 2 − 4 5
F 5pp¯ =1(cid:14)2c(cid:2)o(cid:0)sy siny (cid:1)F 1pp+F 2pp+F(cid:0)3pp−F 4pp +2F 5pp(cid:1)(cid:3) 


where (cid:14) (cid:2) (cid:0) (cid:1) (cid:3) 

m s 4m2
st p p 2t
cosy = ; siny = − sinq ; cosq =1+ ;
s s 4m2p t 4m2p tqt 4m2 s 4m2p
− − p −
−
q - CM scat(cid:0)tering an(cid:1)g(cid:0)le in the(cid:1)s-channel,qm (cid:0)- proton(cid:1)mass. The system (1) shows
p
apparentanalyticalformsforfullsetofamplitudesofelastic pp¯ scattering F pp¯
i
i=1 6
viahelicityamplitudes F pp forcrossing-symmetrical ppchannel.nLetstoostre−ss
j
j=1 5
thattherelationF pp=nF ppohasb−eentakenintoaccountinthe(1)already.Therelation
5 − 6
F pp¯ = F pp¯ corrects according to G-parity conservation in elastic pp¯ collisions [3].
5 6
B(cid:12) uto(cid:12)nth(cid:12)eoth(cid:12)erhandtheexactcorrelationbetween F pp¯ and F pp¯ isopenquestionin
(cid:12) (cid:12) (cid:12) (cid:12) 5 6
(cid:12) (cid:12) (cid:12) (cid:12)
general casebecauseofOdderon polecontributioni(cid:12)sstill(cid:12)acon(cid:12)tenti(cid:12)oustopic.
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Derivative relations
The pp elastic scattering under study in order to obtain some additional correlations
for set of helicity amplitudes F pp . Usually the following formula is suggested
j
j=1 5
for spin non-flip amplitudesF npp =oF pp.−The derivativerelations allow to express spin-
1 3
flipamplitudeF pp and spindouble-flip amplitudeF pp viaF pp [4]:
5 4 1
¶ ¶ 2
F pp(s,t)=Cpp(s) F pp(s,t); F pp(s,t)=Cpp(s) F pp(s,t), (2)
5 1 ¶ (√ t) 1 4 2 ¶ (√ t)2 1
− −
pp pp pp
whereC (s)=C (s)+iC (s), k=1,2 - complex parameters in general. The some
k k1 k2
versions of additional correlation for spin double-flip helicity amplitude F pp were dis-
2
cussedin[1,2,5].WewouldliketoemphasizethatcomplexparametersC (s)(k=1,2)
k
must be defined for exact knowledge of spin-flip and double-spin amplitudes for pp
elasticscattering.
Determination of the free parameters
It seems that the combination of sets of helicity amplitudes for pp and pp¯ elastic
reactions namely in the framework of the method described above and some additional
pp
equationsresult in several ways for analyticdeterminationof parametersC (k=1,2).
k
The Odderon hypothesis is crucial important for definition of unknown parameters
pp
C (k=1,2) in the derivativerelations (2). We suggest to use the asymptotic behavior
k
of total cross section, differential cross section, r and B parameters in order to obtain
thecomplexunknownparameters inhigh-energy domain.
The most general case corresponds to possibility for Odderon exchange as well as
forPomeronone.Thegeneral PomeranchuktheoremissatisfiedinframeworkOdderon
hypothesis, but the original Pomeranchuk theorem is violated. Thus one can get the
pp
followingequationsystemfordefinitionofC (k=1,2)parameters:
k
D s tot ≡s tpop¯t−s tpopt (cid:181) ImF 1pp¯(s,t =0)−ImF 1pp(s,t =0) ,
D r =r pp¯ r pp, h i  (3)
D ds /dt−= ds /dt pp¯ ds /dt pp, 
el el el 
D B=Bpp¯ Bpp. − 
(cid:0) −(cid:1) (cid:0) (cid:1) (cid:0) (cid:1)


ThedefinitionoftheparametersCpp(k=1,2)becomesmodeldependentandnontrivial
k
taskbecauseofmodeldependentvaluesareontheleftpartsofequationinthesystem(3).
According to accelerator and cosmic ray experimental data [6] and phenomenological
estimates[7]alsothevaluesofs pp¯ are(very)closetocorrespondingvaluesofs pp upto
tot tot
energy√s=100TeV.Theusualinterpretationoftheexperimentaldataforr -parameter
is that D r is very small. It should be emphasized that even D r = 0 would not exclude
anOdderon,butwouldonlyruleoutspecificmodelsforthesoftOdderon.Thusonecan
suggestsD s tot=0andD r =0withoutfullexcludingofOdderonpolecontributionand
withoutsignificant violation of general description consequently.One can use only two
first equation in the system (3) in the framework of simplesuggestionsthat there is one
complexparameterortwoclearreal / imagineparameters. It shouldbeemphasizedthat
the unknown parameters would be define model independently in these specific cases
eventakingintoaccount possiblepresenceofthe(soft)Odderon contribution.
Asshownabovethenewexperimentaldateandphenomenologicalinvestigationswill
be very important at ultra-high energy √s 100 TeV in particulary for decision some
∼
fundamental problems for hadron interactions and distinguishing different theoretical
models.
TheoriginalPomerahcuktheorems,namely,fortotalcrosssectionandfordifferential
cross section in binary reaction, and Cornille-Martin theorem for the forward slope
parameterB [8] can beused in theframework ofhypothesisfor presenceonly Pomeron
pp
exchange.Thesystemfordefinitionoffree parametersC (k=1,2)isgivenby
k
ImF pp¯(s,t =0)=ImF pp(s,t =0),
1 1
ReF pp¯(s,t =0)=ReF pp(s,t =0),
1 1  (4)
ds el/dt pp¯ = ds el/dt pp, 
Bpp¯ =Bpp. 
(cid:0) (cid:1) (cid:0) (cid:1)


Thissystemallowstodetermineallcomponentsofcomplexfreeparametersbymodel
independent way. Moreover for Pomeron exchange only there is additional correlation
between helicity amplitudes for pp¯ elastic reaction, namely, F pp¯ = F pp¯ just as well
5 6
as for pp scattering. The asymptotic relations and fit of exp(cid:12)erim(cid:12)enta(cid:12)l dep(cid:12)endences at
(cid:12) (cid:12) (cid:12) (cid:12)
low and intermediate energies should be used for determinat(cid:12)ion o(cid:12)f fr(cid:12)ee pa(cid:12)rameters for
thereconstructionmethodunderstudy.
SUMMARY
The main results of this paper are following. We suggest the explicit procedure for the
reconstruction of the the full set of the helicity amplitudes for the elastic pp¯ scattering.
This method is based on fundamental crossing-symmetry property, derivative relations
for helicity amplitudes and selection the anatlytical expression for the spin non-flip
amplitude describing well the pp experimental data. We apply to this spin non-flip
amplitudethe derivativeprocedure for finding all pp helicity amplitudes.After proving
that we get the good description of pp-data we apply the crossing relations in order to
find the helicity amplitudes for the pp¯ elastic scattering. We introduce the minimumof
thefreeparameterswhicharefixedthroughtheasymptoticrelationstakingintoaccount
the Pomeron and Odderon contributions. The unified analysis of the pp and pp¯ data
allowstocheckindetailstheproposedmethod.Itseemsthisanalyticalmethodmightbe
useful for direct reconstruction of elements of the scattering matrix at extremely wide
initialenergy domainand broadrange ofmomentumtransfer.
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