Neglecting terms with a power decrease at high s, the Particle Data Group PDGfitstototalcrosssections[3,4]arethesumofoneconstant componentandanotherrisingaslns2,correspondingtoasimple pol
Trang 1Contents lists available atScienceDirect
www.elsevier.com/locate/physletb
S.M Roy
HBCSE, Tata Institute of Fundamental Research, Mumbai, India
a r t i c l e i n f o a b s t r a c t
Article history:
Received 18 June 2016
Received in revised form 2 November 2016
Accepted 15 November 2016
Available online 18 November 2016
Editor: J.-P Blaizot
I proposeatwo component analyticformula F ( t =F (1) ( t +F (2) ( t for( ab→ab ) + ( a b¯→a b¯)
scattering at energies≥100 GeV,where s t denote squaresof c.m energyand momentumtransfer
It saturates the Froissart–Martin bound and obeys Auberson–Kinoshita–Martin (AKM) [1,2] scaling I choose ImF (1) ( 0)+ImF (2) ( ,0)asgivenbyParticleDataGroup(PDG)fits [3,4]tototalcrosssections, correspondingtosimpleandtriplepolesinangularmomentumplane.ThePDGformulaisextendedto non-zeromomentumtransfersusingpartialwavesofImF (1)andImF (2)motivatedbyPomeronpoleand
‘greydisk’amplitudesand constrainedbyinelastic unitarity.Re F ( t isdeducedfromrealanalyticity:
IprovethatRe F ( t )/ ImF ( 0) → (π/ ln s ) d dτ(τImF ( , t )/ ImF ( 0))fors→ ∞withτ=t lns )2fixed, and apply it to F (2).Using alsothe forwardslope fitby Schegelsky–Ryskin [5],the model givesreal parts,differentialcrosssectionsfor(−t < 3 GeV2,andinelasticcrosssectionsingoodagreementwith data at546 GeV,1.8 TeV,7 TeVand8 TeV.Itpredictsforinelastic crosssections forpp or p p,¯ σinel=
72.7±1.0 mb at7 TeVand74.2±1.0 mb at8 TeVinagreementwithpp Totem [7–10]experimental values73.1±1.3 mb and74.7±1.7 mb respectively,andwithAtlas [12–15]values71.3±0.9 mb and
71.7±0.7 mb respectively The predictions σinel=48.1±0.7 mb at 546 GeV and 58.5±0.8 mb at
1800 GeV alsoagree with p p experimental¯ resultsof Abeet al [47] 48.4± 98 mb at546 GeVand
60.3±2.4 mb at1800 GeV.Themodelyieldsfor√
s >0.5 TeV,withPDG2013 [4]totalcrosssections, andSchegelsky–Ryskinslopes [5]asinput,σinel ( =22.6+ 034lns +.158( lns )2mb,andσinel/σtot→0.56,
s→ ∞,wheres isinGeV2units.Continuationtopositivet indicatesan‘effective’t-channelsingularity
at∼ (1.5 GeV)2,and suggeststhatusualFroissart–Martinbounds arequantitativelyweakas theyonly assumeabsenceofsingularitiesupto4m2
π
©2016TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense
(http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3
1 Introduction
Precisionmeasurementsofpp crosssectionsatLHC[7–16],and
incosmic rays [17] motivate me topresenta modelforab→ab
scatteringamplitude atc.m energies√
s>100 GeV describedby
an analytic formula containing very few parameters Neglecting
terms with a power decrease at high s, the Particle Data Group
(PDG)fitstototalcrosssections[3,4]arethesumofoneconstant
componentandanotherrisingas(lns)2,correspondingtoasimple
poleandatriplepoleat J=1 intheangularmomentumplane,
σtot ab= σtot (1), ab+ σtot (2), ab,
σtot (1), ab=P ab, σtot (2), ab=H(ln s/s ab M)2. (1)
E-mail address:smroy@hbcse.tifr.res.in
Iproposethat,analogously,thefullamplitude F( ,t) =F (1)( ,t) +
F (2)( ,t),where, F (1) isaPomeronsimplepoleamplitude, ImF (2)
has partial waves with a smooth cut-off at impact parameter
b=R( ) corresponding to a grey disk and Re F (2)( ,t) is calcu-latedfromatheoremIproveusingrealanalyticityandAuberson– Kinoshita–Martin(AKM)[1,2]scalingfors→ ∞withfixedt(lns)2 Inelastic unitarity is tested using inputs of total cross sections, forward slopes and Pomeron parameters Only inputs leading to unitary amplitudes are accepted Model predictions for inelastic cross sections,near forwardreal parts anddifferential cross sec-tionsagreewithexistingdataandcanbetestedagainstfutureLHC experiments
2 Froissart–Martin bound basics
Froissart [18], from the Mandelstam representation, and Mar-tin [19],fromaxiomatic field theory,proved that the total cross-http://dx.doi.org/10.1016/j.physletb.2016.11.025
0370-2693/©2016 The Author Published by Elsevier B.V This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ) Funded by
3
Trang 2section σtot( ) fortwo particlesa,b togo toanythingmust obey
thebound,
σtot( ) ≤s→∞C[ln( /s0) ]2, (2)
whereC,s0 are unknownconstants Itwas provedlater [20] that
C=4π /(t0), wheret0 isthe lowest singularity inthe t-channel.
Thisboundhasbeenextremelyusefulintheoreticalinvestigations
[21,22]andhighenergymodels[23–32].Analogousboundsonthe
inelasticcrosssectionhavebeenobtainedbyMartin[33]andWu
et al.[34];forpion–pioncase,MartinandRoyobtainedboundson
energyaveraged total[35] andinelasticcross sections[36] which
alsofixthescalefactors0 inthesebounds
3 Normalization
Forthe ab→ab scattering amplitude F( ,t),a=b, withk=
c.m.momentum,andz=1+t/(2k2),
F( ,t) = √s/(4k)
∞
l=0
(2l+1)Pl(z)al( ),
σtot( ) =4π /(k2)
∞
l=0
(2l+1)Ima l( )
dσ
dt = π
k2
dσ
d ( ,t) = π
k24F( √ ,t)
s 2
withtheinelasticunitarity constraint Ima l( ) ≥ |a l( ) |2.For
iden-ticalparticles a=b,thepartialwavesa l( ) →2a l( ) intheabove
partialwave expansionsfor F( ,t), and σtot( ), butthe odd
par-tialwavesarezero.We havethesameformulae fortheunitarity
constraint,andthedifferentialcrosssectionasgivenabove
Athighenergy,usinga l( ) ≡a(b,s ,l=bk,whereb isthe
im-pactparameter,andP l(cosθ ) ∼ J0
(2l+1)sin(θ/2)
+O(sin2(θ/2)),
wehavetheimpactparameterrepresentation,
F( ,t) =k√
s/
∞
0
bdba(b, )J0(b√
−t)
σtot=8π
∞
0
bdbIma(b, ); σel=8π
∞
0
bdb|a(b, )|2
dσ /dt=4π ∞
0
bdba(b, )J0(b√
−t)2
There exist very goodfits to highenergy data[37,38] with a
very large number of free parameters There are also very good
eikonal based models involving several free parameters [23–32]
TherecenteikonalbasedmodelofBlockandHalzen (BH)[39,40]
uses highenergy data to guess the glue-ball massand to probe
whethertheprotonisablackdisk
4 A two component partial wave model
Ipresent a two component model with very few parameters
andwithanalytic formulaefor the total amplitude incorporating
unitarity-analyticityconstraints, PDG total cross sections andthe
AKMscalingtheorem
4.1 Imaginary parts
I use the two component PDG total cross section fit I
pro-pose that in the impact parameter picture, the Imaginary part
Ima(b,s of the partial waves at fixed s is also a sum of two components, one part Ima (1)(b,s a Gaussian corresponding to a Pomeronpole,andtheotherIma (2)(b,s apolynomialofdegree2n
inb2 withasmoothcut-offatb=R( ),n beingapositiveinteger,
sothat Ima (2)(b,s iscontinuousandhascontinuousderivativeat
b=R( ).Thesecondcomponentcorrespondstoa“grey”diskwith crosssectionrisingas(ln s)2,
Ima(b, ) =Ima (1)(b, ) +Ima (2)(b, ), Ima (1)(b, ) =C( )exp(−2b2/D2( )), Ima (2)(b, ) =E( )(1−b2/ 2( ))2nθ (R( ) −b), (5) where θ (x) =1, for x≥0, and 0 otherwise The unitarity con-straintsare,
InEq.(5)wetakethesimplestchoicen=1 inthispaper.Usingthe ansatzforIma (1)(b,s,integratingoverb,andmatchingtheresult forImF (1)( ,t)withthestandardsmallt Pomeronamplitude,
F (1)( ,t) =k
√
s
(1)
tot exp(tb P+tα ln s)(i+tπ
weobtain,
D2( ) =8(b P+ α ln s),C( ) = σtot (1)/(2πD2( )). (8) Since σtot (1) is a constant, C( ) →const/(ln s), s→ ∞ for α =0 Similarly,theansatzforIma (2)(b,s withn=1 yields,
ImF (2)( ,t) =E( ) 4k
√
s
q3R( )J3(qR( )),q≡ √ −t, (9) where J m(x)denotestheBesselfunctionoforderm.Hence,
σtot (2)( ) =16π
k√
s ImF
(2)( ,0) =4π
Thus,C( )D2( )andE( )R2( )aredeterminedfromthePDF to-tal cross section fits using Eqs (8) and(10) respectively A nice featureofthemodelisthat theaboveunitarityconstraints(6)as wellasastrongerversionincludingrealpartscanbereadilytested, andprovideacceptability criteriaforextrapolationsof experimen-taldataforpp scattering.
4.2 Theorem on real parts
Let F( ,t) =F(y;t), y≡ ((s−u)/2)2 be an s−u
symmet-ric amplitude, with asymptotic behaviour |s|(ln|s|)γ|φ( τ ) |, τ ≡
t(ln|s/s0|)β, where φ () is a real analytic function of it’s argu-ment andφ (0) =1 Forfixed physicalt, F isrealanalytic in the
cut- y plane with only a right-hand cut from (2ma m b+t/2)2 to
∞ F mustberealfor y= |y|exp(iπ ),i.e.s→ |s|exp(iπ /2),and hencereplacing|s| →sexp( −iπ /2),wehavefors→ ∞, τ fixed,
F( ,t) ∼ −C s exp( −iπ / )(ln( /s0) −iπ / )γ
× φ(t(ln( /s0) −iπ / )β) (11) Expandinginpowersof1/ln s atfixed τ weget,
ImF( ,t)
Re F( ,t)
2 ln( /s0)
γ φ ( τ ) + β τ φ ( τ )
Re F( ,t)
s → ( π / )( ∂(ImF( ,t)/s
Rea(b, ) → ( π / ) ∂(Ima(b, ))
Trang 3where,duetolinearity,thelasttwo equationsalsoholdfora
su-perposition ofterms ofthe form(11),e.g F (1)+F (2) Notethat,
(i) Re F( ,0)/ImF( ,0) agrees with the Khuri–Kinoshita theorem
[41],(ii) thecaseβ = γ =1 agreeswithMartin’sgeometrical
scal-ingformula[42,43] (iii) When σtot∼ (ln s)2, γ = β =2,the AKM
theoremandAuberson–Roytheorem[1,2]guaranteethescalingof
ImF( ,t)/ImF( ,0) with φ ( τ ) being an entire function of order
half The crucial new result is the formula (13) for Re F( ,t) In
turn,thisyieldsforthepartialwavesofF (2),ifb2Ima (2)(b,s →0
forb→ ∞,
Re a (2)(b, ) → − π
2 ln( /s0)b
∂
∂b Im a
(2)(b, ),s→ ∞. (16) However, in view of the slow approach to asymptotics, the
for-mula(15)for Rea(b,s involvingderivative over ln s ispreferable
forcomputations,asitholdsalsofor F (1)+F (2)
4.3 The total amplitude
Consistentwith(13)for γ = β =2,i.e. τ =t(ln|s/s0|)2,I adopt
theansatz,
Re F (2)( ,t)
ImF (2)( ,0) = π
ln( /s0)
d
dτ
τImF
(2)( ,t) ImF (2)( ,0)
Forsimplicity, I choose the scale factor s0 to be the same as in
thePDG (2005)[3]fitfor pp totalcrosssection, √
s0=5.38 GeV
SubstitutingtheexpressionforImF (2)( ,t)Iobtain,
k√
s F
(2)( ,t) = σtot (2)( ) π
ln( /s0)
×8 J2(qR( )) −16 J4(qR( ))
q2R2( ) +i 48 J3(qR( ))
(qR( ))3
The total amplitude F( ,t) = F (1)( ,t) +F (2)( ,t) is now
com-pletelyspecified(analytically)byaddingF (1)( ,t)givenby(7).The
important parameter R2( ) is determined from the experimental
slope parameter B( ) = (d/dt)
ln dσ /dt
|t=0, if the Pomeron pa-rametersb P, α areknown,
R2( )
( ) σtot (2)( )2+1
2σtot (2)( ) σtot( )
=4B( )
( ) σtot (2)( )2+ σtot( )2
− σtot (1)σtot( )D2( ) −4π α
( ) σtot (1)σtot (2)( ), (19) where,wedenote√
( ) ≡ π /ln( /s0).Fortheexperimentalslope parameterIshallusethefits B(M,s toallpp data,withM=1,2,
B(1,s byOkorokov[6]andB(2,s bySchegelsky–Ryskin[5],
B(1, ) =8.81+0.396lns+0.013(lns)2GeV−2,
B(2, ) =11.03+0.0286(lns)2GeV−2, (20)
where√
s isinGeVunits.Forpp,p p total¯ crosssectionsIusethe
PDGfitsof(2005)and(2013),
σtot (2005)( ) =35.63+0.308
ln( s
28.94)
2
mb
σtot (2013)( ) =33.73+0.2838
ln( s
15.618)
2
4.4 Elastic and inelastic cross sections
Theintegralsoverimpactparameterneededtocalculate σelcan
bedoneexactly.Weobtain,
σel( ) = ( π / )C2( )D2( )(2+ (β ( ))2)
+4πR2( )E2( )(3+2 ( ))/15
+2πR2( )C( )E( )δ−3( )
exp(−2δ(s))
× (−1+2β ( )
( )(2δ2( ) +3δ(s +2) ) +
(2β ( )
( )(δ(s −2) +2δ2( ) −2δ(s +1)
, δ(s ≡R2( )/D2( ), β ( ) ≡4π α /D2( ). (22)
5 Predictions of the model versus experimental data forpp and
¯
p p scattering
5.1 Differential cross sections
Remarkably, asinglepairofvaluesofthePomeronparameters
b P, α,
gives very good agreement of model predictions in the entire range|t| <0.3 GeV2withtheexperimentalTotem[7–10]andAtlas [12–15] pp differentialcrosssectionsat7 TeV and8 TeV, experi-mental p p differential¯ cross sections at 546 GeV from UA4 col-laborations, D Bernard et al.[44] and M Bozzo et al [45], and
at1800 GeV fromAmoset al [46] andAbeet al [47].(See also the compilation in [48].) This agreement is independent of the choice between PDG (2005) andPDG (2013) total crosssections, andthechoicebetweenslopes B(1,s andB(2,s.Weexhibitthis
inFigs 1,2,3,4forforwardslopechoice B=B(2,s [5]andthe two choicesoftotalcrosssectionsPDG(2005)[3](dashedcurve), and PDG (2013) [4] (solid curve) (Differential cross sections for
( −t) >0.3 GeV2arenotusedindeterminationofPomeron param-eters b P, α as they make negligible contributions to σel in this energyrange;e.g.inthismodel,about0.2 mbat7 TeVand8 TeV.) For the choice B=B(2,s [5] andPDG (2013) [4] total cross sections,wegivebelowthreeparameterfitstopredicted differen-tialcrosssectionsinthisrangeoft atc.m.energiesupto14 TeV,
ln((dσ /dt)/(dσ /dt)t=0)
=19.5t−11.9t2+43.5(−t)3,7 TeV
=19.7t−13.2t2+47.3(−t)3,8 TeV
=20.5t−19.2t2+64.2(−t)3,13 TeV
forreadycomparisonswithexistingandfuturedata
5.2 Inelastic cross sections
Fig 5 depicts the predicted inelastic cross sections up to
100 TeV andtheirasymptotic fits.Tables 1 and 2give model pa-rameters anddetailedpredictions from 546 GeVto 14 TeV,with input total cross sections P DG2013 and P DG2005 respectively.
The predicted ρ = Re F( ,t)/ImF( ,t) |t=0 and the predicted in-elastic crosssections (e.g.forinput totalcross section P DG2013,
ρ =0.136, σinel=74.2 mb, at 8 TeV) are very close to available experimental values [49,50,7–10,12–15] The predicted inelastic cross sections are fairly robust, changing by lessthan 0.5 mb in the range (7 TeV, 14 TeV) when the slope parameter is changed from B(1,s toB(2,s andbylessthan1 mbwhentheinput σtot
is changed from PDG (2005) to PDG (2013) Model results give
∂ σinel/∂B∼1.07 mb GeV2,∂ σinel/∂ σtot∼0.46,andusinginput er-rorsofPDG2013fits,andδB∼0.3 GeV−2 upto100 TeV[5],I have theerrorestimate,δ σ ∼ 47+ 0021
ln( /15.618) 2
mb
Trang 4Fig 1 Modelpredictions for pp elastic differential cross sectionsd σ / dt at7 TeV,
with parametersb P=3.8 GeV−2 ,α=0.07 GeV−2 , forward slope from Schegelsky–
Ryskin fit [5] , inputσ tot from PDG (2005) [3] (dashed curve), and inputσ tot from
PDG (2013) [4] (solid curve), show excellent agreement with experimental values
from the Totem [7–10] and Atlas [12–15] collaborations for|t|<0.3 GeV2.
Fig 2 Modelpredictions for pp elastic differential cross sectionsd σ / dt at8 TeV,
with parametersb P=3.8 GeV−2 ,α=0.07 GeV−2 , forward slope from Schegelsky–
Ryskin fit [5] , inputσ tot from PDG (2005) [3] (dashed curve), and inputσ tot from
PDG (2013) [4] (solid curve), show excellent agreement with experimental values
from the Totem [7–10] and Atlas [12–15] collaborations for|t|<0.3 GeV 2
Fig 3 Modelpredictions forp p elasticdifferential cross sectionsd σ / dt at546 GeV,
with parametersb P=3.8 GeV−2,α=0.07 GeV−2, forward slope from Schegelsky–
Ryskin fit [5] , inputσ tot from PDG (2005) [3] (dashed curve), and inputσ tot from
PDG (2013) [4] (solid curve), show good agreement with experimental values from
UA4 collaborations, D Bernad et al [44] and M Bozzo et al [45] for|t|<0.3 GeV 2
Fig 4 Model predictions for p p elastic differential cross sections d σ / dt at
1800 GeV, with parametersb P=3.8 GeV−2 ,α =0.07 GeV−2 , forward slope from Schegelsky–Ryskin fit [5] , inputσ totfrom PDG (2005) [3] (dashed curve), and input
σ tot from PDG (2013) [4] (solid curve), show good agreement with experimental values from Amos et al [46] and Abe et al [47] for|t|<0.3 GeV2.
Fig 5 Plotsofpp inelasticcross sectionsσ inel ( q , M )computed from the model with
q=1 andq=2 signifying inputs ofσ total ( P D G−2005)[3] andσ total ( P DG−2013)
[4] respectively and M=1 and M=2 signifying inputs of Okorokov [6] and Schegelsky–Ryskin [5] slopes respectively Input Pomeron parameters are b P=
3.8 GeV−2 ,α=0.07 GeV−2 Three parameter fits to these inelastic cross sections are also shown.
In thec.m energy rangefrom0.5 TeVto 100 TeV, the model parametersareverywellapproximatedbythefollowingfits
Inputσtot (2005)( ) :
Inputσtot (2013)( ) :
where,x≡ln s.
Remarkably,fitsforinput σtot (2005)( )showthatthechoiceM=1 givesE( )whichisbarelybelowtheunitaritylimitfors→ ∞.The inelasticcrosssectionfitsinFig 5yield,
Trang 5Table 1
Detailed results at 546 GeV, 1.8 TeV, 7 TeV, 8 TeV, 13 TeV and 14 TeV from the model using inputsb P=3.8,α =
.07 GeV−2 , PDG 2013 values ofσ tot ( pp )[4] , and Schegelsky–Ryskin extrapolations(M=2, i.e. B=B (2, s [5] for forward slopes The output parametersC and E showexplicitly that inelastic unitarity is obeyed The output values of
R2 show a slowly expanding size of the proton with increasing energy The output results forσ inel / σ tot, 16π σ el B σ2
tot, andρ=Re F ( t=0)/ ImF ( t=0), which would be 1/2, 1 and 0 respectively in the black disk limit, give quantitative measures for deviations from that limit The outputρ agrees with available experiments [49,50] The output values of
σ inelagree within errors with Totem results [7–10] and Atlas results [12–15] forpp scatteringat 7 TeV and 8 TeV, and with the results of [47] forp p scatteringat 546 GeV and 1800 GeV Model predictions at higher energies can be tested
in future experiments.
Table 2
Same as Table 1 , but for inputσ tot (PDG-2005) Comparison shows that the predicted inelastic cross section at 7 TeV (8 TeV) increases by about 0.7 mb, when the inputσ totincreases by 1.8 mb (1.9 mb).
Inputσtot (2013)( ) :
M=1: σinel
σtot →0.449;M=2: σinel
σtot →0.556
Inputσtot (2005)( ) :
M=1: σinel
σtot →0.431;M=2: σinel
σtot →0.536 (27) Theseresultsareclosetotheblackdiskvalue of1/2 favoured
by BH [39,40] Recent detailed analysis of highenergy data [51]
concluded that, although consistent with experimental data, the
blackdiskdoesnotrepresentanuniquesolution
5.3 Phenomenological lowest t-channel singularity
If continued to complex t, |F( ,t) | given by this model is
boundedbyConst.s2fors→ ∞and
|t| <t1=min[(1 α ),lim s→∞(ln s/ ( ))2]. (28)
JinandMartin[52] provedthat for|t| <t0,wheret0 isthe
low-estt-channelsingularity,twicesubtracteddispersionrelationsins
hold Hence t1 may be thoughtof asa phenomenologicallowest
t-channel singularity.UsingtheformulaeforR2( )givenabove,
Inputσtot (2013)( ) :
Inputσtot (2005)( ) :
Our√
t1∼1.4–1.8 GeV isreminiscentof,butdifferentfromthe glue-ballmassofBH[39,40].Giventheinstabilityofanalytic con-tinuations,itsmainfunctionistosuggestthattheusualLukaszuk– Martin bound [20] is quantitatively poor as it assumes lack of
t-channelsingularitiesonlyupto4m2π whichismuchsmallerthan
t1
6 Conclusion
Ipresentedananalyticformulaforthehighenergyelastic am-plitude F( ,t) = F (1)( ,t) +F (2)( ,t) given by Eqs (7), (18) for
√
s>100 GeV, exhibiting Froissart bound saturation, AKM scal-ing [1,2], inelastic unitarity, predicting differential cross sections for( −t) <0.3 GeV2 andtotalinelasticcrosssections,at546 GeV,
1800 GeV, 7 TeV and8 TeV in agreementwith experimental re-sults, as well as the real parts and inelastic cross sections upto
100 TeV An ‘effective’ t-channel singularity at √
t∼1.4–1.8 GeV
issuggestedbyanalyticcontinuation topositivet.Detailedtables andgraphsofmodelparameters,realpartsandcrosssectionsupto
100 TeV will be published separately.The ‘grey disk’ component could be generalized usinga smootherimpact parameter cut-off, i.e.n>1 inEq.(5)
Trang 6Ipresentedanearlierversionwithablackdisksecond
compo-nent in 2015to André Martinand T.T Wu atCERN; their
insis-tencethat a sharpimpact parametercut-off is too‘brutal’led to
theblackdiskbeingreplaced bythe greydisk.IthankG
Auber-son for remarks concerning instability of analytic continuation,
D Atkinson,G MahouxandV Singhforhelpfulcommentsonthe
manuscript;IalsothankGilbertoColangeloandHeiriLeutwylerfor
veryhelpfuldiscussions,andaseminarinvitationatUniv.ofBern,
andIrinelCapriniandJuergGasserfordiscussionsonavery
stim-ulatingansatz for highenergy pion–pion scattering [53] Ithank
therefereesforthecrucialsuggestionofcomparisonwiththe
lat-estdifferentialcrosssection dataandthe IndianNationalScience
AcademyforanINSAseniorscientistgrant
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...forreadycomparisonswithexistingandfuturedata
5.2 Inelastic cross sections< /i>
Fig depicts the predicted inelastic cross sections up to
100 TeV andtheirasymptotic fits.Tables and 2give model pa-rameters...
issuggestedbyanalyticcontinuation topositivet.Detailedtables andgraphsofmodelparameters,realpartsandcrosssectionsupto
100 TeV will be published separately.The ? ?grey disk? ?? component could be generalized usinga... andtotalinelasticcrosssections,at546 GeV,
1800 GeV, TeV and8 TeV in agreementwith experimental re-sults, as well as the real parts and inelastic cross sections upto
100 TeV An ‘effective’ t-channel