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1. Introduction

2. Limitations involved in West and Yennie approach

3. West and Yennie approach and experimental data

4. Approaches based on impact parameter representation

5. General formula and interpretation of experimental data

6. Conclusion

Coulomb and hadronic scattering in elastic high-energy nucleon collisions

V. Kundrát, M. Lokajíček Institute of Physics, AS CR, Praha, CR

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1. IntroductionElastic nucleon collisions:

extensive and precise experimental data accurate analysis

mainly due to hadronic interactions at all t

charged hadrons: due to Coulomb interactions mainly at small |t|

influence of both interactions (spins neglected)

… Coulomb (QED), …. hadronic amplitudes

mutually bound by relative phase ,

West–Yennie (1968) (for large s )

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• some approximations; adding dipole form factors simplified West-Yennie formula

with relative phase ,

• fitting differential cross section data

constant and averaged values of

what approximations used?

their contradictory character t independence of untenable

(V.K., M. Lokajíček, Phys. Lett. B 611 (2005) 102)

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2. Fundamental limitations involved in West-Yennie general belief: no limitation of relative phase

phase should be real imaginary part of integral should be zero for

all t

quantity should be independent on t in the whole kinematically allowed region of t values!!!

for any t, t’for any t, t’

(mentioned among assumptions in: Amaldi et al., Phys. Lett. B 43 (1973) 231)

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phase real:phase real:

introducing phase and modulus ( s depressed)

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• both I1(t), I2(t) proper integrals if ζ(t) has bounded derivatives for all admissible t

• modulus F(t) is non-zero in interval (-4p2,0)

• similar for higher derivatives

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• key question: what ζ (t) solves

• representing infinite number of conditions → only one solution

• ζ(t) = const

• define functions (heuristic way)

→

→

generally near

• contradiction: asume for ε small, ζ(t)’ ≠0 →

f(t,τ)≠0 for any t,τ → ζ(t) = const …any solution !!!

0 ≠ ε [f(t1,τ1) + f(t2,τ2)] positive number!!! …for finite ε

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3. West-Yennie approach and data3. West-Yennie approach and data

• t t independence of independence of ρρ first derivative is zero first derivative is zero

valid for all valid for all tt

data: data: existence of diffractive minimum atexistence of diffractive minimum at derivative is zero derivative is zero

……contradiction contradiction existence of diffr. minimum, i.e.,existence of diffr. minimum, i.e.,

data excludes possibility for to be data excludes possibility for to be tt independent !!! independent !!!

purely exponential t dependence of the modulus at allpurely exponential t dependence of the modulus at all t t contradicts experimental data;contradicts experimental data; satisfied only “approximately” in satisfied only “approximately” in limited region of limited region of t t

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under what assumptions simplified WY derived?

in addition to being constant: purely exponential t dependence of in the whole kinematically allowed region of t

integral from WY:

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• high and

as

and

• small |t| can be neglected

are assumptions fulfilled by data ?are assumptions fulfilled by data ?

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change of magnitude from optical point to diffractive minimum:change of magnitude from optical point to diffractive minimum:

~ 8~ 899 ~ 7~ 78 ~ 58 ~ 56 ~ 46 ~ 4 change of change of | | | | ::

~ 4. ~ 4. 4.5 4.5 3.5 3.5 4. 4. 2.5 2.5 3 3 2 2

““approximately” purely exponential for small approximately” purely exponential for small region region becomes narrower when energy increases with becomes narrower when energy increases with ||t| ; !!!t| ; !!!

-t [GeV ] -t [GeV ] (0., 1.5) (0., 1.5) (0., 1.35) (0., 0.8) (0., 0.4) (0., 1.35) (0., 0.8) (0., 0.4) 22

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deviations from purely exponential behavior rise strongly with deviations from purely exponential behavior rise strongly with increasing energy; measure of deviations: diffractive slopeincreasing energy; measure of deviations: diffractive slope

constant slope constant slope B(s)B(s) corresponding to WY corresponding to WY formula for formula for

• simplified WY formula commonly used in simplified WY formula commonly used in inconsistentinconsistent way if way if applied to region of small applied to region of small |t||t| only; discrepancy cannot be removed only; discrepancy cannot be removed even if two even if two differentdifferent formulas are used: simplified WY at small formulas are used: simplified WY at small |t||t| and phenomelogically constructed hadronic amplitude at large and phenomelogically constructed hadronic amplitude at large |t||t|

• integral and simplified WY formulas contradict experimental dataintegral and simplified WY formulas contradict experimental data

• however: before ISR experiments nothing was known about diffractive however: before ISR experiments nothing was known about diffractive structure structure WY amplitude might be used WY amplitude might be used

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4. Approaches based on impact param. representation4. Approaches based on impact param. representation (Franco (1966,1973), Lapidus et al. (1978), Cahn (1982))(Franco (1966,1973), Lapidus et al. (1978), Cahn (1982))

used eikonal models based on approximate form of Fourier-Bessel used eikonal models based on approximate form of Fourier-Bessel transformation valid at asymptotic transformation valid at asymptotic ss and small and small ||t|t|

mathematically rigorous formulation (valid at any mathematically rigorous formulation (valid at any ss and and tt ) ) (Adachi et al., Islam (1965 – 1976)) (Adachi et al., Islam (1965 – 1976))

additivity of potentialsadditivity of potentials additivity of eikonals additivity of eikonals (Franco (1973)) (Franco (1973))

total scattering amplitudetotal scattering amplitude

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convolution integralconvolution integral

• equation describes simultaneous actions of both Coulomb and equation describes simultaneous actions of both Coulomb and hadronic interactions; to the sum of both amplitudes new complex hadronic interactions; to the sum of both amplitudes new complex function (convolution integral) is addedfunction (convolution integral) is added

• at difference with WY amplitude (Coulomb amplitude multiplied at difference with WY amplitude (Coulomb amplitude multiplied by phase factor only)by phase factor only)

• valid at any valid at any ss and and tt

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• general formula general formula valid up to terms linear in valid up to terms linear in αα (V. K., M. Lokaj(V. K., M. Lokajíček (1994))íček (1994))

• expression expression [1 [1 ± i± iαα G(s,t)] G(s,t)] … two terms of Talyor series expansion of … two terms of Talyor series expansion of

complex complex G(s,t)G(s,t) cannot be simply interpreted as mere change of phasecannot be simply interpreted as mere change of phase

• real real ↔↔ constant in constant in tt

• use: (i) determine FN(s,t) (ii) predict FC+N(s,t)

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5. General formula and interpretation of exp. data5. General formula and interpretation of exp. data absence of any actual theory absence of any actual theory must be derived from data must be derived from data

complex elastic hadronic amplitude complex elastic hadronic amplitude characterized by pair of real modulus and phase characterized by pair of real modulus and phase

only one experimentally determined function only one experimentally determined function

elastic hadron amplitude can be hardly derived from mere exper. data elastic hadron amplitude can be hardly derived from mere exper. data t t dependence of phase needed dependence of phase needed

distribution of elastic scattering in impact parameter space distribution of elastic scattering in impact parameter space

if small in Coulomb and interference regions and if small in Coulomb and interference regions and increases monotony for higher increases monotony for higher |t||t| and and

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has Gaussian shape with maximum at has Gaussian shape with maximum at b = 0b = 0 central behavior of elastic collisions ( )central behavior of elastic collisions ( )

current status: elastic diffraction … central current status: elastic diffraction … central inelastic diffraction … peripheral inelastic diffraction … peripheral

peripheral distribution of elastic hadron scattering: peripheral distribution of elastic hadron scattering: maximum at maximum at b > 0 ;b > 0 ; e.g., e.g., increases increases

with rising with rising |t||t| (e.g., (e.g., ); vanishes at ); vanishes at |t| <|t| <

similar behavior investigated by: Franco and Yin (1985-6)similar behavior investigated by: Franco and Yin (1985-6)

analysis of experimental data (pp at 53 GeV, pp at 541 GeV)analysis of experimental data (pp at 53 GeV, pp at 541 GeV)

• … … tt dependent dependent

• influence of Coulomb scattering cannot be neglected at higher influence of Coulomb scattering cannot be neglected at higher |t||t|

• peripheral feature of elastic scattering: statistically preferredperipheral feature of elastic scattering: statistically preferred

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M. Islam (2005) model predictionsM. Islam (2005) model predictions

(J. Ka(J. Kašpar: Diploma thesis (2005))špar: Diploma thesis (2005))

• ddσ/dt (difference at t = -.0005 GeVσ/dt (difference at t = -.0005 GeV22 …. 4.5 %) …. 4.5 %)

• ratio of interference to hadronic terms [%]ratio of interference to hadronic terms [%] influence of influence of Coulomb scattering cannot be neglected at higher Coulomb scattering cannot be neglected at higher |t||t|

• tottot=109.17 mb, =109.17 mb, elel=21.99 mb, =21.99 mb, B B = 31.5 GeV= 31.5 GeV-2-2, , =0.123=0.123

totaltotal

WYWY

eikonaleikonal

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Bourrely-Soffer-Wu model Bourrely-Soffer-Wu model (Eur. Phys. J. C 28 (2003) 97)(Eur. Phys. J. C 28 (2003) 97)

• tottot=103.64 mb, =103.64 mb, elel=28.51 mb, =28.51 mb, B B = 20.19 GeV= 20.19 GeV-2-2, , =0.121=0.121

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LuminosityLuminosity

• different total (eikonal and WY) amplitudes different total (eikonal and WY) amplitudes different different luminosity determinations luminosity determinations

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6. Conclusion6. Conclusion

WY integral formula admits only WY integral formula admits only hadronic amplitudes with constant hadronic amplitudes with constant quantity quantity

simplified WY amplitude issimplified WY amplitude is in contradiction with datain contradiction with data

WY approach leads to WY approach leads to false false results at high energiesresults at high energies

approach based on eikonal model approach based on eikonal model suitable tool for analyzing high- suitable tool for analyzing high-energy elastic hadron scattering energy elastic hadron scattering t dependence of its modulus and phase needed at all allowed tt dependence of its modulus and phase needed at all allowed t

dynamical characteristics of elastic hadronic amplitude determined dynamical characteristics of elastic hadronic amplitude determined by its by its t t dependence, i.e., are dependence, i.e., are model dependent quantitiesmodel dependent quantities

influence of Coulomb scattering influence of Coulomb scattering cannot be neglected at highercannot be neglected at higher |t||t|