Real and Imaginary

Amplitudes in proton-proton and

proton-antiprotonscattering

Purposes

• To show results for Imaginary and Real amplitudes with details on slopes, zeros, dips.

• Tests of DDR – derivative dispersion relations beyond |t|=0.

• Predictions and remarks for LHC energies.

Universal tail

Real amplitudes with one or two zeros

• Imaginary amplitudes are similar for pp and p-ap scattering, with one detected zero in the region of the dip in dsigma/dt.

• Due to different signs of the real tail (e.g. 3 gluon exchange), pp real amplitudes have a second zero, near the dip region.

• The p-ap real amplitudes do not have the second zero, and the dips are flatter.

proton-proton real amplitude

p-antiproton real amplitude

Martin’s theorem for the real zero

Please, pay attention to slopes

Energy B(real) B(imag)

p-proton 23.5 20.0 12.130.7 25.3 12.244.7 25.8 12.752.8 26.2 12.962.5 26.9 13.1

p-antiproton 30.4 25.5 12.752.6 26.3 13.262.5 26.6 13.2541 30.2 15.1

1800 32.3 16.6

Energy dependence of slopes

Measurement of B(real)• B(real) is nearly twice B(imag). • Both B(real) and B(imag) slopes are important for

the analysis of the Coulomb interference region. • Measurement of B(real): P.Gauron, B.Nicolescu,

O.V.Selyugin, PLB629 (2005) 83, find the real amplitude for pp at 52.8 GeV at a point t(min) where a certain quantity extracted from data at the interference region is minimized. The method confirms large values of B(real) as in the table.

• Usual global slope of dsig/dt is an average B=[rho**2*B(real)+B(imag)]/[rho**2+1] .

DDR

Derivative Dispersion Relations

beyond |t|=0

EVEN equation

(BR*rho*sig)pp + (BR*rho*sig)p-ap = tan(pi/2 d/dlogs)[(BI*sig)pp+(BI*sig)p-ap]

ODD equation

(pi/2 d/dlogs)[(BR*rho*sig)pp - (BR*rho*sig)p-ap] =-[1-1/3 (pi/2 d/dlogs)- ...][(BI*sig)pp-(BI*sig)p-ap]

DDR for SLOPES

Even DDR for slopes