Reflective elastic scattering at LHC

Sergey Troshin, IHEP, Protvino, Russia

(in collaboration with Nikolay Tyurin)

EDS'09: 13th International Conference on Elastic & Diffractive Scattering, CERN, Tuesday 30 June 2009

Elastic scattering, amplitude and S-matrix of this process:

)(21)( sifsS ll ),(21),( bsifbsS

),(|),(|),(Im 2 bsbsfbsf Structure of S(s,b) for the pure imaginary amplitude f(s,b) in the s and b plane:

b

s

S(s,b)>0

S(s,b)<0

S(s,b)=0

Reflective scattering

0)0,( bsS R

Absorptive scattering

Impact parameter representation (unitarity):

1)0,( bsS

S-matrix representation

),(1),(1),(bsiUbsiUbsS

US 2|),(1|

),(Im),(bsiUbsUbs

One-to-one transform

01-1

Various models for U(s,b) (Regge-type, geometrical, quark…): exponential decrease with impact parameter and power-like increase with energy.

1|),(lim 0 bs bsS

Light reflection off a dense medium; the phase of reflected light is changed by 180o. Appearance of a reflecting ability of scatterer due to increase of its density beyond some critical value.

Maximal absorption

241),(dbdbs inel

4/1 4/1 4/1

b b bRss 1 Rss

Rss 2 )(sRb

1))(,( sRbsU

4/1))(,( sRbs

Peripheral inelastic overlap function at large s and small impact parameters

sM

sR ln1)(

Energy evolution

21 sss R

0b

4/1 Rss

21 ssss

0 2/

I

11

S

Inelastic scattering gives subleading contribution to the total cross-section at asymptotical energies

sss totel2ln)()(

ssinel ln)(

Analogy with Berry phase

Phases (critical behavior)),(1),(1ln

21),(

bsUbsU

ibs

))((2

),( bsRbsR

),(2),( bsiebsS ),(),(),( bsibsbs IR

b

b

)(sRb

)(sRb

2/I R

b

s

),(2),( bsIebsS

),(2),( bsIebsS

Rs

I

Phase transition line

0)0,( bsI

),(41),( bsbsS

),(41),( bsbsS

2|),(| bsS 2/),( bsR2|),(|1 bsS

)(sRb Physical scattering picture (beyond the black disk) evolves with energy by simultaneous increase of the reflective ability and decrease of the absorptive ability at the impact parameters

The generic geometric picture at fixed energy (beyond the black disk limit) can be described as a scattering off the partially reflective and partially absorptive disk surrounded by the black ring which becomes grey at larger values of the impact parameter.

Reflective scattering and diffractive pattern in differential cross-section

),(),(),(Im tsHtsHtsF inelel

)(),(),(0

0,2,

tbJbsbdbhstsH inelelinelel

)0,(1)( ,, tsHs

s inelelinelel

b bR(s) R(s)

1

1/4

4/1Black disk limit

),( bshel),( bshinel

ttRRJtsH el

)(),( 1 )(),( 0 tRRJtsH inel

:0 t )()( 2 sRsel )()( sRsinel

)()()()(

)()()( 222 sb

sssb

sssb

ineltot

inelel

tot

eltot

)()( 2

,

2 sRsbinelel

Forward elastic scattering

Slope of diffraction cone:

)()()( sBsBsB inelel

)()( 2 sRsBel )()( sRsBinel

Subleading contribution of inelastic channels

0|)(ln)( tdtd

dtdsB

totbsB 2)(

:0 t )(2/1 sRH el )(2/1 sRH inel

dtd

Enhancement of large –t region by factor t

Diffraction picture with dips and bumps in the differentialcross-section will be kept in the case when the reflective scattering dominates.

Non-forward elastic scattering

Dips and bumps in

Reflective scattering leads to power-like dependence

N

sdtd

1

when –t/s-fixed and s∞ (fixed angles).

Absence of diffraction cone (and Orear type behavior) in the backward scattering, e.g. for -scattering, power-like dependence on –u in the backward hemisphere.

N

Phenomenology (based on chiral quark model for U-matrix)

Total, elastic and inelastic cross-sections;Differential cross-section; Knee in cosmic rays energy spectrum and other phenomena in this

field;

Estimate for the gap survival probability is 0.2.

Large values of mb and ratio at the LHC energies (14 TeV), while mb. Standard background estimations remain valid. Note the strong model dependence of numerical estimates.

230)( stot3/2)(/)( ss totel

77)( sinel

Consequences for the LHC heavy-ion studies

At very high energies a certain part of central hadron collisions would look like collision of hard spheres. It will affect initial condition in nuclear reactions (with total number N of hadrons in the initial state) reducing the volume V of the available space which the hadrons are able to occupy in the initial stage of nuclear collision:

V 2)()( NsVspV RR 3D volume of reflective

scatteringAveraged over volume probability of reflective scattering

RVxdrsS

sVsp

sVRR

R

3

)(

2|),(|)(

1)(

)()3/4()( 3 sRsVR

),()(1),(),(

Tns

TnTnR

2/)()()( sVsps RR2|)0,(|)( bsSspR

sMsTnR33 ln/)(/1),(

Limiting dependence for the hadron density appears due to the presence of reflective scattering. It corresponds to saturation of unitarity (and the Froissart-Martin bound for the total cross-section).

),( bsf

Unitarity in models for

),( tsF2|),(|),(Im bsfbsf

?

Inconsistency of maximal odderon with unitarity saturation

Additional check is needed

),( tsF ?

Appearance of the reflective scattering at very high energies is not promptly dictated by any specific dynamical model, it is a result of S-matrix unitarity saturation related to the total cross section growth.

The unitarity saturation can be related in its turn to the principle of the maximal strength of strong interactions proposed by Chew and Frautchi.

Concluding remarks

Reflective scattering is just a hypothesis as well as unitarity saturation.

Presence or absence of reflective scattering can be checked at the LHC.